Representations and Cohomology of Finite Categories (DRAFT)
Representations and Cohomology of Finite Categories (DRAFT)
Representations and Cohomology of Finite Categories (DRAFT)
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Fei Xu<br />
<strong>Representations</strong> <strong>and</strong> <strong>Cohomology</strong> <strong>of</strong> <strong>Finite</strong><br />
<strong>Categories</strong> (<strong>DRAFT</strong>)<br />
Category Algebras & Simplicial Methods<br />
September 1, 2011
Preface<br />
These are exp<strong>and</strong>ed lecture notes that I used for a short course<br />
<strong>of</strong> the same title at Universitat Autònoma de Barcelona in<br />
2010-2011 academic year. The subtitle reveals the main tools<br />
we are using. It means that we shall study representations<br />
<strong>and</strong> cohomology <strong>of</strong> category algebras via simplicial modules.<br />
I assume the reader to have good background on homological<br />
algebra from classical books such as “A Course in Homological<br />
Algebra” by Hilton <strong>and</strong> Stammbach. The main theme <strong>of</strong> these<br />
notes is to answer the question: what we may do with an<br />
abstract finite category Here we regard a finite category as<br />
a generalization <strong>of</strong> an abstract finite group <strong>and</strong> <strong>of</strong> an abstract<br />
finite partially ordered set. Starting from a finite group <strong>and</strong><br />
a base ring, there are well developed group representation<br />
theory <strong>and</strong> group cohomology theory. Parallel theories have<br />
been introduced to finite partially ordered sets as well. This<br />
book presents a theory that extending all <strong>of</strong> the above.<br />
Category cohomology theory is a place where simplicial<br />
methods, homotopy theory <strong>and</strong> representation theory naturally<br />
meet. The objectives <strong>of</strong> our theory may be summarized<br />
in the following diagram.
vi<br />
C<br />
classifying space<br />
functor category<br />
nerve<br />
category algebra <strong>and</strong> modules<br />
BC<br />
geo.<br />
bar resolution<br />
<br />
NC<br />
RC−mod<br />
realiz.<br />
<strong>of</strong> the trivial RC−module R<br />
singular cohomology<br />
simplicial cohomology<br />
cohomology <strong>of</strong> modules<br />
H ∗ (C; N) ∼ ∗<br />
= lim ←−<br />
C N ∼ = Ext ∗ RC(R, N)<br />
Ext ∗ RC(M, N)<br />
Preface<br />
functor cohomology<br />
In the diagram, C is a finite category, NC is its nerve, a combinatorial<br />
construction, <strong>and</strong> BC is the geometric realization <strong>of</strong><br />
NC (or C) which is a CW-complex. Also R is a commutative<br />
ring with identity <strong>and</strong> M, N ∈ (R-mod) C are two covariant<br />
functors. (A functor in (R-mod) C should be considered as a<br />
diagram <strong>of</strong> R-modules.) In this picture, RC is the so-called<br />
category algebra. It was shown by B. Mitchell [56] that there<br />
exists an equivalence between (R-mod) C <strong>and</strong> RC-mod (the<br />
category <strong>of</strong> left RC-modules). Thus a functor from C to R-<br />
mod is always an RC-module. Notably when C is a group, the<br />
theory becomes the usual group cohomology theory. When C<br />
is a transporter category (see Chapter 6) defined over a group<br />
G, we recover the equivariant cohomology theory.<br />
We can similarly study category homology by replacing<br />
the last two rows in the picture by H∗(C; N) ∼ ∗<br />
= lim −→ N ∼ =<br />
C<br />
Tor RC<br />
∗ (R, N) <strong>and</strong> Tor RC<br />
∗ (M ′ , N) (M ′ is a right RC-module).<br />
In these notes we shall focus on cohomology theory because<br />
homology theory can be developed parallel to it.<br />
Much <strong>of</strong> the general theory is indeed established for all small<br />
categories. To make this book as useful as we can, we shall<br />
state results in their general forms, for small, not just finite,<br />
categories whenever it is the case. It is perhaps a good point<br />
to explain why small categories are <strong>of</strong> particular interests. In<br />
fact this is a set-theoretic issue. In the above diagram, if we<br />
<br />
(R−mod
Preface<br />
consider a small category C then we can still construct a simplicial<br />
set NC, a topological space BC, an algebra RC <strong>and</strong><br />
a functor category (R-mod) C . In all these constructions, explicitly<br />
defined in the main text, the class <strong>of</strong> morphisms in C,<br />
Mor C, has to be a set, which is precisely the smallness condition<br />
on C. It lies in the applications <strong>of</strong> category cohomology<br />
<strong>and</strong> representations that we have in mind that C is frequently<br />
finite (Mor C is a finite set). The chief reason is that we underst<strong>and</strong><br />
quite well the representations <strong>of</strong> RC, which becomes<br />
an associative algebra with identity <strong>and</strong> is <strong>of</strong> finite R-rank.<br />
There exists another functor (co)homology theory [25], used<br />
in Steenrod algebras <strong>and</strong> cohomology <strong>of</strong> finite group schemes.<br />
In that (co)homology theory, one mainly investigates representations<br />
<strong>and</strong> cohomology <strong>of</strong> the concrete (essentially small<br />
but infinite) category <strong>of</strong> finite-dimensional vector spaces over<br />
a field. The relationship between these two theories is comparable<br />
to that between the cohomology theories <strong>of</strong> abstract<br />
finite groups <strong>and</strong> <strong>of</strong> general linear groups. They start <strong>of</strong>f the<br />
same foundation but are <strong>of</strong> different flavors <strong>and</strong> use quite different<br />
methods, even though they supply important ideas <strong>and</strong><br />
results to each another. Thus although there are something in<br />
common between [25] <strong>and</strong> this book, the main bulks <strong>of</strong> these<br />
two are different <strong>and</strong> to some extent complementary to each<br />
other.<br />
The ingredients shown in the previous diagram have been<br />
studied since the early stage <strong>of</strong> homological algebra in one<br />
way or another. In this book we collect many existing materials<br />
to form a source for self-learning as well as a reference<br />
for researchers. The major inputs from the author are firstly<br />
to provide a systematic introduction to the uses <strong>of</strong> simplicial<br />
methods, <strong>and</strong> secondly to develop tools for comparing<br />
cohomology <strong>of</strong> two small categories connected by a functor.<br />
vii
viii<br />
Preface<br />
Simplicial methods help us to construct some (complexes <strong>of</strong>)<br />
modules combinatorially while the tools we have mentioned<br />
tell us how to compare these (complexes <strong>of</strong>) modules (<strong>and</strong><br />
their cohomology). We shall explain these two features next.<br />
Simplicial methods should be considered as natural tools<br />
for algebraists. If A is a ring, then a simplicial A-module M<br />
is a collection <strong>of</strong> A-modules M n , n ≥ 0, equipped with certain<br />
maps among them. By Dold-Kan Correspondence, the<br />
category <strong>of</strong> non-negatively (or non-positively) graded chain<br />
complexes <strong>of</strong> A-modules is equivalent to the category <strong>of</strong> simplicial<br />
(or cosimplicial) A-modules. It also implies the category<br />
<strong>of</strong> first quadrant double complexes is equivalent to the<br />
category <strong>of</strong> bisimplicial A-modules. The complexes <strong>of</strong> modules<br />
have long been used in (at least the homological aspects<br />
<strong>of</strong>) the representation theory <strong>of</strong> algebras, so there is no reason<br />
why we do not consider their combinatorial predecessors.<br />
In this book we shall focus on the case for A = RC, where<br />
simplicial methods seem to be truly powerful.<br />
Let u : D → C be a functor between small categories. We<br />
construct a simplicial RC-module R[N(u/−)] (a bar construction).<br />
It has the properties that, for each n ≥ 0, R[N(u/−)] n<br />
is a left projective RC-module, <strong>and</strong> that, for each x ∈ Ob C,<br />
R[N(u/x)] is a simplicial R-module. Here u/x is the category<br />
over x <strong>and</strong> N(u/x) is its nerve, a simplicial set. Much <strong>of</strong> category<br />
(co)homology is developed based on this construction.<br />
For instance, R[N(Id C /−)] gives rise to the bar resolution<br />
<strong>of</strong> the trivial RC-module R such that R[N(Id C /−)] = RC.<br />
This is the reason why simplicial methods are indispensable<br />
to us. Due to the existence <strong>of</strong> these constructions, the Kan<br />
extensions are used, as generalizations <strong>of</strong> the usual induction<br />
<strong>and</strong> co-induction on which one relies to investigate group<br />
(co)homology, to compare (co)homology <strong>of</strong> C <strong>and</strong> D. In fact,
Preface<br />
let LK u : RD-mod → RC-mod be the left Kan extension<br />
along u. Then LK u R[N(Id D /−)] ∼ = R[N(u/−)]. Accordingly<br />
underst<strong>and</strong>ing various overcategories becomes an important<br />
issue as they show up in the constructions <strong>of</strong> simplicial modules<br />
as well as in the definitions <strong>of</strong> Kan extensions. An illuminating<br />
example is as follows. Suppose G is a finite group<br />
<strong>and</strong> H is a subgroup. Consider them as categories with one<br />
object •, equipped with the inclusion i : H → G. Then the<br />
only overcategory i/• is equivalent to the set <strong>of</strong> left cosets<br />
G/H. The left Kan extension <strong>of</strong> the bar resolution <strong>of</strong> the<br />
trivial RH-module is a projective resolution <strong>of</strong> the induced<br />
module R(G/H) = R ↑ G H . As in algebraic topology where<br />
they originate <strong>and</strong> their applications confine, we shall show<br />
that simplicial methods now become crucial theoretical <strong>and</strong><br />
computational tools in algebra too.<br />
What we do here in this book is really to take up the algebraic<br />
<strong>and</strong> combinatorial tools invented by topologists <strong>and</strong><br />
apply them to algebra. We shall see that we get satisfactory results.<br />
Simplicial methods were introduced by algebraic topologists<br />
<strong>and</strong> lie in the center <strong>of</strong> homotopy theory. In homotopy<br />
theory, one <strong>of</strong>ten considers diagrams <strong>of</strong> spaces, which are simply<br />
functors from an index category C to T op, the category<br />
<strong>of</strong> topological spaces. When u : D → C <strong>and</strong> F : C → T op are<br />
two functors, one can naturally obtain a new functor F ◦ u.<br />
This is called the restriction along u, written as Res u F. Indeed<br />
Res u is a functor from the category T op C to T op D . On the<br />
other h<strong>and</strong> if G : D → T op is a functor, we can construct<br />
two functors C → T op. These two functors are called the left<br />
<strong>and</strong> right Kan extensions <strong>of</strong> G along u. It is intuitive if we<br />
record topologists’ notations<br />
C ⊗ D G <strong>and</strong> Hom D (C, G).<br />
ix
x<br />
Preface<br />
These two functors C ⊗ D −, Hom D (C, −) are the left <strong>and</strong> right<br />
adjoints <strong>of</strong> Res u . In Hollender-Vogt [37], one can see that<br />
these two functors enjoy many properties that an algebraist<br />
would expect for similar functors, induced by a ring homomorphism<br />
R 1 → R 2 , between R 1 -mod <strong>and</strong> R 2 -mod. In fact<br />
in [37], a covariant functor in T op C is called a left C-module<br />
while a contravariant functor is called a right C-module. I<br />
would like to point out here that since a functor u : D → C<br />
does not lead to an algebra homomorphism from RD to RC,<br />
one has to be very careful when constructing RC ⊗ RD − <strong>and</strong><br />
Hom RD (RC, −). This is the reason why I refrain from using<br />
topologists’ notations <strong>of</strong> the Kan extensions. Plus the intuitive<br />
notations do not seems to be convenient in computations<br />
as they are in other places such as group (co)homology.<br />
However the Kan extensions are truly generalizations <strong>of</strong> well<br />
known functors. When i : H → G is the inclusion functor at<br />
the end <strong>of</strong> last paragraph, the left <strong>and</strong> right Kan extensions<br />
along i are isomorphic to the induction RG ⊗ RH − <strong>and</strong> coinduction<br />
Hom RH (RG, −) (which happen to be isomorphic<br />
to each other).<br />
Let us now recall the history <strong>of</strong> (co)homology theory <strong>of</strong> small<br />
categories <strong>and</strong> related works. Homology theory <strong>of</strong> small categories,<br />
H∗(C; N), can be found in Gabriel-Zisman [28] (1967),<br />
as a special case <strong>of</strong> homology <strong>of</strong> simplicial sets with coefficients,<br />
in the appendices. There they also showed H∗(C; N) ∼ =<br />
lim<br />
−→ ∗ N, where lim<br />
C −→ ∗ are the derived functors <strong>of</strong> direct limit<br />
C<br />
functor −→C<br />
lim . By contrast, although the special case <strong>of</strong> partially<br />
ordered sets was studied by J. E. Roos [66] (1961), cohomology<br />
<strong>of</strong> small categories began with Baues-Wirsching’s<br />
theory [3] (1985). They actually studied H ∗ (C; N ), where N<br />
is not a functor from C. Instead, it is a functor from another<br />
category F (C), the category <strong>of</strong> factorizations in C. In their
Preface<br />
situation cohomology theory <strong>of</strong> small category can also be<br />
∗<br />
introduced by ←−<br />
lim , derived functors <strong>of</strong> inverse limit functor<br />
C<br />
lim . Simplicial methods are used to define various chain or<br />
←−C<br />
cochain complexes whose (co)homology is the (co)homology<br />
<strong>of</strong> C.<br />
In these theories the coefficients <strong>of</strong> (co)homology are provide<br />
by functors. Thus analyzing structures <strong>of</strong> functors should be<br />
<strong>of</strong> great importance. A significant achievement in this direction<br />
was obtained by Lück [51] <strong>and</strong> tom Dieck [] (1987),<br />
where they classified simple <strong>and</strong> projective functors under<br />
reasonable assumptions on the small category. An influential<br />
approach to functor categories interestingly began with P.<br />
Gabriel’s thesis [27] (1962). Given a small category, he made<br />
it into an additive category by linearizing all morphism sets.<br />
Then he showed that the category <strong>of</strong> addictive functors from<br />
such an additive category to a module category is equivalent<br />
to another module category, <strong>of</strong> modules over an algebra<br />
that now we may call the additive category algebra <strong>of</strong> a small<br />
category. Later on B. Mitchell [56] (1972) introduced a different<br />
construction <strong>and</strong> defined an associative ring over a small<br />
category. Consequently he established connections between<br />
(R-mod) C <strong>and</strong> RC-mod. This is the result we pictured in the<br />
diagram.<br />
At this point, one perhaps does not see a connection between<br />
Mitchell’s work <strong>and</strong> category (co)homology. These<br />
two seemingly unrelated subjects were pieced up together<br />
by P. J. Webb (around 2000) who named these rings appeared<br />
in Mitchell’s paper, as “category algebras”. Representation<br />
theory <strong>of</strong> categories generalizes both representation<br />
theory <strong>of</strong> groups <strong>and</strong> <strong>of</strong> quivers, <strong>and</strong> thus is <strong>of</strong> great interest.<br />
<strong>Representations</strong> <strong>of</strong> categories are important to category<br />
(co)homology in the same way as group representations to<br />
xi
xii<br />
Preface<br />
group (co)homology. From category algebra point <strong>of</strong> view, we<br />
can put all necessary ingredients <strong>of</strong> category (co)homology<br />
under one framework, as shown in the diagram. Furthermore<br />
from the intrinsic structure <strong>of</strong> a category algebra, one can<br />
see the similarities with <strong>and</strong> differences from a groups algebra.<br />
Hence it explains why some classical results in group<br />
(co)homology may be generalized to category (co)homology,<br />
while others may not.<br />
At the early stage <strong>of</strong> category (co)homology, it seemed<br />
like a purely theoretic construction with few calculations.<br />
The thrust <strong>of</strong> recent development was brought in by representation<br />
<strong>and</strong> homotopy theorist working on locally determined<br />
structures. Their work completely reshaped the whole<br />
(co)homology theory <strong>and</strong> provided many interesting concrete<br />
categories to work with. We shall comment on it in Chapter<br />
6. In the last decade many interesting results on both abstract<br />
<strong>and</strong> concrete small categories have been obtained. For<br />
instance, a pivotal discovery is that the category <strong>of</strong> factorizations,<br />
F (C), first used by Quillen to show homotopy equivalence<br />
between BC <strong>and</strong> BC op , <strong>and</strong> then by Baues-Wirsching<br />
to introduce their cohomology theory, possesses the property<br />
that all (co)homology theories we consider here are indeed<br />
(co)homology <strong>of</strong> F (C) with appropriate coefficients. However<br />
when one tries to teach oneself about category (co)homology,<br />
one finds materials scattered in the literature <strong>and</strong> different<br />
writers have different background <strong>and</strong> styles. As an example<br />
the existing introduction to this subject by Webb [80]<br />
emphasizes its module-theoretic aspects. Lack <strong>of</strong> a comprehensive<br />
treatment makes the theory daunting for whoever<br />
wants to learn, <strong>and</strong> even for a researcher who uses category<br />
(co)homology theory it may cause inconvenience. Thus<br />
a book, introducing basic ideas <strong>of</strong> category (co)homology the-
Preface<br />
ory, presenting st<strong>and</strong>ard techniques <strong>and</strong> addressing the interactions<br />
between representation <strong>and</strong> homotopy theories, seems<br />
necessary. Such a book should provide a clear view <strong>of</strong> basic<br />
ideas, key methods, known results <strong>and</strong> unsolved conjectures,<br />
being a h<strong>and</strong>y introduction that can be used to foster further<br />
investigations <strong>and</strong> to search for future applications.<br />
Finally we turn to the structure <strong>of</strong> this book. The first<br />
two chapters consists <strong>of</strong> preliminaries needed in category<br />
(co)homology. It means one can find them in various classical<br />
books. The reason why I collect them here are, firstly it<br />
is convenient for the reader, secondly I try to provide some<br />
concrete examples to illustrate many abstract constructions<br />
which are in the center <strong>of</strong> this book. The first chapter recalls<br />
some basic definitions from category theory. The main<br />
focuses are limits <strong>of</strong> functors <strong>and</strong> their generalizations, that<br />
is, the Kan extensions. The purpose <strong>of</strong> the second chapter<br />
is to equip the reader with necessary knowledge about simplicial<br />
methods. We begin with a review <strong>of</strong> chain complexes.<br />
It is followed by an introduction to simplicial sets <strong>and</strong> the<br />
nerve <strong>of</strong> a small category where many combinatorially constructed<br />
chain complexes appear. Simplicial (co)homology is<br />
defined <strong>and</strong> several examples are given. To tell how to compare<br />
(co)homology <strong>of</strong> small categories, we have to inform the<br />
reader how to compare small categories <strong>and</strong> their nerves as<br />
well as classifying spaces. Thus some important categorical<br />
constructions are provided, which are needed throughout this<br />
book, for example to state Quillen’s Theorem A. For future<br />
references, <strong>and</strong> for the interested reader, we end Chapter 2<br />
with bisimplicial sets <strong>and</strong> several key results. Although only<br />
the statements will be used in future, we nonetheless present<br />
their pro<strong>of</strong>s.<br />
xiii
xiv<br />
Preface<br />
The third chapter introduces category algebras <strong>and</strong> their<br />
representations. Examples are served at the beginning to motivate<br />
the reader. We shall classify projective <strong>and</strong> injective<br />
modules under mild assumptions. Moreover we give an intrinsic<br />
characterization <strong>of</strong> category algebras so that they are<br />
comparable with cocommutative bialgebras. It explains why<br />
category algebras possess interesting homological properties.<br />
The fourth chapter studies (ordinary) (co)homology <strong>of</strong> category<br />
algebras. We begin with the most economical way by<br />
using derived functors to define category (co)homology. Then<br />
we recall Baues-Wirsching’s construction on the way to introduce<br />
the bar resolution. The bar resolution is simplicially<br />
constructed <strong>and</strong> it leads to various important modules by applying<br />
Kan extensions on it. We will see they are the most<br />
powerful tools for us. In this chapter we also define the extension<br />
<strong>of</strong> a category by a group <strong>and</strong> the Grothendieck spectral<br />
sequences.<br />
The fifth chapter discusses Hochschild (co)homology <strong>of</strong> category<br />
algebras. The key result here is a theorem to interpret<br />
Hochschild (co)homology <strong>of</strong> category algebras by their ordinary<br />
(co)homology, <strong>and</strong> vice versa. A theorem we prove here<br />
shows that all the previously mentioned (co)homology theories<br />
are just (co)homology <strong>of</strong> F (C) with coefficients. Some<br />
examples are given to demonstrate explicit calculations.<br />
The sixth chapter talks about connections between category<br />
<strong>and</strong> group cohomology. It contains mostly unpublished<br />
results. We bring up the notion <strong>of</strong> a local category <strong>of</strong> a finite<br />
group. Local categories are the motivating cases for research<br />
in category representations <strong>and</strong> cohomology. We put<br />
it in the end because we do need techniques developed earlier.<br />
This chapter contains many concrete categories <strong>and</strong> we<br />
shall see clearly how one can compute using various abstract
Preface<br />
machineries introduced before. A key concept in this chapter<br />
is a transporter category. We show how closely related<br />
are the transporter categories to the groups on which they<br />
are defined. The most important result is perhaps the finite<br />
generation <strong>of</strong> cohomology <strong>of</strong> modules <strong>of</strong> finite transporter categories.<br />
Also we will construct transfer maps for ordinary <strong>and</strong><br />
Hochschild cohomology <strong>of</strong> transporter categories. This chapter<br />
should help the reader to carry on further readings in<br />
advanced research papers.<br />
There may be many errors or even mistakes in this unfinished<br />
manuscript, all <strong>of</strong> which are my responsibility.<br />
xv<br />
Universitat Autònoma de Barcelona,<br />
June 2011<br />
Fei Xu<br />
xu@mat.uab.cat
Contents<br />
1 Functors <strong>and</strong> their Kan extensions . . . . . . . . . . . 1<br />
1.1Functors <strong>and</strong> limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />
1.1.1Basic category theory . . . . . . . . . . . . . . . . . . . . . . 1<br />
1.1.2Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
1.1.3Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
1.2Restriction <strong>and</strong> Kan extensions . . . . . . . . . . . . . . . . . 21<br />
1.2.1Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21<br />
1.2.2Overcategories <strong>and</strong> undercategories . . . . . . . . . . 23<br />
1.2.3Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26<br />
2 Simplicial methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
2.1Complexes <strong>and</strong> homology . . . . . . . . . . . . . . . . . . . . . . 31<br />
2.1.1Chain complexes, homology <strong>and</strong> chain homotopy 32<br />
2.1.2Double complexes <strong>and</strong> operations on chain<br />
complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35<br />
2.2Nerves, classifying spaces <strong>and</strong> cohomology . . . . . . . . 39<br />
2.2.1Simplicial sets <strong>and</strong> nerves <strong>of</strong> small categories . . 39<br />
2.2.2Classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 58<br />
2.2.3Cup product <strong>and</strong> cohomology ring . . . . . . . . . . . 62<br />
2.3Quillen’s work on classifying spaces. . . . . . . . . . . . . . 67<br />
2.3.1Quillen’s Theorem A . . . . . . . . . . . . . . . . . . . . . . . 67<br />
2.3.2Constructions over categories <strong>and</strong> relevant<br />
functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70<br />
2.4Further categorical <strong>and</strong> simplicial constructions . . . 74
xviii<br />
Contents<br />
2.4.1Grothendieck constructions . . . . . . . . . . . . . . . . . 75<br />
2.4.2Bisimplicial sets <strong>and</strong> homotopy colimits . . . . . . . 80<br />
2.4.3Pro<strong>of</strong>s <strong>of</strong> Quillen’s Theorem A <strong>and</strong><br />
Thomason’s theorem . . . . . . . . . . . . . . . . . . . . . . . 83<br />
3 Category algebras <strong>and</strong> their representations . 89<br />
3.1Basic concepts <strong>and</strong> examples . . . . . . . . . . . . . . . . . . . 89<br />
3.1.1Category algebras . . . . . . . . . . . . . . . . . . . . . . . . . 89<br />
3.1.2<strong>Representations</strong> <strong>of</strong> categories <strong>and</strong> Mitchell’s<br />
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91<br />
3.1.3Three examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 94<br />
3.2A closed symmetric monoidal category . . . . . . . . . . . 96<br />
3.2.1Tensor structure <strong>and</strong> an intrinsic<br />
characterization <strong>of</strong> category algebras . . . . . . . . . 96<br />
3.2.2The internal hom . . . . . . . . . . . . . . . . . . . . . . . . . . 103<br />
3.3Functors between module categories . . . . . . . . . . . . . 105<br />
3.3.1Restriction on algebras <strong>and</strong> modules . . . . . . . . . 105<br />
3.3.2Kan extensions <strong>of</strong> modules . . . . . . . . . . . . . . . . . . 108<br />
3.3.3Dual modules <strong>and</strong> Kan extensions . . . . . . . . . . . 114<br />
3.4EI categories, projectives <strong>and</strong> simples . . . . . . . . . . . . 116<br />
3.4.1EI condition <strong>and</strong> its implications . . . . . . . . . . . . . 116<br />
3.4.2Some representation theory . . . . . . . . . . . . . . . . . 118<br />
3.4.3Classifications <strong>of</strong> projectives <strong>and</strong> simples . . . . . . 125<br />
3.4.4Projective covers, injective hulls <strong>and</strong> their<br />
restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131<br />
4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules. . . . . . . 133<br />
4.1General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133<br />
4.1.1<strong>Cohomology</strong> <strong>of</strong> modules . . . . . . . . . . . . . . . . . . . . 133<br />
4.1.2<strong>Cohomology</strong> <strong>of</strong> a small category with<br />
coefficients in a functor . . . . . . . . . . . . . . . . . . . . . 140
Contents<br />
4.1.3Extensions <strong>of</strong> categories <strong>and</strong> low dimension<br />
cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150<br />
4.2Classical methods for computation . . . . . . . . . . . . . . 157<br />
4.2.1Minimal resolutions <strong>and</strong> reduction . . . . . . . . . . . 157<br />
4.2.2Examples using classifying spaces . . . . . . . . . . . . 160<br />
4.3Computation via adjoint functors . . . . . . . . . . . . . . . 162<br />
4.3.1Adjoint restrictions . . . . . . . . . . . . . . . . . . . . . . . . 162<br />
4.3.2Kan extensions <strong>of</strong> resolutions . . . . . . . . . . . . . . . . 164<br />
4.4Grothendieck spectral sequences . . . . . . . . . . . . . . . . 170<br />
4.4.1Grothendieck spectral sequences for a functor . . 170<br />
4.4.2Spectral sequences <strong>of</strong> category extensions . . . . . 175<br />
5 Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . 185<br />
5.1Hochschild homology <strong>and</strong> cohomology . . . . . . . . . . . 185<br />
5.1.1Definition <strong>and</strong> general properties. . . . . . . . . . . . . 185<br />
5.1.2Ring homomorphisms from the Hochschild<br />
cohomology ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 190<br />
5.2Hochschild (co)homology <strong>of</strong> category algebras . . . . . 193<br />
5.2.1Basic ideas <strong>and</strong> examples . . . . . . . . . . . . . . . . . . . 193<br />
5.2.2Hochschild (co)homology as ordinary<br />
(co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195<br />
5.2.3EI categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204<br />
5.3Examples <strong>of</strong> the Hochschild cohomology rings <strong>of</strong><br />
categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208<br />
5.3.1The category E 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 208<br />
5.3.2The category E 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 211<br />
5.3.3The category E 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 212<br />
5.3.4The category E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 214<br />
6 Connections with group representations <strong>and</strong><br />
cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217<br />
6.1Local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217<br />
xix
xx<br />
Contents<br />
6.1.1G-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219<br />
6.1.2Homology representations <strong>of</strong> kG . . . . . . . . . . . . . 220<br />
6.1.3Transporter categories as Grothendieck<br />
constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221<br />
6.1.4Local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 224<br />
6.2Properties <strong>of</strong> local categories . . . . . . . . . . . . . . . . . . . 226<br />
6.2.1Two diagrams <strong>of</strong> categories . . . . . . . . . . . . . . . . . 226<br />
6.2.2Frobenius Reciprocity . . . . . . . . . . . . . . . . . . . . . . 227<br />
6.3The functor π: group representations via<br />
transporter categories . . . . . . . . . . . . . . . . . . . . . . . . . 229<br />
6.3.1Homology representations via transporter<br />
categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229<br />
6.3.2On finite generation <strong>of</strong> cohomology . . . . . . . . . . . 238<br />
6.3.3Transfer for ordinary cohomology . . . . . . . . . . . . 242<br />
6.4The functor ρ: invariants <strong>and</strong> coinvariants . . . . . . . . 252<br />
6.4.1Orbit categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 253<br />
6.4.2Brauer categories, fusion <strong>and</strong> linking systems . . 255<br />
6.4.3Puig categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258<br />
6.4.4Orbit categories <strong>of</strong> fusion systems . . . . . . . . . . . . 259<br />
6.5Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . . . 260<br />
6.5.1<strong>Finite</strong> generation . . . . . . . . . . . . . . . . . . . . . . . . . . 260<br />
6.5.2Transfer for Hochschild cohomology . . . . . . . . . . 265<br />
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
List <strong>of</strong> symbols<br />
Conventions: <strong>Categories</strong> are denoted by C, D in general unless<br />
otherwise specified; x, y are objects; α, β are morphisms<br />
in a category; F, G are general functors; u, v are functors between<br />
small categories; natural transformations between two<br />
functors are Φ, Ψ etc. Groups are written as G, H even when<br />
they are regarded as categories with one object.<br />
In a functor category we normally use M, N to denote the<br />
objects.<br />
• the trivial category <strong>of</strong> only one object <strong>and</strong> one morphism<br />
Set the category <strong>of</strong> sets<br />
T op the category <strong>of</strong> topological spaces<br />
Cat the category <strong>of</strong> small categories<br />
T C the category <strong>of</strong> all (if not specified, covariant) functors<br />
from a small category C to a category T<br />
Res u the restriction along a functor u<br />
LK u , RK u the left <strong>and</strong> right Kan extensions along u<br />
u/x, x\u categories over <strong>and</strong> under x ∈ Ob C, for a given<br />
u : D → C<br />
△ n the st<strong>and</strong>ard n-simplex<br />
n the totally ordered set 0 < 1 < · · · < n, a combinatorial<br />
construction <strong>of</strong> △ n<br />
△ the category <strong>of</strong> st<strong>and</strong>ard simplicies
xxii<br />
List <strong>of</strong> symbols<br />
∆ the diagonal functor C → C × C <strong>and</strong> various maps<br />
induced by it<br />
C e = C × C op the enveloping category <strong>of</strong> C<br />
F (C) the category <strong>of</strong> factorizations in C<br />
∇ : F (C) → C e the skew diagonal functor<br />
NC ∗ the nerve <strong>of</strong> C, a simplicial set<br />
BC the classifying space <strong>of</strong> C, geometric realization <strong>of</strong><br />
NC ∗ , a CW complex<br />
C ∗ , D ∗ , ... chain complexes<br />
C ∗,∗ , D ∗,∗ , ... double complexes<br />
Gr C M Grothendieck construction over M : C → Cat<br />
G ∝ P transporter category over a G-poset P (a Grothendieck<br />
construction)<br />
R a commutative ring with identity<br />
R-Mod the category <strong>of</strong> all R-modules<br />
R-mod the category <strong>of</strong> finitely generated R-modules<br />
k a field, usually algebraically closed<br />
Vect k the category <strong>of</strong> all k-vector spaces<br />
V ect k the category <strong>of</strong> finite-dimensional k-vector spaces<br />
RC the R-category algebra <strong>of</strong> C<br />
R the trivial RC-module, a constant functor<br />
H n (C, R), Hn(C, R) the nth simplicial (co)homology <strong>of</strong> C with<br />
coefficients in R<br />
H n (C; M), Hn(C; M) the nth (co)homology <strong>of</strong> C with coefficients<br />
in a functor/module M<br />
H n (BC, R), Hn(BC, R) the nth singular (co)homology <strong>of</strong> BC<br />
with coefficients in R<br />
H ∗ (C, R) the simplicial cohomology ring<br />
H ∗ (BC, R) the singular cohomology ring<br />
Ext ∗ RC(R, R) the ordinary cohomology ring <strong>of</strong> RC<br />
HH ∗ (RC) = Ext ∗ RCe(RC, RC) the Hochschild cohomology ring<br />
<strong>of</strong> RC
Chapter 1<br />
Functors <strong>and</strong> their Kan extensions<br />
Abstract We recall some basics from category theory <strong>and</strong> homological<br />
algebra. In this way we set conventions. The main<br />
constructions we discuss here are limits <strong>and</strong> their generalizations,<br />
the Kan extensions. We shall include some examples to<br />
illustrate these abstract concepts.<br />
1.1 Functors <strong>and</strong> limits<br />
1.1.1 Basic category theory<br />
In this section we recall some st<strong>and</strong>ard definitions <strong>and</strong> constructions.<br />
One purpose is to settle the conventions <strong>and</strong> terminologies<br />
that we shall follow in these notes.<br />
Definition 1.1.1. A category C consists <strong>of</strong> a class <strong>of</strong> objects<br />
Ob C <strong>and</strong> a class <strong>of</strong> morphisms Mor C such that<br />
1. each morphism α ∈ Mor C has a source <strong>and</strong> a target, written<br />
as s(α) <strong>and</strong> t(α), which are two objects in Ob C.<br />
2. for any pair <strong>of</strong> object x, y ∈ Ob C,<br />
Hom C (x, y) := {α ∣ ∣ s(α) = x <strong>and</strong> t(α) = y}<br />
is a set.<br />
3. for any α ∈ Hom C (x, y), β ∈ Hom C (y, z), their composite<br />
exists <strong>and</strong> is written as β ◦ α ∈ Hom C (x, z);<br />
4. for each object x there is an identity morphism 1 x ∈<br />
Hom C (x, x) satisfying the condition that 1 x ◦ α = α <strong>and</strong>
2 1 Functors <strong>and</strong> their Kan extensions<br />
β ◦ 1 x = β, if α ∈ Hom C (y, x) <strong>and</strong> β ∈ Hom C (x, z) for some<br />
y, z ∈ Ob C;<br />
5. if α ∈ Hom C (w, x), β ∈ Hom C (x, y) <strong>and</strong> γ ∈ Hom C (y, z),<br />
then (γ ◦ β) ◦ α = γ ◦ (β ◦ α).<br />
If α ∈ Hom C (x, y) is a morphism in C, very <strong>of</strong>ten we will picture<br />
it as α : x → y. A morphism α ∈ Mor C is a monomorphism<br />
if αβ = αγ for two morphisms β, γ ∈ Mor C, then<br />
β = γ. A morphism α ∈ Mor C is an epimorphism if βα = γα<br />
for two morphisms β, γ ∈ Mor C, then β = γ. A morphism<br />
α ∈ Hom C (x, y) is an isomorphism if there exists a morphism<br />
β ∈ Hom C (y, x) such that αβ = 1 y <strong>and</strong> βα = 1 x . An<br />
isomorphism is always both monomorphic <strong>and</strong> epimorphic.<br />
An isomorphism is also called an invertible morphism.<br />
We say two objects x <strong>and</strong> y are isomorphic, written as x ∼ =<br />
y, if there exists an isomorphism α ∈ Hom C (x, y). Let x be<br />
an object in a category C. Then the isomorphism class <strong>of</strong> x<br />
consists <strong>of</strong> all y ∈ Ob C that are isomorphic to x. The class<br />
<strong>of</strong> objects isomorphic to x is denoted by [x] ⊂ Ob C.<br />
A category is called a groupoid if every morphism is an<br />
isomorphism.<br />
An object x in C is initial if to any other object y, there<br />
exists a unique morphism x → y. An object x in C is terminal<br />
if to any other object y, there exists a unique morphism y →<br />
x. An object is a zero object if it is both initial <strong>and</strong> terminal.<br />
All initial (or terminal or zero) objects are isomorphic.<br />
Suppose C has a zero object 0, then for any two objects<br />
x, y we call the composite x → 0 → y the zero morphism<br />
in Hom C (x, y), written as 0 x,y . Note that the composite <strong>of</strong><br />
a zero morphism with any other morphism is a zero morphism.<br />
Let α ∈ Hom C (x, y) be a morphism. A kernel <strong>of</strong> α is<br />
a morphism β ∈ Hom C (w, x) satisfying αβ = 0 w,y such that<br />
if β ′ ∈ Hom C (w ′ , x) satisfies αβ ′ = 0 w ′ ,x then there exists a
1.1 Functors <strong>and</strong> limits 3<br />
unique morphism µ : w ′ → w such that β ′ = βµ. A cokernel<br />
<strong>of</strong> α is a morphism γ ∈ Hom C (y, z) satisfying γα = 0 x,z such<br />
that if γ ′ ∈ Hom C (x, z ′ ) satisfies γ ′ α = 0 x,z<br />
′ then there exists<br />
a unique morphism ν : z → z ′ such that γ ′ = νγ.<br />
Let I ⊂ Ob C be a subset. Then the coproduct <strong>of</strong> objects<br />
in I, written as ⊕ I x i, is an object X ∈ Ob C such that each<br />
x i ∈ I is equipped with a morphism α i : x i → X <strong>and</strong> if<br />
Y is another object equipped with β i : x i → Y then there<br />
exists a unique morphism γ : X → Y such that β i = γα i .<br />
The product <strong>of</strong> objects in I, written as ∏ I x i, is an object<br />
X ′ ∈ Ob C such that each x i ∈ I is equipped with a morphism<br />
α ′ i : X → x i <strong>and</strong> if Y ′ is another object equipped with β ′ i :<br />
Y → x i then there exists a unique morphism γ ′ : Y ′ → X ′<br />
such that α ′ i = β ′ i γ′ .<br />
Definition 1.1.2. A category C is called preadditive if every<br />
Hom C (x, y) is an abelian group <strong>and</strong> compositions are bilinear.<br />
A preadditive category is additive if furthermore all finite<br />
coproducts <strong>and</strong> all products exists in C.<br />
An additive category C is said to be preabelian if<br />
1. it has a zero object;<br />
2. every morphism has a kernel <strong>and</strong> a cokernel.<br />
A preabelian category is abelian if every monomorphism is a<br />
kernel <strong>of</strong> some morphism <strong>and</strong> every epimorphism is a cokernel<br />
<strong>of</strong> some morphism.<br />
There are abundance <strong>of</strong> categories in mathematics, many <strong>of</strong><br />
them are very natural while one can also cook up all kinds <strong>of</strong><br />
abstract categories.<br />
Example 1.1.3.1. The trivial category • has exactly one object<br />
• <strong>and</strong> one morphism 1 • ;<br />
2. A group G gives rise to two categories: the first one has<br />
only one object • <strong>and</strong> its morphism set is G; the sec-
4 1 Functors <strong>and</strong> their Kan extensions<br />
ond, denoted by EG, has Ob EG = G <strong>and</strong> Mor EG =<br />
⊎ g1 ,g 2 ∈Ob EGHom EG (g 1 , g 2 ) with Hom EG (g 1 , g 2 ) = {g 2 g1 −1 }.<br />
The former will usually be written just as G while the latter<br />
is called the Cayley graph <strong>of</strong> G;<br />
3. A partially ordered set (poset in short) P is naturally a category,<br />
still named P, if we let Ob P be the set <strong>of</strong> elements in<br />
the poset <strong>and</strong> Mor P = ⊎ x,y∈Ob C Hom C (x, y) in which the set<br />
Hom P (x, y) = {x ≤ y} if the two objects are comparable, or<br />
empty otherwise. Posets can be characterized as categories<br />
such that there exists at most one morphism between any<br />
two objects.<br />
4. The category <strong>of</strong> sets <strong>and</strong> set maps is denoted by Set, ;<br />
5. Let R be a commutative ring with identity <strong>and</strong> A an associative<br />
R-algebra. Then A-Mod <strong>and</strong> A-mod are the categories<br />
<strong>of</strong> all left A-modules <strong>and</strong> finitely generated left A-modules;<br />
6. The category Z-Mod is <strong>of</strong>ten written as Ab, the category <strong>of</strong><br />
abelian groups;<br />
7. The category <strong>of</strong> topological spaces <strong>and</strong> continuous maps is<br />
written as T op.<br />
In these notes, for an R-algebra A, if we do not specify, any<br />
A-module will be a left A-module.<br />
Definition 1.1.4. A category C is a small category if Ob C<br />
is a set, <strong>and</strong> is a finite category if Mor C is a finite set.<br />
A finite category is necessarily small by Definition 1.1.1 (4).<br />
In Example 1.1.3, the first three are small while the rest are<br />
abelian <strong>and</strong> not small.<br />
Definition 1.1.5. Let C be a category. Then a subcategory<br />
D ⊂ C consists <strong>of</strong> a subclass <strong>of</strong> objects Ob D ⊂ Ob C <strong>and</strong> a<br />
subclass <strong>of</strong> morphisms Mor D ⊂ Mor C, satisfying the axioms<br />
<strong>of</strong> a category with composition laws inherited from C.
1.1 Functors <strong>and</strong> limits 5<br />
A subcategory D ⊂ C if full, if for any pair <strong>of</strong> objects x, y ∈<br />
Ob D, we always have Hom D (x, y) = Hom C (x, y).<br />
Definition 1.1.6. A covariant functor F from a category<br />
D to another category C assigns to each x ∈ Ob D an object<br />
F(x) ∈ Ob C <strong>and</strong> to each α ∈ Hom D (x, y) a morphism F(α) ∈<br />
Hom C (F(x), F(y)) satisfying the conditions that<br />
1. F(1 x ) = 1 F(x) for every x ∈ Ob D;<br />
2. if α, β ∈ Mor D <strong>and</strong> β◦α exists, then F(β◦α) = F(β)◦F(α).<br />
A contravariant functor F ′ from a category D to another<br />
category C assigns to each x ∈ Ob D an object F ′ (x) ∈<br />
Ob C <strong>and</strong> to each α ∈ Hom D (x, y) a morphism F ′ (α) ∈<br />
Hom C (F ′ (x), F ′ (y)) satisfying the conditions that<br />
1 ′ . F ′ (1 x ) = 1 F ′ (x) for every x ∈ Ob D;<br />
2 ′ . if α, β ∈ Mor C <strong>and</strong> β◦α exists, then F ′ (β◦α) = F ′ (α)◦F ′ (β).<br />
Example 1.1.7. Let C be a category <strong>and</strong> x ∈ Ob C an object.<br />
Then<br />
1. there exists a covariant functor Hom C (x, −) : C → Set such<br />
that for any y ∈ Ob C, Hom C (x, −)(y) := Hom C (x, y) <strong>and</strong><br />
for any α ∈ Hom C (y, y ′ ) the composition Hom C (x, −)(α) =<br />
α ◦ − : Hom C (x, y) → Hom C (x, y ′ ); <strong>and</strong><br />
2. there exists a contravariant functor Hom C (−, x) : C → Set<br />
such that for any y ∈ Ob C, Hom C (−, x)(y) := Hom C (y, x)<br />
<strong>and</strong> for any α ∈ Hom C (y, y ′ ) the composition Hom C (−, x)(α) =<br />
− ◦ α : Hom C (y ′ , x) → Hom C (y, x).<br />
An object x ∈ Ob C is projective if Hom C (x, −) : C → Set<br />
preserves epimorphisms. An object x ∈ Ob C is injective if<br />
Hom C (−, x) : C → Set preserves monomorphisms. If C is an<br />
abelian category, we say that C has enough projective objects<br />
if for any y ∈ Ob C there exists a projective object x along<br />
with an epimorphism x → y <strong>and</strong> that C has enough injective
6 1 Functors <strong>and</strong> their Kan extensions<br />
objects if for any y ∈ Ob C there exists an injective object x<br />
along with a monomorphism y → x.<br />
Definition 1.1.8. Suppose C is a category. Its opposite category,<br />
named C op , share the same objects with C. Each α ∈<br />
Hom C (x, y) defines a unique morphism α op ∈ Hom C op(y, x). If<br />
α op ∈ Hom C op(y, x) <strong>and</strong> β op ∈ Hom C op(z, y), their composite<br />
is α op ◦ β op := (β ◦ α) op ∈ Hom C op(z, x).<br />
Passing from a category to its opposite has a dualizing effect<br />
on many categorical concepts, constructions <strong>and</strong> properties.<br />
For instance, it interchanges initial objects with terminal objects,<br />
monomorphisms with epimorphisms, projective objects<br />
with injective objects, products with coproducts etc. Moreover<br />
one can readily verify that a covariant functor F : C → D<br />
naturally determines a contravariant functor, the dual functor,<br />
F ∧ : C op → D, <strong>and</strong> vice versa. Thus what we learn about<br />
covariant functors can be directly translated to contravariant<br />
functors. From now on, if not specified, all functors will be<br />
covariant in these notes.<br />
Definition 1.1.9. A functor F : D → C is full if F(D) ⊂ C<br />
is a full subcategory.<br />
A functor F : D → C is faithful if for any α, β ∈ Mor D<br />
such that F(α) = F(β) ∈ Mor C, then α = β.<br />
If D <strong>and</strong> C are two preadditive categories, then a functor F :<br />
D → C is called additive if Hom D (x, y) → Hom C (F(x), F(y))<br />
is a homomorphism between abelian groups.<br />
Definition 1.1.10. Suppose F, G : D → C are two functors.<br />
A natural transformation Φ : F → G assigns to each object<br />
x ∈ Ob D a morphism Φ x : F(x) → G(x) so that we have a<br />
commutative diagram
1.1 Functors <strong>and</strong> limits 7<br />
F(x) Φ x <br />
G(x)<br />
F(α)<br />
<br />
G(α)<br />
F(y) Φy<br />
<br />
G(y)<br />
for any given α ∈ Hom D (x, y). If every Φ x is an isomorphism<br />
in C, we call such Φ a natural equivalence <strong>and</strong> write F ∼ = G<br />
in this situation.<br />
Definition 1.1.11. Two categories C <strong>and</strong> D are equivalent<br />
, written as C ≃ D, if there exist functors F : C → D <strong>and</strong><br />
G : D → C such that FG ∼ = Id D <strong>and</strong> GF ∼ = Id C .<br />
Definition 1.1.12. Let C be a category. If we take exactly<br />
one object from each isomorphism class <strong>of</strong> objects in C, then<br />
we can form a full subcategory consisting <strong>of</strong> these chosen objects.<br />
This subcategory is called a skeleton <strong>of</strong> C.<br />
Since skeletons <strong>of</strong> a category C are naturally equivalent to<br />
each other, we can speak about the skeleton <strong>of</strong> C. We shall<br />
always denote by [C] the skeleton <strong>of</strong> a category C. Then one<br />
can verify that [C] ≃ C.<br />
It is a fact that when D is small all the covariant functors<br />
from D to C form a category whose objects are these functors<br />
<strong>and</strong> morphisms are the natural transformations. We call it a<br />
functor category, written as C D . Sometimes we call such a<br />
D an index category. Functor categories are <strong>of</strong> pivotal importance<br />
in these notes. We <strong>of</strong>ten consider functor categories<br />
whose index categories are finite (such as a finite group G)<br />
<strong>and</strong> whose target categories are large (such as R-mod), in order<br />
to underst<strong>and</strong> finite categories via their representations<br />
(see Chapter 3) in certain large categories.<br />
The functors in Example 1.1.7 are very important to us.<br />
Here we present a crucial property <strong>of</strong> those functors.
8 1 Functors <strong>and</strong> their Kan extensions<br />
Definition 1.1.13. A covariant (respectively, contravariant)<br />
functor F : C → Set is called representable if it is naturally<br />
equivalent to Hom C (x, −) (respectively, Hom C (−, x)) for some<br />
x ∈ Ob C.<br />
Lemma 1.1.14 (Yoneda Lemma). Let F : C → Set be a<br />
functor. Then we have<br />
for any x ∈ Ob C.<br />
Hom Set<br />
C(Hom C (x, −), F) ∼ = F(x),<br />
Pro<strong>of</strong>. Each natural transformation Φ in the left side is uniquely<br />
determined by the morphism<br />
Φ x : Hom C (x, x) → F(x)<br />
which is uniquely determined by the image <strong>of</strong> Φ x (1 x ) ∈ F(x).<br />
Hence we can define a bijection Φ → Φ x (1 x ) <strong>and</strong> then the<br />
statement follows.<br />
⊓⊔<br />
In the end, we record a construction that we will use in the<br />
next section.<br />
Definition 1.1.15. Suppose C <strong>and</strong> D are two categories.<br />
Then we can define their product category C × D, whose objects<br />
are {(x, y) ∣ x ∈ Ob C, y ∈ Ob D} <strong>and</strong> whose morphisms<br />
are {(α, β) ∣ α ∈ Mor C, β ∈ Mor D}.<br />
A functor from a product category C × D to some target<br />
category sometimes is called a bifunctor.<br />
1.1.2 Adjoint functors<br />
We take the opportunity to record the definition <strong>of</strong> an adjoint<br />
functor <strong>and</strong> its basic properties for future reference.<br />
Definition 1.1.16. Let F : C → D <strong>and</strong> G : D → C be two<br />
functors such that there is a natural equivalence
1.1 Functors <strong>and</strong> limits 9<br />
Ω : Hom D (F(−), −) ∼= →Hom C (−, G(−))<br />
<strong>of</strong> functors C op ×D → Set. We say that F is left adjoint to G,<br />
G is right adjoint to F <strong>and</strong> write Ω : F ⊣ G for the adjunct.<br />
Example 1.1.17. If F : C → D <strong>and</strong> G : D → C give rise to<br />
category equivalences as in Definition 1.1.11, then F is both<br />
a left <strong>and</strong> a right adjoint <strong>of</strong> G.<br />
Another example is for U, V, W ∈ V ect k , there is an isomorphism<br />
Hom k (U ⊗ k V, W ) ∼ = Hom k (U, Hom k (V, W )).<br />
Next we deduce some fundamental properties <strong>of</strong> the adjunction.<br />
1. Naturality <strong>of</strong> Ω: for any α : x ′ → x <strong>and</strong> β : y → y ′ the<br />
following diagram commutes<br />
Hom D (F(x), y) Ω x,y <br />
Hom C (x, G(y))<br />
β◦(−)◦F(α)<br />
<br />
G(β)◦(−)◦α<br />
Hom D (F(x ′ ), y ′ )<br />
<br />
Ωx<br />
Hom<br />
′ ,y ′ C (x ′ , G(y ′ )).<br />
Equivalently, we have for any ϕ : F(x) → y<br />
Ω x ′ ,y ′ (β ◦ ϕ ◦ F(α)) = G(β) ◦ Ω x,y (ϕ) ◦ α.<br />
We shall repeatedly use the above formula. Very <strong>of</strong>ten, the<br />
subscript <strong>of</strong> Ω will be omitted for the sake <strong>of</strong> convenience.<br />
For instance, when x ′ = x <strong>and</strong> α = 1 x , we have<br />
Ω(β ◦ ϕ) = G(β) ◦ Ω(ϕ),<br />
<strong>and</strong> when y = F(x) <strong>and</strong> ϕ = 1 F(x) , we get<br />
Ω(β ◦ F(α)) = G(β) ◦ α.
10 1 Functors <strong>and</strong> their Kan extensions<br />
One should try to deduce similar formulas for future applications.<br />
2. Unit <strong>and</strong> counit: we define, for each x ∈ Ob C, Σ x =<br />
Ω(1 F(x) ) : x → GF(x) <strong>and</strong>, for each y ∈ Ob D, Λ y =<br />
Ω −1 (1 G(y) ) : FG(y) → y. By naturality <strong>of</strong> Ω, we actually<br />
obtain two natural transformations<br />
Σ : Id C → GF <strong>and</strong> Λ : FG → Id D .<br />
For example, for any α : x ′ → x, using various formulas we<br />
obtain from the naturality <strong>of</strong> Ω, we get<br />
GF(α) ◦ Σ x<br />
′ = GF(α) ◦ Ω(1 F(x ′ ))<br />
= Ω(F(α) ◦ 1 F(x ′ ))<br />
= Ω(1 F(x) ◦ F(α))<br />
= Ω(1 F(x) ) ◦ α<br />
= Σ x ◦ α.<br />
Thus we have a commutative diagram<br />
x ′<br />
α<br />
x<br />
Σ x ′<br />
<br />
GF(x ′ )<br />
Σx<br />
GF(α)<br />
<br />
GF(x).<br />
We can deduce that the following<br />
F FΣ<br />
−→FGF ΛF<br />
−→F,<br />
G ΣG<br />
−→GFG GΛ<br />
−→G<br />
compose to identities. For example, given some x ∈ Ob C,<br />
one can verify Λ F(x) ◦ F(Σ x ) = 1 F(x) by showing<br />
Ω(Λ F(x) ◦ F(Σ x )) = Ω(Λ F(x) ) ◦ Σ x = Σ x = Ω(1 F(x) ).<br />
Moreover one can recover the adjunct by unit <strong>and</strong> counit:<br />
for any ϕ : F(x) → y <strong>and</strong> ψ : x → G(y)<br />
Ω(ϕ) = G(ϕ) ◦ Σ x<br />
<strong>and</strong> Ω −1 (ψ) = Λ y ◦ F(ψ).
1.1 Functors <strong>and</strong> limits 11<br />
Theorem 1.1.18. Suppose F : C → D <strong>and</strong> G : D → C<br />
are two functors. If there exist two natural transformations<br />
Σ : Id C → GF <strong>and</strong> Λ : FG → Id D such that the following<br />
two composites<br />
F FΣ<br />
−→FGF ΛF<br />
−→F,<br />
G ΣG<br />
−→GFG GΛ<br />
−→G<br />
are identities, then F is a left adjoint <strong>of</strong> G. Moreover<br />
Ω : Hom D (F(x), y) → Hom C (x, G(y)) defined by Ω(ϕ) =<br />
G(ϕ) ◦ Σ x is the adjunct, with Σ, Λ the unit <strong>and</strong> counit <strong>of</strong><br />
the adjunction.<br />
Pro<strong>of</strong>. For x ∈ Ob C, y ∈ Ob D, φ ∈ Hom D (F(x), y) <strong>and</strong><br />
ψ ∈ Hom C (x, G(y)), we define<br />
Ω x,y (φ) = G(φ) ◦ Σ x <strong>and</strong> ¯Ωx,y (ψ) = Λ y ◦ F(ψ).<br />
Then we can verify that they define two natural transformations<br />
<strong>of</strong> functors C op × D → Sets. In fact for any α ∈<br />
Hom C (x, x ′ ), β ∈ Hom D (y, y ′ ) <strong>and</strong> φ ∈ Hom D (F(x), y) we<br />
have<br />
G(β) ◦ Ω(φ) ◦ α = G(β) ◦ G(φ) ◦ Σ x ◦ α<br />
= G(β ◦ φ) ◦ GF(α) ◦ Σ x<br />
′<br />
= G(β ◦ φ ◦ F(α)) ◦ Σ x<br />
′<br />
= Ω(β ◦ φ ◦ F(α)),<br />
which implies Ω is a natural transformation. The assertion<br />
for ¯Ω is proved in a similar way.<br />
Moreover Ω <strong>and</strong> ¯Ω provide natural equivalences because we<br />
can show<br />
( ¯Ω ◦ Ω)(φ) = ¯Ω(G(φ) ◦ Σx )<br />
= Λ y ◦ F(G(φ) ◦ Σ x )<br />
= Λ y ◦ FG(φ) ◦ F(Σ x )<br />
= φ ◦ Λ F(x) F(Σ x )<br />
= φ,
12 1 Functors <strong>and</strong> their Kan extensions<br />
<strong>and</strong> similarly (Ω ◦ ¯Ω)(ψ) = ψ.<br />
This theorem also implies that the adjoint <strong>of</strong> a functor is<br />
unique up to natural equivalence.<br />
Corollary 1.1.19. If G <strong>and</strong> G ′ are right adjoints <strong>of</strong> F, then<br />
they are naturally equivalent. Similarly if F <strong>and</strong> F ′ are two<br />
left adjoints <strong>of</strong> G, then they are naturally equivalent.<br />
Pro<strong>of</strong>. We only prove the first assertion. For any x ∈ Ob C<br />
<strong>and</strong> y ∈ Ob D we have two isomorphisms<br />
Hom C (x, G(y)) Ω<br />
←−Hom D (F(x), y) Ω′<br />
−→Hom C (x, G ′ (y)).<br />
Then (Ω◦Ω ′−1 )(1 G (y)) : ′ G ′ (y) → G(y) <strong>and</strong> (Ω ′ ◦Ω −1 )(1 G(y) ) :<br />
G(y) → G ′ (y) induce a natural equivalence between G <strong>and</strong><br />
G ′ .<br />
⊓⊔<br />
⊓⊔<br />
1.1.3 Limits<br />
In these notes, direct <strong>and</strong> inverse limits <strong>of</strong> functors are key<br />
concepts so we introduce their definitions here. Suppose M ∈<br />
Ob(T C ) is an object in a functor category. Then it can be<br />
identified with a commutative diagram <strong>of</strong> objects {M(x) ∣ ∣<br />
x ∈ Ob C} <strong>and</strong> morphisms {M(α) ∣ ∣ α ∈ Mor C} in T . For<br />
convenience <strong>and</strong> future reference, we call this the diagram <strong>of</strong><br />
M.<br />
Definition 1.1.20. Let C be a small category <strong>and</strong> T an arbitrary<br />
category. Consider the functor category T C . Suppose<br />
M ∈ Ob(T C ). Then it has an inverse limit, denoted by<br />
lim M ∈ Ob T , if for each x ∈ Ob C there exists a morphism<br />
θ x : ←−C<br />
←−C<br />
lim M → M(x) such that, by adding the object<br />
lim M <strong>and</strong> the morphisms {θ ∣<br />
←−C<br />
x x ∈ Ob C} to the diagram<br />
<strong>of</strong> M, we obtain an enlarged commutative diagram, <strong>and</strong> this
1.1 Functors <strong>and</strong> limits 13<br />
object lim ←−C<br />
M is universal in the sense that if t is another object<br />
in Ob T that enjoys the same properties as lim ←−C<br />
M then<br />
we must have a unique morphism Θ t : t → lim ←−C<br />
M making<br />
the whole diagram commutative as pictured as follows<br />
t<br />
θ ′ y<br />
Θ t<br />
θ ′ x<br />
<br />
lim M θ x<br />
←−C<br />
θ y <br />
M(α)<br />
M(x)<br />
M(y) .<br />
Dually the functor M has a direct limit in T , denoted by<br />
lim M ∈ Ob T , if for each x ∈ Ob C there exists a morphism<br />
−→C<br />
τ x : M(x) → lim ←−C<br />
M such that, by adding the object −→C<br />
lim M<br />
∣<br />
<strong>and</strong> the morphisms {τ x x ∈ Ob C} to the diagram <strong>of</strong> M,<br />
we obtain an enlarged commutative diagram, <strong>and</strong> this object<br />
lim M is universal in the sense that if t is another object in<br />
−→C<br />
Ob T that enjoys the same properties as lim −→C<br />
M then we must<br />
have a unique morphism Ξ t : −→C<br />
lim M → t making the whole<br />
diagram commutative as pictured as follows<br />
M(α)<br />
M(y) τy<br />
<br />
M(x)<br />
τ ′ y<br />
τ x<br />
τ ′ x<br />
lim<br />
−→C M Ξ t<br />
t<br />
It follows directly from the definition that, when a limit<br />
exists, it is unique up to isomorphism. If either the index<br />
category or the functor itself is structurally simple, we may<br />
explicitly compute the limits.
14 1 Functors <strong>and</strong> their Kan extensions<br />
Example 1.1.21.1. Let C = Z <strong>and</strong> T = Set. Then a functor<br />
M : Z → Set is represented by a chain <strong>of</strong> set maps<br />
· · · → M(−1) → M(0) → M(1) → M(2) → · · · .<br />
In particular if M 0 is given by the following diagram in which<br />
each map is an inclusion<br />
· · · ⊂ M 0 (−1) ⊂ M 0 (0) ⊂ M 0 (1) ⊂ M 0 (2) ⊂ · · · ,<br />
then lim ←−Z<br />
M 0 = ⋂ i∈Z M 0(i) <strong>and</strong> −→Z<br />
lim M 0 = ⋃ i∈Z M 0(i).<br />
2. If C = Z <strong>and</strong> T = V ect k , the category <strong>of</strong> finite-dimensional<br />
k-vector spaces. Suppose N is represented by the following<br />
· · · → 0 N(1) N(2) N(3) · · · .<br />
Then −→Z<br />
lim N does not exist in V ect k , the category <strong>of</strong> finitedimensional<br />
k-vector spaces, but lim ←−Z<br />
N = {0}.<br />
3. Let T be a category <strong>and</strong> I a set considered as a discrete<br />
category. Then a functor M : I → T is simply an I-indexed<br />
set <strong>of</strong> objects in T . The the coproduct <strong>of</strong> these objects is<br />
defined as ⊕ I x i = −→I<br />
lim M <strong>and</strong> the product <strong>of</strong> these objects<br />
is ∏ I x i = ←−I<br />
lim M.<br />
4. Let C be the following poset<br />
x<br />
α<br />
<br />
y β<br />
<br />
z<br />
<strong>and</strong> T = Vect k , the category <strong>of</strong> all k-vector spaces. Each<br />
functor M ∈ Ob T C is represented by<br />
M(y) M(β)<br />
M(x)<br />
M(α)<br />
<br />
M(z).
1.1 Functors <strong>and</strong> limits 15<br />
Then ←−C<br />
lim M is called the pullback <strong>of</strong> the latter diagram <strong>and</strong><br />
lim M = M(z).<br />
−→C<br />
Dually if D is the following poset<br />
a f <br />
g<br />
c<br />
<strong>and</strong> N ∈ Ob T D then −→D<br />
lim N is called the pushout <strong>of</strong> the<br />
diagram<br />
N(a)<br />
N(g)<br />
<br />
N(c)<br />
b<br />
N(f) <br />
N(b)<br />
<strong>and</strong> lim ←−D<br />
N = N(a).<br />
The definition <strong>of</strong> limits can be rewritten by using a simple,<br />
yet very important, construction.<br />
Definition 1.1.22. There is a constant functor<br />
K : T → T C<br />
such that, for any t ∈ Ob T , K(t) is defined by K(t)(x) = t<br />
<strong>and</strong> K(α) = 1 t for any x ∈ Ob C <strong>and</strong> α ∈ Mor C.<br />
In the literature, the functor K : T → T C is <strong>of</strong>ten named the<br />
“diagonal functor”. Since the terminology is used for another<br />
purpose, see Definition 2.2.34, we shall stick with our notion<br />
which seems to be more appropriate.<br />
Now we present alternative characterizations <strong>of</strong> limits. The<br />
definition <strong>of</strong> an inverse limit can be rephrased as saying that<br />
there exists lim ←−C<br />
M ∈ Ob C with a natural transformation<br />
Γ : K(lim ←−C<br />
M) → M which is universal in the sense that if<br />
there is another object t, along with a natural transformation<br />
Υ : K(t) → M, then there exists a unique morphism θ t :
16 1 Functors <strong>and</strong> their Kan extensions<br />
t → lim ←−C<br />
M hence a natural transformation K(θ t ) : K(t) →<br />
K(lim ←−C<br />
M) such that Γ ◦ K(θ t ) = Υ .<br />
K(θ t )<br />
K(t)<br />
K(lim ←−C<br />
M) Γ<br />
Especially we obtain a morphism<br />
Ω : Hom T C(K(t), M) → Hom T (t, lim ←−C<br />
M),<br />
given by Ω(Υ ) = θ t . It is an isomorphism because each t →<br />
lim M extends to a functor K(t) → M. Thus if every M ∈<br />
←−C<br />
Ob(T C ) has a limit in T , then lim ←−<br />
: T C → T is the right<br />
adjoint <strong>of</strong> K <strong>and</strong> Ω becomes the corresponding adjunct. We<br />
characterize the direct limit lim −→C<br />
in a similar way but we leave<br />
it to the reader. When T is “large” enough with respect to C,<br />
we can introduce the limits in an economical way.<br />
Proposition 1.1.23. If K has a right adjoint R : T C → T ,<br />
then for each M ∈ Ob(T C ), R(M) is the inverse limit <strong>of</strong><br />
M. If K has a left adjoint L : T C → T , then for each<br />
M ∈ Ob(T C ), L(M) is the direct limit <strong>of</strong> M.<br />
For convenience, we introduce the following concepts.<br />
Definition 1.1.24. A category T is called complete if for<br />
any small category C <strong>and</strong> any functor M ∈ T C the inverse<br />
limit lim ←−C<br />
M exists. A category T is called cocomplete if for<br />
any small category C <strong>and</strong> any functor M ∈ T C the direct<br />
limit lim −→C<br />
M exists.<br />
A category T is called finitely complete (respectively finitely<br />
cocomplete) if for any finite category C <strong>and</strong> any functor<br />
Υ<br />
M
1.1 Functors <strong>and</strong> limits 17<br />
M ∈ T C the inverse limit ←−C<br />
lim M (respectively the direct limit<br />
lim M) exists.<br />
−→C<br />
The categories Set <strong>and</strong> R-Mod are both complete <strong>and</strong> cocomplete.<br />
By Example 1.1.21, a coproduct is a direct limit <strong>and</strong> a product<br />
in an inverse limit. In abelian categories, the existence <strong>of</strong><br />
coproducts, respectively product, is equivalent to the cocompleteness,<br />
respectively completeness, condition on a category<br />
T .<br />
Proposition 1.1.25. Let T be an abelian category. Then<br />
1. it is complete (or finitely complete) if <strong>and</strong> only if all products<br />
(or all finite products) exist; <strong>and</strong><br />
2. it is cocomplete (or finitely cocomplete) if <strong>and</strong> only if all<br />
coproducts (or all finite coproducts) exist.<br />
Pro<strong>of</strong>. We only prove (1). Since a product is an inverse limit,<br />
we just have to demonstrate the opposite direction. Suppose<br />
C is a small (or finite) category <strong>and</strong> M : C → T is a functor.<br />
Let<br />
C 0 (C; M) = {f : Ob C →<br />
∏<br />
M(x) ∣ f(x) ∈ M(x)}<br />
<strong>and</strong><br />
C 1 (C; M) = {f : Mor C →<br />
x∈Ob C<br />
∏<br />
α∈Mor C<br />
We define δ : C 0 (C; M) → C 1 (C; M) by<br />
M(y) ∣ ∣ f(x α →y) ∈ M(y)}.<br />
δ(f)(x α →y) = f(y) − M(α)[f(x)].<br />
Then we can verify that the kernel <strong>of</strong> δ is ←−C<br />
lim M. ⊓⊔<br />
For example the category <strong>of</strong> finitely generated R-modules,<br />
R-mod, is both finitely complete <strong>and</strong> finitely cocomplete.
18 1 Functors <strong>and</strong> their Kan extensions<br />
Completeness <strong>and</strong> cocompleteness pass to functor categories.<br />
We record this <strong>and</strong> another result that shall be useful in the<br />
next chapter.<br />
Proposition 1.1.26. Let C be a small category. If T is complete<br />
(resp. cocomplete), then T C is complete (resp. cocomplete).<br />
Pro<strong>of</strong>. Let D be another small category <strong>and</strong> M : D → T C<br />
a functor. Since (T C ) D ∼ = T C×D , M can be identified with a<br />
bifunctor from ˜M(−, −) : C × D → T .<br />
We may define a functor N : C → T by N(x) = ←−D<br />
lim ˜M(x, −)<br />
for every x ∈ Ob C. Since [lim ←−D ˜M(−, −)](x) ∼ = lim ←−D<br />
[ ˜M(x, −)]<br />
for all x ∈ Ob D, we can show N is the limit ←−D<br />
lim M. It means<br />
T C is complete. The cocompleteness <strong>of</strong> T C can be proved in<br />
the same way.<br />
⊓⊔<br />
The above result tells us that Set C <strong>and</strong> (R-Mod) C are both<br />
complete <strong>and</strong> cocomplete if C is small. Furthermore (R-mod) C<br />
is finitely complete <strong>and</strong> cocomplete if C is finite.<br />
We record two results on preserving limits.<br />
Proposition 1.1.27. Let T be both complete <strong>and</strong> cocomplete<br />
<strong>and</strong> C a small category. Given a functor M ∈ Ob T C<br />
<strong>and</strong> T ∈ Ob T , we have<br />
Hom T (T, lim ←−C<br />
M) ∼ = lim ←−C<br />
Hom T (T, M(−))<br />
<strong>and</strong><br />
Hom T (lim −→C<br />
M, T ) ∼ = lim ←−C<br />
Hom T (M(−), T )<br />
Pro<strong>of</strong>. To prove the first isomorphism, we take the diagram <strong>of</strong><br />
M, which determines lim ←−C<br />
M. Applying the covariant functor<br />
Hom T (T, −) to the diagram it results in the diagram <strong>of</strong> the<br />
functor Hom T (T, M(−)). From here one may complete the
1.1 Functors <strong>and</strong> limits 19<br />
pro<strong>of</strong> based by using the universal property <strong>of</strong> an inverse limit.<br />
The second isomorphism may be proved in the same way. ⊓⊔<br />
Proposition 1.1.28. Let F : T → T ′ be the left adjoint <strong>of</strong><br />
G : T ′ → T .<br />
1. Suppose M : C → T has a direct limit. Then F(lim −→C<br />
M) ∼ =<br />
lim F ◦ M. −→C<br />
2. Suppose N : D → T ′ has an inverse limit. Then G(lim ←−D<br />
N) ∼ =<br />
lim G ◦ N. ←−D<br />
Pro<strong>of</strong>. We only prove (1). It is obvious that F(lim −→C<br />
M) fits into<br />
the defining diagram <strong>of</strong> −→C<br />
lim F◦M. We need to show F(lim −→C<br />
M)<br />
is universal. Suppose t ′ ∈ Ob T ′ is another object satisfying<br />
the colimit defining diagram. It follows from the adjunction<br />
Hom T ′(F(lim −→C<br />
M), t ′ ) ∼ = Hom T (lim −→C<br />
M, G(t ′ )),<br />
along with the unit Id → F◦G that G(t ′ ) satisfies the defining<br />
diagram <strong>of</strong> −→C<br />
lim M. From the universal property <strong>of</strong> −→C<br />
lim M,<br />
there exists a unique morphism lim −→C<br />
M → G(t ′ ) which gives<br />
rise to a morphism F(lim −→C<br />
M) → t ′ . The universal property<br />
<strong>of</strong> F(lim −→C<br />
M) follows from it.<br />
⊓⊔<br />
In the end, we give a result that is used in the next section.<br />
Theorem 1.1.29. Let C be a small category. Then any<br />
functor M : C → Set is canonically a colimit <strong>of</strong> a diagram<br />
<strong>of</strong> representable functors.<br />
Pro<strong>of</strong>. We shall establish this result by firstly constructing a<br />
small category D from M : C → Set <strong>and</strong> secondly showing<br />
that M induces a functor ˜M : D → Set C satisfying −→D<br />
lim ˜M ∼ =<br />
M.<br />
The objects in the category D are pairs (x, a) such that x<br />
is an object <strong>of</strong> C <strong>and</strong> a is an element in the set M(x). A
20 1 Functors <strong>and</strong> their Kan extensions<br />
morphism f : (x, a) → (y, b) is a morphism f ∈ Hom C (x, y)<br />
satisfying M(f)(a) = b.<br />
We define a functor ˜M : D → Set C by ˜M(x, a) =<br />
Hom C (x, −). From the Yoneda Lemma, there exists for each<br />
x ∈ Ob C an isomorphism <strong>of</strong> sets Hom Set<br />
C(Hom C (x, −), M) ∼ =<br />
M(x). It implies that each m x ∈ M(x) determines uniquely<br />
a functor f (x,mx ) : ˜M(x, mx ) → M making the enlarged diagram<br />
<strong>of</strong> ˜M commutative<br />
Hom C (x, −) = ˜M(x, mx )<br />
f◦−<br />
˜M(y, m y ) = Hom C (y, −)<br />
f (x,m x)<br />
M<br />
f (y,m y)<br />
Note that the commutativity <strong>of</strong> the diagram forces M(f)(m x ) =<br />
m y .<br />
In order to prove that M is the direct limit we need to<br />
show it is universal. Suppose L is another functor fitting<br />
into the above diagram, <strong>and</strong> for each (x, m x ) ∈ Ob D it<br />
comes with a functor g (x,mx ) : Hom C (x, −) → L. Again by<br />
the Yoneda Lemma, g (x,mx ) determines uniquely an element<br />
l x ∈ L(x). From the commutativity <strong>of</strong> the diagram, we must<br />
have L(f)(l x ) = l y . Now we define a functor h : M → L by<br />
h z (m z ) = l z<br />
if z ∈ Ob C. We may readily verify that h z ′M(f) = L(f)h z<br />
for any f : z → z ′ . It implies that h is well defined. Since by<br />
definition hf (x,mx ) = g (x,mx ) for all objects (x, m x ) ∈ Ob D, M<br />
is universal.<br />
⊓⊔<br />
The construction <strong>of</strong> D from a functor to Set in the pro<strong>of</strong> is<br />
the predecessor <strong>of</strong> the Grothendieck constructions introduced<br />
in Section 2.4.
1.2 Restriction <strong>and</strong> Kan extensions 21<br />
1.2 Restriction <strong>and</strong> Kan extensions<br />
In last section we mentioned that, for a functor category T C ,<br />
if the target category are complete <strong>and</strong> cocomplete, then we<br />
may define the limits using adjoints <strong>of</strong> the constant functor<br />
K : T → T C . When we examine closely, we realize that K itself<br />
is induced by another functor pt : C → •. This observation<br />
generates new ideas for constructing some extremely powerful<br />
functors, called Kan extensions.<br />
1.2.1 Restriction<br />
Definition 1.2.1. Suppose u : D → C is a functor between<br />
two small categories. For any target category T , u induces a<br />
functor Res u : T C → T D via precomposition with u, called<br />
the restriction along u.<br />
Example 1.2.2. The canonical functor pt : C → • induces a<br />
restriction T ∼ = T • → T C , which is identical to K <strong>of</strong> Definition<br />
1.1.22.<br />
In last section, we know explicitly the left <strong>and</strong> right adjoints<br />
<strong>of</strong> K. As a generalization, we shall describe the adjoints <strong>of</strong><br />
an arbitrary restriction. But before doing that, we provide<br />
several simple properties <strong>of</strong> the restriction.<br />
Lemma 1.2.3. Suppose u : D → C is a functor. Then for<br />
any M ∈ Ob(T C ), there are canonical morphisms lim ←−C<br />
M →<br />
lim Res ←−D uM <strong>and</strong> lim −→D<br />
Res u M → lim −→C<br />
M.<br />
Pro<strong>of</strong>. The map between inverse limits follows from the universal<br />
property
22 1 Functors <strong>and</strong> their Kan extensions<br />
lim M ←−C α x<br />
Θ<br />
<br />
lim Res φ x<br />
α y ←−D uM<br />
<br />
Res u M(a) = M(u(a))<br />
φ y<br />
<br />
Res u M(φ)=M(u(φ))<br />
Res u M(b) = M(u(b)) ,<br />
in which φ : a → b is a morphism in D. The map between<br />
direct limits can be established similarly.<br />
⊓⊔<br />
Proposition 1.2.4. Suppose u : D → C has a left adjoint<br />
v : C → D. Then for any T , Res u has a right adjoint Res v .<br />
In particular, if u <strong>and</strong> v are category equivalences, Res u <strong>and</strong><br />
Res v are equivalences.<br />
If T is complete <strong>and</strong> cocomplete, then for any M ∈<br />
Ob(T C ), lim −→C<br />
M ∼ = lim −→D<br />
Res u M, <strong>and</strong> for any N ∈ Ob(T ) D ,<br />
lim N ∼ = lim<br />
←−D ←−C<br />
Res v N.<br />
Pro<strong>of</strong>. We shall prove that for M ∈ Ob(T C ) <strong>and</strong> N ∈ Ob(T D )<br />
there exists an isomorphism<br />
Hom T D(Res u M, N) ∼ = Hom T C(M, Res v N).<br />
Let Σ : Id C → uv <strong>and</strong> Λ : vu → Id D be the unit <strong>and</strong> counit<br />
for the adjunction between u <strong>and</strong> v. We define two natural<br />
transformations Σ ′ : Id T<br />
C → Res v Res u <strong>and</strong> Λ ′ : Res u Res v →<br />
Id T<br />
D as follows. Given M ∈ Ob(T C ), x ∈ Ob C, N ∈ Ob(T D )<br />
<strong>and</strong> a ∈ Ob D,<br />
(Σ ′ M) x := M(Σ x ) : M(x) → Res v Res u M(x) = M(uv(x)),<br />
<strong>and</strong><br />
(Λ ′ N) a := N(Λ a ) : Res u Res v N(a) = N(vu(a)) → N(a).<br />
One can verify that Σ ′ <strong>and</strong> Λ ′ provide the unit <strong>and</strong> counit <strong>of</strong><br />
an adjunction.
1.2 Restriction <strong>and</strong> Kan extensions 23<br />
As for the second statement, we have for all t ∈ Ob T<br />
<strong>and</strong><br />
Hom T (lim −→D<br />
Res u M, t) ∼ = Hom T D(Res u M, K(t))<br />
∼ = HomT C(M, Res v K(t))<br />
∼ = HomT C(M, K(t))<br />
∼ = HomT (lim −→C<br />
M, t).<br />
Hom T (t, lim ←−C<br />
Res v N) ∼ = Hom T C(K(t), Res v N)<br />
∼ = HomT D(Res u K(t), N)<br />
∼ = HomT D(K(t), N)<br />
∼ = HomT (t, lim ←−D<br />
N).<br />
Thus −→D<br />
lim Res u M ∼ = lim −→C<br />
M <strong>and</strong> ←−C<br />
lim Res v N ∼ = lim ←−D<br />
N. ⊓⊔<br />
In many places we will have to consider the adjoint functors<br />
<strong>of</strong> some restriction Res u . When u has an adjoint, we get an<br />
adjoint <strong>of</strong> Res u which is also a restriction, by Proposition 1.2.4.<br />
The truth is that even if u does not admit an adjoint, we can<br />
still construct the adjoints <strong>of</strong> Res u , <strong>and</strong> this is the main result<br />
<strong>of</strong> the upcoming two sections.<br />
1.2.2 Overcategories <strong>and</strong> undercategories<br />
In order to introduce the adjoints <strong>of</strong> a restriction, we have<br />
to provide some important categorical constructions. These<br />
categorical constructions are <strong>of</strong> great importance in both homological<br />
algebra <strong>and</strong> homotopy theory <strong>of</strong> classifying spaces.<br />
We shall be familiar with them as they appear almost everywhere<br />
throughout these notes.<br />
Definition 1.2.5. Let u : D → C be a functor between<br />
(small) categories <strong>and</strong> x ∈ Ob C. The category over x, u/x,<br />
consists <strong>of</strong> objects {(a, α) ∣ ∣ a ∈ Ob D, α ∈ Hom C (u(a), x)}.<br />
For any two objects (a, α), (b, β), a morphism from (a, α) to
24 1 Functors <strong>and</strong> their Kan extensions<br />
(b, β) is a morphism µ ∈ Hom D (a, b) making the following<br />
diagram commutative<br />
u(a)<br />
u(µ)<br />
u(b)<br />
The category under x, written as x\u, is defined in a dual<br />
fashion. It consists <strong>of</strong> objects {(α, a) ∣ ∣ a ∈ Ob D, α ∈<br />
Hom C (x, u(a))}. For any two objects (α, a), (β, b), a morphism<br />
from (α, a) to (β, b) is a morphism µ ∈ Hom D (a, b)<br />
making the following diagram commutative<br />
x<br />
α<br />
β<br />
α<br />
β<br />
u(a)<br />
x,<br />
u(µ)<br />
u(b)<br />
The categories defined above are customarily called overcategories<br />
<strong>and</strong> undercategories, associated with u : D → C. We<br />
will see later on that Id C /x <strong>and</strong> x\Id C , for any x ∈ Ob C, are<br />
already very interesting.<br />
Remark 1.2.6.1. From definition, an object in the overcategory<br />
u/x, (a, α), can be pictured as u(a) →x, α <strong>and</strong> consequently<br />
a morphism µ : (a, α) → (b, β) can be equivalently<br />
interpreted as a sequence u(a) u(µ)<br />
→ u(b) →x. β This kind<br />
<strong>of</strong> rewritings will be useful for us when dealing with chains<br />
<strong>of</strong> morphisms in u/x <strong>and</strong> we shall come back to this point<br />
in later chapters. Similar reinterpretation can be made for
1.2 Restriction <strong>and</strong> Kan extensions 25<br />
objects <strong>and</strong> morphisms in undercategories too but we leave<br />
it to the reader.<br />
2. There is a canonical functor P x : u/x → D (resp. P x :<br />
x\u → D given on objects as projection to the first (resp.<br />
the second) component <strong>and</strong> on morphisms as the identity.<br />
For simplicity, we shall denote such functors just as P.<br />
3. If γ : x → y is a morphism in C, then it naturally induces a<br />
functor γ ∗ : u/x → u/y <strong>and</strong> a functor γ ∗ : y\u → x\u.<br />
Example 1.2.7.1. Let pt : C → • be the canonical functor.<br />
Then pt/• ∼ = •\pt ∼ = C.<br />
2. Let G be a group <strong>and</strong> H a subgroup. Then the inclusion<br />
functor i H : H ↩→ G gives exactly one overcategory i H /•.<br />
By direct calculation, the objects are {(•, g)|g ∈ G}, <strong>and</strong><br />
biject with the elements <strong>of</strong> G. There is a morphism from<br />
one object (•, g 1 ) to another (•, g 2 ) if there exists a h ∈ H<br />
such that g 1 = g 2 h. Obviously h = g2 −1 g 1. Thus there is at<br />
most one morphism from an object to another. Since h is<br />
invertible, there exists a morphism between two objects if<br />
<strong>and</strong> only if their are isomorphic in i H /•. In other words, two<br />
objects (•, g 1 ) to another (•, g 2 ) are isomorphic if <strong>and</strong> only<br />
if g 1 H = g 2 H. Because the category i H /• consists <strong>of</strong> [G : H]<br />
many groupoids, each <strong>of</strong> which is equivalent to the trivial<br />
category, i H /• is equivalent to the discrete set G/H <strong>of</strong> left<br />
cosets (regarded as a category).<br />
Similarly the undercategory •\i H has objects {(g, •)|g ∈<br />
G}. There is a morphism from (g 1 , •) to (g 2 , •) if <strong>and</strong> only<br />
if there exists a (unique) h ∈ H such that hg 1 = g 2 or<br />
equivalently h = g 2 g1 −1 . The undercategory is equivalent<br />
to H\G, the set <strong>of</strong> right cosets. We have an isomorphism<br />
i H /• → •\i H given by<br />
(•, g) ↦→ (g −1 , •) <strong>and</strong> (•, g 1 ) g−1 2 g 1<br />
→ (•, g 2 ) ↦→ (g −1<br />
1 , •)g−1<br />
2 g 1<br />
→ (g2 −1 , •).
26 1 Functors <strong>and</strong> their Kan extensions<br />
When G = H, i G /• = Id G /• ∼ = •\Id G = •\i G is the Cayley<br />
graph <strong>of</strong> G.<br />
Based on our observations, by Proposition 1.2.4, if M ∈<br />
Ob(V ect H k<br />
) (commonly called a k-representation <strong>of</strong> H),<br />
lim M ∼ −→i H<br />
= lim<br />
/• −→G/H<br />
M ∼ = ⊕ g∈G/H gH ⊗ k M. One can see that<br />
G permutes these direct summ<strong>and</strong>s <strong>and</strong> the limit lim M −→iH /•<br />
is isomorphic to kG ⊗ kH M.<br />
3. Suppose u : D → C is a functor between two posets. Then<br />
for any x ∈ Ob C, u/x is isomorphic to the subposet <strong>of</strong><br />
D consisting <strong>of</strong> objects {a ∈ Ob D ∣ Hom C (u(a), x) ≠ ∅},<br />
while x\u is isomorphic to the subposet <strong>of</strong> D consisting <strong>of</strong><br />
objects {b ∈ Ob D ∣ Hom C (x, u(b)) ≠ ∅}.<br />
The following observation follows directly from definitions<br />
<strong>and</strong> will be useful to us. For any functor u : D → C one can<br />
define a (covariant) opposite functor u op : D op → C op such<br />
that u op (x) = x <strong>and</strong> u op (α op ) = u(α) op . Be aware that it is<br />
different from the dual functor given before Definition 1.1.9.<br />
Lemma 1.2.8. Suppose u : D → C is a functor. Consider<br />
its opposite functor u op : D op → C op . Then for any<br />
x ∈ Ob C = Ob C op we have (u/x) op ∼ = x\u op <strong>and</strong> (x\u) op ∼ =<br />
u op /x.<br />
1.2.3 Kan extensions<br />
In this section, we assume T to be a complete <strong>and</strong> cocomplete<br />
abelian category. The reader should bear Example 1.2.7 (1)<br />
in mind in order to see that Kan extensions generalize direct<br />
<strong>and</strong> inverse limits.<br />
Theorem 1.2.9. Let u : D → C be a functor between small<br />
categories. Then the restriction Res u : T C → T D admits a<br />
left adjoint LK u , called the left Kan extension along u, as
1.2 Restriction <strong>and</strong> Kan extensions 27<br />
well as a right adjoint RK u , called the right Kan extension<br />
along u.<br />
Pro<strong>of</strong>. We only sketch the constructions <strong>of</strong> the Kan extensions<br />
<strong>and</strong> leave details to be filled by the reader.<br />
Given M ∈ Ob(T D ) we define its left <strong>and</strong> right Kan extensions<br />
along u as<br />
LK u M = lim −→u/−<br />
Res P M<br />
<strong>and</strong> RK u M = lim ←−−\u<br />
Res P M,<br />
where P is the functor in Remark 1.2.6 (2). Here we only<br />
prove the statement for the left Kan extension because the<br />
pro<strong>of</strong> for the right Kan extension follows the same pattern.<br />
Step 1, we show LK u M is a functor from C to T . If<br />
γ ∈ Hom C (x, y), then we have a functor γ ∗ : u/x → u/y. Since<br />
Res γ∗ Res P M = Res P M as functors over u/x, by Lemma<br />
1.2.3, it determines a canonical morphism<br />
lim<br />
−→ γ ∗ : LK u M(x) = −→u/x<br />
lim Res P M → LK u M(y) = −→u/y<br />
lim Res P M.<br />
Hence LK u M = lim −→u/−<br />
Res P M ∈ Ob(T C ).<br />
Step 2, we state that LK u is a functor from T D to T C . For<br />
any natural transformation Ψ : M → M ′ between two objects<br />
<strong>of</strong> T D . We can use the universal property to build a canonical<br />
natural transformation LK u Ψ : LK τ M → LK u M ′ .<br />
Step 3, we construct an adjunct<br />
Ω : Hom T C(LK u M, N) → Hom T D(M, Res u N).<br />
For each M ∈ Ob(T D ) we define the counit<br />
by the defining map <strong>of</strong> a limit<br />
Σ M : M → Res u LK u M<br />
(Σ M ) a : M(a) = Res P M[(a, 1 u(a) )] → lim −→u/u(a)<br />
Res P M,
28 1 Functors <strong>and</strong> their Kan extensions<br />
for any a ∈ Ob D. For each N ∈ Ob(T C ) we put the counit<br />
Λ N : LK u Res u N → N<br />
such that, for every x ∈ Ob C, lim −→u/x<br />
Res P Res u N → N(x)<br />
comes from the universal property <strong>of</strong> direct limit. We need to<br />
prove that the following are identities<br />
Res u<br />
Res u Σ<br />
−→ Res u LK u Res u<br />
ΛRes u<br />
−→Resu <strong>and</strong> LK u<br />
ΣLK u<br />
−→ LKu Res u LK u<br />
LK u Λ<br />
−→LK u<br />
For the first we compute for any N ∈ Ob(T C ) <strong>and</strong> a ∈ Ob D<br />
the following composite<br />
(Res u N)(a) [Res u(Σ N )] a<br />
−→ (Resu LK u Res u N)(a) (Λ ResuN) a<br />
−→ (Res u N)(a).<br />
This composite is<br />
(Res u N)(a) = (Res P Res u N)[(a, 1 a )] → lim −→u/u(a)<br />
Res P Res u N → (Res u N<br />
which is really an identity by the universal property <strong>of</strong> lim −→u/u(a)<br />
Res P Re<br />
For the second we compute for any M ∈ Ob(T D ) <strong>and</strong> x ∈<br />
Ob C that<br />
(LK u M)(x) [LK uΛ M ] x<br />
−→ (LKu Res u LK u M)(x) (Λ LKuM) x<br />
−→ (LK u M)(x)<br />
gives the identity. But it is rewritten as<br />
lim Res −→u/x<br />
PM → lim −→u/x<br />
(Res P Res u LK u M) → lim −→u/x<br />
Res P M,<br />
which in turn equals −→u/x<br />
lim Res P applying to<br />
M → Res u LK u M → M.<br />
However the preceding morphisms compose to the identity<br />
because the following composite<br />
M(a) → lim −→u/u(a)<br />
Res P M → M(a)
1.2 Restriction <strong>and</strong> Kan extensions 29<br />
is an identity for every a ∈ Ob D, due to the fact that (a, 1 u(x) )<br />
is a terminal object <strong>and</strong> that Res P M(a, 1 u(a) ) = M(a). ⊓⊔<br />
Remark 1.2.10.1. One may choose T to be Ab, or R-Mod etc,<br />
for practical applications. When the index categories are<br />
finite, we can even use R-mod, the category <strong>of</strong> finitely generated<br />
R-modules.<br />
2. When u : D → C is a full embedding <strong>and</strong> M ∈ Ob(Ab D ),<br />
then LK u M ∈ Ob(Ab C ) restricted on u(D) is identified with<br />
M, which means LK u M(u(d)) = M(d) for any d ∈ Ob D.<br />
This is why we call such functors discovered by D. M. Kan,<br />
“the Kan extensions”.<br />
Example 1.2.11. In Example 1.2.7 (2) where the over- <strong>and</strong><br />
under-categories associated with i : H ↩→ G are computed,<br />
we can continue to verify that Res i is the usual restriction<br />
↓ G H , LK i is equivalent to the induction ↑ G H = kG ⊗ kH − <strong>and</strong><br />
RK i is equivalent to the coinduction ⇑ G H = Hom kH(kG, −).<br />
If G is finite, the two Kan extensions are well known to be<br />
equivalent.<br />
Corollary 1.2.12. Let u : D → C <strong>and</strong> v : E → D be two<br />
functors between small categories. Suppose T is a complete<br />
<strong>and</strong> cocomplete category, M ∈ Ob(T E ) <strong>and</strong> N ∈ Ob(T C ).<br />
Then Res v Res u = Res uv <strong>and</strong> consequently LK u LK v<br />
∼ = LKuv ,<br />
RK u RK v<br />
∼ = RKuv .<br />
Pro<strong>of</strong>. The equality between restrictions follows directly from<br />
definition. Then we have<br />
Hom T E(M, Res v Res u N) ∼ = Hom T D(LK v M, Res u N)<br />
∼ = HomT C(LK u LK v M, N).<br />
Hence LK u LK v<br />
∼ = LKuv . The isomorphism between right<br />
Kan extensions can be proved similarly.<br />
⊓⊔
Chapter 2<br />
Simplicial methods<br />
Abstract We begin with chain complexes <strong>and</strong> their homology,<br />
the fundamental concepts <strong>of</strong> homological algebra. Then<br />
we review simplicial constructions in algebra <strong>and</strong> topology<br />
as they provide concrete examples <strong>and</strong> motivating ideas. Out<br />
main interest lies in the nerve <strong>of</strong> a small category. This particular<br />
simplicial set allows us to define simplicial homology<br />
<strong>and</strong> cohomology <strong>of</strong> a category with coefficients in a commutative<br />
ring. The nerve <strong>of</strong> a small category has a geometric<br />
realization, called the classifying space. Hence we can also<br />
consider the singular homology <strong>and</strong> cohomology <strong>of</strong> a classifying<br />
space. We shall show these two theories agree. We will<br />
introduce various important categorical constructions. Meanwhile<br />
we develop techniques for comparing categoires, their<br />
nerves <strong>and</strong> classifying spaces. A major theorem is Quillen’s<br />
Theorem A. For future references <strong>and</strong> better underst<strong>and</strong>ing<br />
<strong>of</strong> simplicial methods, we also include a description <strong>of</strong> bisimplicial<br />
sets <strong>and</strong> several relevant results.<br />
2.1 Complexes <strong>and</strong> homology<br />
Here we recall basics about chain <strong>and</strong> cochain complexes as<br />
well as operations on them.
32 2 Simplicial methods<br />
2.1.1 Chain complexes, homology <strong>and</strong> chain homotopy<br />
Suppose A is an associative ring with identity <strong>and</strong> Z is the<br />
totally ordered set <strong>of</strong> integers. From Example 1.1.3 (3) we<br />
may deem Z as a category.<br />
Definition 2.1.1. A chain complex <strong>of</strong> A-modules is an object<br />
C ∈ Ob(A-Mod) Zop such that C(n → n + 2) = 0 for<br />
all n ∈ Z. In other words it consists <strong>of</strong> a collection <strong>of</strong> objects<br />
in A-Mod, {C n = C(n)} n∈Z , together with morphisms<br />
∂ n = C((n − 1) → n) : C n → C n−1 , called the differentials,<br />
such that ∂ n ◦ ∂ n+1 = 0.<br />
A cochain complex <strong>of</strong> objects in A-Mod is an object C ∈<br />
Ob(A-Mod) Z such that C(n → n + 2) = 0, for all n ∈ Z.<br />
Alternatively it consists <strong>of</strong> a collection <strong>of</strong> objects in A, {C n =<br />
C(n)} n∈Z , together with morphisms ∂ n = C(n → (n + 1)) :<br />
C n → C n+1 , called the differentials, such that ∂ n ◦ ∂ n−1 = 0.<br />
If {C n , ∂ n } n∈Z (respectively, {C n , ∂ n } n∈Z ) is a chain complex<br />
(respectively, cochain complex) <strong>of</strong> A-modules, <strong>and</strong> x ∈ C n<br />
(respectively x ∈ C n ), then we say x has degree n, written as<br />
deg x = n.<br />
We call a complex bounded below if C n (or C n ) vanishes<br />
when n is sufficiently small. A complex bounded above if C n<br />
(or C n ) vanishes when n is sufficiently large. A complex if<br />
bounded if it is both bounded below <strong>and</strong> bounded above (that<br />
is, there are only finitely many non-zero terms). A complex is<br />
called a stalk complex if it has exactly one non-zero term.<br />
Suppose C = {C n , ∂ n } is a chain complex. Then we usually<br />
write Z n (C) = ker ∂ n ⊂ C n . Any element x ∈ Z n (C) is called<br />
an n-cycle. We also write B n (C) = Im∂ n+1 ⊂ C n . Any element<br />
x ∈ B n (C) is called an n-boundary. Similarly suppose<br />
C = {C n , ∂ n } is a cochain complex. Then we usually write<br />
Z n (C) = ker ∂ n ⊂ C n . Any element x ∈ Z n (C) is called an n-
2.1 Complexes <strong>and</strong> homology 33<br />
cocycle. We also write B n (C) = Im∂ n−1 ⊂ C n . Any element<br />
x ∈ B n (C) is called an n-coboundary.<br />
Definition 2.1.2. The i-th homology <strong>of</strong> a chain complex<br />
{C n , ∂ n } n∈Z is<br />
Hi(C) = Z i (C)/B i (C).<br />
The i-th cohomology <strong>of</strong> a cochain complex {C n , ∂ n } n∈Z is<br />
H i (C) = Z i (C)/B i (C).<br />
Definition 2.1.3. Let C <strong>and</strong> D be two chain complexes (respctively,<br />
cochain complexes). A chain map (respectively,<br />
cochain map) φ : D → C is a natural transformation. In<br />
other words, it consists <strong>of</strong> maps φ n : D n → C n (respectively,<br />
φ n : D n → C n ) such that the following diagram commutes<br />
D n<br />
φ n <br />
∂ D n <br />
D n−1<br />
φ n−1<br />
respectively, D n ∂n D <br />
φ n <br />
C n ∂<br />
C<br />
n<br />
<br />
C n−1 C n ∂ n C<br />
D n+1<br />
φ n+1<br />
<br />
C n+1<br />
The chain map φ is a chain isomorphism if it is a natural<br />
equivalence.<br />
Clearly a chain map between chain complexes φ : D → C<br />
induces maps between homology groups φ ∗ : Hn(D) → Hn(C)<br />
<strong>and</strong> a cochain map between cochain complexes φ : D → C<br />
induces maps between cohomology groups φ ∗ : H n (D) →<br />
H n (C).<br />
For convenience, we shall call both chain maps <strong>and</strong> cochain<br />
maps just chain maps. Given a chain complex {C n , ∂ n } n∈Z <strong>of</strong><br />
objects in an abelian category A, we can obtain a cochain<br />
complex {C n , ∂ n } n∈Z , simply by asking C n = C −n <strong>and</strong> ∂ n =<br />
∂ −n . Thus in some sense, chain <strong>and</strong> cochain complexes are<br />
the same. In practice, we will regard a complex as a chain
34 2 Simplicial methods<br />
complex or a cochain complex depending on where it comes<br />
from.<br />
Definition 2.1.4. If φ, ψ : D → C are two chain maps, we<br />
say φ <strong>and</strong> ψ are chain homotopic, written as φ ≃ ψ, if there<br />
are maps h n : D n → C n+1 such that Φ n − Ψ n = h n−1 ∂ D n +<br />
∂ C n+1h n· · ·<br />
<br />
D n+1<br />
∂ D n+1<br />
D n<br />
∂ D n<br />
D n−1<br />
· · ·<br />
φ n+1 −ψ n+1<br />
h n<br />
φ n −ψ n<br />
h n−1<br />
φ n−1 −ψ n−1<br />
· · ·<br />
<br />
C n+1 ∂<br />
C<br />
n+1<br />
<br />
C n<br />
∂ C n<br />
<br />
C n−1<br />
<br />
· · ·<br />
We say two complexes D <strong>and</strong> C are chain homotopy equivalent,<br />
written as D ≃ C if there are chain maps φ : D → C<br />
<strong>and</strong> ψ : C → D such that φ ◦ ψ ≃ Id C <strong>and</strong> ψ ◦ φ ≃ Id D .<br />
We say a complex C is contractible, if it is chain homotopy<br />
equivalent to the zero complex. We say a complex C is acyclic<br />
if Hn(C) = 0 for all n. Acyclicity is strictly weaker than contractibility.<br />
Proposition 2.1.5. If φ, ψ : D → C are chain homotopic,<br />
<strong>and</strong> if F : A-Mod → B-Mod is an additive functor, then<br />
1. φ ∗ = ψ ∗ : Hn(D) → Hn(C). Thus a chain homotopy equivalence<br />
D ≃ C induces isomorphisms Hn(D) ∼ = Hn(C).<br />
2. F(φ) ≃ F(ψ) : F(D) → F(C), <strong>and</strong> F(φ) ∗ = F(ψ) ∗ :<br />
Hn(F(D)) → Hn(F(C)).<br />
Let C be a bounded below chain complex <strong>of</strong> A-modules.<br />
A projective resolution <strong>of</strong> C is a (bounded below) chain<br />
complex P <strong>of</strong> projective A-modules, along with a chain map<br />
P → C which induces isomorphisms on homology groups. Any<br />
two projective resolutions <strong>of</strong> C are chain homotopy equiva-
2.1 Complexes <strong>and</strong> homology 35<br />
lent. Similarly we can consider an injective resolution I <strong>of</strong> a<br />
bounded above complex <strong>of</strong> A-modules D.<br />
2.1.2 Double complexes <strong>and</strong> operations on chain complexes<br />
Definition 2.1.6. A double complex <strong>of</strong> A-modules is an object<br />
C ∈ Ob(A-Mod) Zop ×Z op such that C(p → (p + 2), 1 q ) =<br />
C(1 p , q → (q + 2)) = 0 <strong>and</strong><br />
C[(1 p+1 , q → (q + 1))(p → (p + 1), 1 q )] + C[(p → (p + 1), 1 q+1 )(1 p , q → (q + 1))] = 0<br />
for all p, g ∈ Z.<br />
Alternatively it consists <strong>of</strong> a collection <strong>of</strong> objects {C p,q =<br />
C(p, q)} p,q∈Z , together with maps<br />
∂ h = C((p − 1) → p, 1 q ) : C p,q → C p−1,q ; ∂ v = C(1 p , (q − 1) → q) : C p,q → C p,q−1<br />
such that ∂ h ◦ ∂ h = ∂ v ◦ ∂ v = ∂ v ◦ ∂ h + ∂ h ◦ ∂ v = 0.<br />
A map between two double complexes D → C is just a<br />
natural transformation. It is an isomorphism if it is a natural<br />
equivalence.<br />
Double complexes are also called bicomplexes in the literature.<br />
A double complex C = {C p,q } p,q∈Z is called bounded if<br />
it has only finitely many non-zero terms along each diagonal<br />
line p + q = n. An example will be the first quadrant double<br />
complex where C p,q = 0 unless both p <strong>and</strong> q are non-negative.<br />
Definition 2.1.7. The total complexes <strong>of</strong> a double complex<br />
C are given by<br />
Tot ⊕ (C) n = ⊕<br />
<strong>and</strong> Tot ∏ (C) n = ∏<br />
p+q=n<br />
C p,q<br />
with differentials ∂ = ∂ h + ∂ v .<br />
p+q=n<br />
C p,q<br />
These two constructions are usually different, but when C is<br />
bounded they are equal. Now we present a classical construction.
36 2 Simplicial methods<br />
Let A be an associative ring. Suppose D is a chain complex <strong>of</strong><br />
right A-modules <strong>and</strong> C is a chain complex <strong>of</strong> left R-modules.<br />
We define their product complex D ⊗ A C by<br />
(D ⊗ A C) n = ⊕<br />
D p ⊗ A C q .<br />
p+q=n<br />
It is a double complex with ∂ h p,q = ∂ D p ⊗1 <strong>and</strong> ∂ v p,q = (−1) p (1⊗<br />
∂ C q ), pictured as<br />
D p−1 ⊗ A C q<br />
∂ h p,q<br />
D p ⊗ A C q<br />
∂ v p−1,q <br />
D p−1 ⊗ A C q−1<br />
∂ h p,q−1<br />
∂ v p,q<br />
D p ⊗ A C q−1<br />
Hence the differential <strong>of</strong> the total complex Tot ⊕ (D ⊗ A C) is<br />
given by<br />
∂(x ⊗ y) = ∂ D p x ⊗ y + (−1) p x ⊗ ∂ C q y,<br />
for x ∈ D p <strong>and</strong> y ∈ C q .<br />
Suppose both E <strong>and</strong> C are chain complexes <strong>of</strong> left A-<br />
modules. Then we introduce another double complex Hom A (E, C)<br />
by (there are different conventions in the literature)<br />
Hom A (E, C) n = ∏<br />
Hom A (E −p , C q ),<br />
pictured as<br />
q+p=n<br />
Hom A (E −p+1 , C q )<br />
∂ h p,q<br />
Hom A (E −p , C q )<br />
∂ v p−1,q <br />
∂ v p,q<br />
Hom A (E −p+1 , C q−1 ) Hom A (E −p , C q−1 )<br />
∂ h p,q−1<br />
with ∂ h p,q(f) = f∂ E −p+1 <strong>and</strong> ∂ v p,q(f) = (−1) p+q ∂ C q f.
2.1 Complexes <strong>and</strong> homology 37<br />
The differential <strong>of</strong> the total complex Tot ∏ (Hom A (E, C)) is<br />
∂ n = ∂ h p,q + ∂ v p,q : Hom A (E, C) n → Hom A (E, C) n−1 given by<br />
for any f ∈ Hom A (E −p , C q ).<br />
∂ n f = f∂ E −p+1 + (−1) n ∂ C q f<br />
Theorem 2.1.8. Suppose R is a commutative ring. Let C,<br />
D <strong>and</strong> E be complexes <strong>of</strong> R-modules. We have an isomorphism<br />
<strong>of</strong> double complexes<br />
Hom R (C ⊗ R D, E) ∼ = Hom R (C, Hom R (D, E))<br />
The following theorem is well known <strong>and</strong> its pro<strong>of</strong> can be<br />
found in many places. Here we record it for future references.<br />
Theorem 2.1.9 (Künneth formula). Suppose A is a ring.<br />
Let D be a chain complex <strong>of</strong> right A-modules <strong>and</strong> C, E chain<br />
complexes <strong>of</strong> left A-modules. Then<br />
1. if D n <strong>and</strong> ∂(D n ) are flat for all n, there is a short exact<br />
sequence<br />
0 → ⊕ p+q=n Hp(D) ⊗ A Hq(C) → Hn(D ⊗ A C)<br />
→ ⊕ p+q=n−1 TorA 1 (Hp(D), Hq(C))<br />
2. if E n <strong>and</strong> ∂(E n ) are projective for all n, there is a short<br />
exact sequence<br />
0 → ∏ q−p=n+1 Ext1 A(Hp(E), Hq(C)) → Hn(Hom A (E, C))<br />
→<br />
∏<br />
q−p=n Hom A(Hp(E), Hq(C)) →<br />
Corollary 2.1.10 (Universal Coefficient Theorem). Let<br />
D be a chain complex <strong>of</strong> right A-modules over a ring A, E<br />
a chain complex <strong>of</strong> left A-modules <strong>and</strong> M a left A-module<br />
considered as a complex concentrated in degree zero. Then
38 2 Simplicial methods<br />
1. if D n <strong>and</strong> ∂(D n ) are flat for all n, there is a short exact<br />
sequence<br />
0 → Hp(D) ⊗ A M → Hn(D ⊗ A M) → Tor A 1 (Hn−1(D), M) → 0,<br />
2. if E n <strong>and</strong> ∂(E n ) are projective for all n, there is a short<br />
exact sequence<br />
0 → Ext 1 A(Hn−1(E), M) → H−n(Hom A (E, M)) → Hom A (Hn(E), M)<br />
In Corollary 2.1.10 (2), the middle term is <strong>of</strong>ten written as<br />
cohomology H n (Hom A (E, M)).<br />
The dual complex <strong>of</strong> C, denoted by C ∧ , is defined to be<br />
Hom A (C, A), where the second A is regarded as a complex<br />
concentrated in degree zero. This is a complex <strong>of</strong> right<br />
A-modules. Note that if P is a projective A-module then<br />
Hom A (P, A) is a projective right module. We can define an<br />
evaluation map<br />
ev : C ∧ ⊗ A C → A.<br />
It is non-zero only at degree zero. More explicitly it is given<br />
by f ⊗ x ↦→ f(x) for any base elements f ⊗ x ∈ (C ∧ ⊗ C) 0 .<br />
Here x ∈ C n <strong>and</strong> f ∈ C ∧ −n.<br />
Proposition 2.1.11. Suppose k is a field. Let D <strong>and</strong> C be<br />
two complexes <strong>of</strong> finite-dimensional k-vector spaces. If for<br />
every n, Hom k (D, C) n is finite-dimensional, then we have<br />
an isomorphism <strong>of</strong> complexes<br />
D ∧ ⊗ k C ∼ = Hom k (D, C).<br />
Pro<strong>of</strong>. Note that if M, N are two stalk complexes <strong>of</strong> k-vector<br />
spaces concentrated in degrees m <strong>and</strong> n respectively, then we<br />
have M ∧ ⊗ k N ∼ = Hom k (M, N) as stalk complexes concentrated<br />
in degree n − m. Consequently we have an isomorphism<br />
for every integer n, (D ∧ ⊗ k C) n<br />
∼ = Homk (D, C) n since<br />
Hom k (D, C) n is finite-dimensional <strong>and</strong> then Hom k (D, C) n =
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 39<br />
⊕ q+p=n Hom k (D −p , C q ). Hence in order to finish the pro<strong>of</strong>, we<br />
only need to verify that the two differentials are identified under<br />
the vector space isomorphisms.<br />
⊓⊔<br />
In the next section, we will see many combinatorially constructed<br />
complexes <strong>and</strong> double complexes.<br />
2.2 Nerves, classifying spaces <strong>and</strong> cohomology<br />
By Dold-Kan Correspondence, the category <strong>of</strong> non-negatively<br />
graded complexes <strong>of</strong> A-modules is equivalent to the category<br />
<strong>of</strong> simplicial A-modules. It implies also that the category<br />
<strong>of</strong> first quadrant double complexes <strong>of</strong> A-modules is<br />
equivalent to the category <strong>of</strong> bisimplicial A-modules. Since in<br />
these notes, we are mainly interested in various non-negatively<br />
graded complexes such as projective resolutions <strong>of</strong> modules,<br />
<strong>and</strong> moreover these complexes come from corresponding simplicial<br />
sets, it is useful to introduce simplicial sets <strong>and</strong> modules.<br />
2.2.1 Simplicial sets <strong>and</strong> nerves <strong>of</strong> small categories<br />
We recall the fundamental idea in algebraic topology <strong>of</strong> singular<br />
(co)homology theory. From here we shall see how algebraic<br />
<strong>and</strong> combinatorial methods enter the study <strong>of</strong> spaces. Then<br />
we will apply the same methods, called simplicial methods, to<br />
investigate small categories (instead <strong>of</strong> spaces). Our first definition<br />
<strong>of</strong> category (co)homology is given soon after the basic<br />
definitions are recorded.<br />
Let us start with the topological simplicies. For every integer<br />
n ≥ 0, a st<strong>and</strong>ard n-simplex is defined as a subspace <strong>of</strong> the<br />
(n + 1)-dimensional real vector space R n+1
40 2 Simplicial methods<br />
△ n = {(x 0 , x 1 , · · · , x n ) ∈ R n+1 ∣ ∣ xi ≥ 0 <strong>and</strong><br />
n∑<br />
x i = 1}.<br />
An i-face <strong>of</strong> △ n for 0 ≤ i ≤ n is a subspace such that there are<br />
exactly i+1 chosen entries that are not constantly zero. Each<br />
i-face is isomorphic to the st<strong>and</strong>ard i-simplex. For example<br />
in the following picture, △ 2 has exactly one 2-face (e.g. △ 2 ),<br />
three 1-faces (isomorphic to △ 1 ) <strong>and</strong> three 0-faces (isomorphic<br />
to △ 0 which is a point).<br />
i=0<br />
X 3<br />
(0,0,1)<br />
△ 2 ⊂ R 3<br />
X 2<br />
(1,0,0) (0,1,0)<br />
X 1<br />
Fix an integer n, there are many natural maps among all<br />
the faces <strong>of</strong> △ n . Since each face is isomorphic to a st<strong>and</strong>ard<br />
simplex, these maps can be described as maps among all st<strong>and</strong>ard<br />
simplicies. The most distinguished are the face maps<br />
d i : △ n−1 → △ n (inserting a zero) <strong>and</strong> the degeneracy maps<br />
s i : △ n+1 → △ n (adding up two adjacent entries), given explicitly<br />
by<br />
d i (x 0 , x 1 , · · · , x n−1 ) = (x 0 , · · · , x i , 0, x i+1 , · · · , x n−1 ), 0 ≤ i ≤ n;<br />
s i (x 0 , x 1 , · · · , x n+1 ) = (x 0 , · · · , x i + x i+1 , · · · , x n+1 ), 0 ≤ i ≤ n.<br />
Given a topological space X, in order to define <strong>and</strong> compute<br />
its singular homology H∗(X, Z) we first form the sets <strong>of</strong><br />
continuous maps S(X) n = Hom T op (△ n , X) <strong>and</strong> then produce<br />
free abelian groups ZS(X) n on top <strong>of</strong> them. Each face map
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 41<br />
d i : △ n−1 → △ n induces a map d i : S(X) n → S(X) n−1 , <strong>and</strong><br />
moreover ∂ n = ∑ n<br />
i=0 (−1)i d i : ZS(X) n → ZS(X) n−1 satisfies<br />
∂ n+1 ∂ n = 0. In this way we obtain a non-negatively graded<br />
chain complex<br />
{ZS(X) ∗ ; ∂ ∗ } ∗≥0 ,<br />
in which every differential ∂ n is determined by an alternating<br />
sum <strong>of</strong> face maps. The homology <strong>of</strong> this complex is defined<br />
to be the singular homology <strong>of</strong> X, written as H∗(X, Z). The<br />
degeneracy maps are important to us as well, <strong>and</strong> we shall<br />
discuss their roles shortly.<br />
In order to allow further applications <strong>of</strong> such a fundamental<br />
construction, we propose some alternative descriptions <strong>of</strong> the<br />
st<strong>and</strong>ard simplices based on the following two observations.<br />
Firstly, to specify a st<strong>and</strong>ard complex, it suffices to provide its<br />
vertices because △ n is the convex hull <strong>of</strong> the set <strong>of</strong> its vertices.<br />
Moreover since, in R n+1 , one can give the lexicographic order<br />
to its elements by asking (x 0 , · · · , x n ) < (y 0 , · · · , y n ) if there<br />
exists an integer 0 ≤ k ≤ n such that x i = y i for i < k <strong>and</strong><br />
x k < y k , particularly there is an order on the set <strong>of</strong> vertices <strong>of</strong><br />
△ n . This totally ordered set <strong>of</strong> vertices uniquely determines<br />
△ n . For example in R 3 we have (0, 0, 1) < (0, 1, 0) < (1, 0, 0),<br />
<strong>and</strong> we can certainly identify this totally ordered set with △ 2 .<br />
Thus giving △ n is equivalent to giving the totally ordered set<br />
(0, · · · , 0, 1) < (0, · · · , 1, 0) < · · · < (1, 0, · · · , 0) <strong>of</strong> points in<br />
R n+1 . Moreover there exists a natural one-to-one correspondence<br />
between the set <strong>of</strong> all totally ordered subsets <strong>of</strong> vertices<br />
<strong>and</strong> the set <strong>of</strong> faces <strong>of</strong> △ n .<br />
Secondly, because the face <strong>and</strong> degeneracy maps on △ n<br />
are completely determined by their values on the vertices,<br />
we can translate the face <strong>and</strong> degeneracy maps accordingly.<br />
For example, we can write out values <strong>of</strong> d 0 : △ 2 → △ 3 <strong>and</strong><br />
s 0 : △ 2 → △ 1 on the vertices
42 2 Simplicial methods<br />
d 0 (0, 0, 1) = (0, 0, 0, 1), d 0 (0, 1, 0) = (0, 0, 1, 0), d 0 (1, 0, 0) = (0, 1, 0, 0<br />
s 0 (0, 0, 1) = (0, 1), s 0 (0, 1, 0) = (1, 0), s 0 (1, 0, 0) = (1, 0).<br />
In general it is easy to see that d i <strong>and</strong> s i always send vertices<br />
to vertices. Furthermore they (weakly) preserve the order, in<br />
the sense that if a ≤ b then d i (a) ≤ d i (b) <strong>and</strong> s i (a) ≤ s i (b).<br />
We shall illustrate it by an example. Let us examine △ 1 =<br />
{(0, 1) < (1, 0)} <strong>and</strong> △ 2 = {(0, 0, 1) < (0, 1, 0) < (1, 0, 0)}.<br />
The face maps △ 1 → △ 2 , adapted to our new combinatorial<br />
expression, are given by embeddings<br />
d 0 : (0, 1) < (1, 0) ↦→ (0, 0, 1) < (0, 1, 0)<br />
d 1 : (0, 1) < (1, 0) ↦→ (0, 0, 1) < (1, 0, 0)<br />
d 2 : (0, 1) < (1, 0) ↦→ (0, 1, 0) < (1, 0, 0).<br />
The degeneracy maps △ 2 → △ 1 are the same as projecting<br />
the vertex (0, 1, 0) to one <strong>of</strong> the other two, upon identifying<br />
△ 1 with the line segment (0, 0, 1) − (1, 0, 0)<br />
s 0 : (0, 0, 1) < (0, 1, 0) < (1, 0, 0) ↦→ (0, 1) < (1, 0) = (1, 0) (0, 1) <<br />
s 1 : (0, 0, 1) < (0, 1, 0) < (1, 0, 0) ↦→ (0, 1) = (0, 1) < (1, 0) (0, 1) <<br />
In summary, the face <strong>and</strong> degeneracy maps are indeed weakly<br />
monotonic maps among those totally ordered sets corresponding<br />
to the st<strong>and</strong>ard simplices. As a matter <strong>of</strong> fact, all weakly<br />
monotonic functions among those totally ordered sets, coming<br />
from st<strong>and</strong>ard simplices, are composites <strong>of</strong> these face <strong>and</strong><br />
degeneracy maps.<br />
For various good reasons we continue to work on the combinatorial<br />
characterizations <strong>of</strong> st<strong>and</strong>ard simplices. Previously<br />
we have identified △ n with the poset (0, · · · , 0, 1) < (0, · · · , 1, 0) <<br />
· · · < (1, 0, · · · , 0), <strong>and</strong> have rewritten the face <strong>and</strong> degeneracy<br />
maps. Now we abstract the totally ordered set as<br />
0 < 1 < · · · < n. Consequently the face <strong>and</strong> degeneracy
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 43<br />
maps can be reformulated <strong>and</strong> will be denoted by d i <strong>and</strong> s i ,<br />
respectively. This reformulation allows us to forget the geometric<br />
definition <strong>of</strong> △ n , <strong>and</strong> thus all relevant constructions<br />
can be made in an entirely combinatorial fashion. For future<br />
applications we write out the ith face map d i (corresponding<br />
to d i ) for 0 ≤ i ≤ n<br />
{0 < 1 < · · · < n − 1} → {0 < 1 < · · · < n}<br />
{<br />
d i j, if j < i<br />
(j) =<br />
j + 1 , if j ≥ i<br />
<strong>and</strong> the ith degeneracy map s i (corresponding to s i ) for 0 ≤<br />
i ≤ n<br />
{0 < 1 < · · · < n + 1} → {0 < 1 < · · · < n}<br />
{<br />
s i j, if j ≤ i<br />
(j) =<br />
j − 1 , if j > i<br />
One can verify that these maps satisfy the relations (the<br />
cosimplicial identities)<br />
d j d i = d i d j−1 i < j<br />
s j d i = d i s j−1 i < j<br />
s j d i = 1 i = j or j + 1<br />
s j d i = d i−1 s j i > j + 1<br />
s j s i = s i s j+1 i ≤ j .<br />
For each n ∈ {0} ∪ N, we define an ordered set (which happens<br />
to be a finite category) n = 0 < 1 < 2 < · · · < n<br />
(or rather 0 → 1 → 2 → · · · → n). We denote by △ the<br />
category consisting <strong>of</strong> all such n, in which morphisms are<br />
weakly monotonic functions (equivalently, functors) among<br />
these ordered sets. Recall the motivating example at the<br />
very beginning <strong>of</strong> this section. Given a space X, in order
44 2 Simplicial methods<br />
to make use <strong>of</strong> △, we associated to each n, the combinatorial<br />
model <strong>of</strong> △ n , a set Hom T op (△ n , X), as well as an abelian<br />
group ZHom T op (△ n , X). If we put all pieces together, we realize<br />
that we actually constructed a contravariant functor<br />
Hom T op (−, X) : △ → Set, as well as a ZHom T op (−, X) :<br />
△ → Ab. Since these functors give us the singular homology<br />
<strong>of</strong> X, contravariant functors from △ to other categories may<br />
lead to interesting constructions too.<br />
Definition 2.2.1. A simplicial object in a category T is a<br />
contravariant functor X : △ → T . Two simplicial objects<br />
X, Y in T are said to be isomorphic, written as X ∼ = Y , if<br />
as functors they are naturally equivalent.<br />
A simplicial set is a simplicial object in Set.<br />
Equivalently, since monotonic maps are compositions <strong>of</strong> d i<br />
<strong>and</strong> s i , a simplicial object X in T consists <strong>of</strong> a set <strong>of</strong> objects<br />
X n := X([n]) ∈ Ob T (an element <strong>of</strong> X n is called an n-<br />
simplex) <strong>and</strong> morphisms among these simplicies which are<br />
composites <strong>of</strong> two special kinds <strong>of</strong> morphisms: d i := X(d i ) :<br />
X n → X n−1 , 0 ≤ i ≤ n, <strong>and</strong> s i := X(s i ) : X n → X n+1 ,<br />
0 ≤ i ≤ n, satisfying (the simplicial identities)<br />
d i d j = d j−1 d i i < j<br />
d i s j = s j−1 d i i < j<br />
d i s j = 1 i = j or j + 1<br />
d i s j = s j d i−1 i > j + 1<br />
s i s j = s j+1 s i i ≤ j .<br />
In order to define a simplicial object X, it suffices to specify<br />
{X n } n≥0 ⊂ Ob T , together with some maps d i <strong>and</strong> s i satisfying<br />
the above relations.<br />
We shall be interested in the cases where T = Set or R-Mod<br />
for some commutative ring R with identity. By Proposition
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 45<br />
1.1.27, the two categories SimpSet <strong>and</strong> Simp(R-Mod) are<br />
both complete <strong>and</strong> cocomplete.<br />
Example 2.2.2.1. Let C be a small category. We define the<br />
nerve <strong>of</strong> C to be a simplicial set NC = Hom Cat (−, C) such<br />
that NC n = C n (indeed Ob(C n ) since we only need the underlying<br />
set). Alternatively NC n can be identified with the<br />
set <strong>of</strong> n-chains <strong>of</strong> morphisms in C if n > 0 <strong>and</strong> Ob C if n = 0.<br />
When n > 0, the ith face map d i : NC n+1 → NC n is<br />
α<br />
d i (x 0 → · · · → x<br />
i α i+1<br />
i−1→xi → xi+1 → · · · → x n+1 )<br />
α<br />
= x 0 → · · · → x<br />
i α i+1<br />
i−1→ ̂xi → xi+1 → · · · → x n+1 ,<br />
where ∧ means removing an object. For instance when n = 1<br />
α<br />
the three face maps NC 2 → NC 1 are d 0 (x<br />
1 α 2<br />
0→x1→x2 ) =<br />
α<br />
x<br />
2<br />
α<br />
1→x2 , d 1 (x<br />
1 α 2<br />
α<br />
0→x1→x2 ) = x<br />
2 α 1<br />
α<br />
0 → x2 <strong>and</strong> d 0 (x<br />
1 α 2<br />
0→x1→x2 ) =<br />
α<br />
x<br />
1<br />
0→x1 . The ith degeneracy map s i : NC n → NC n+1 is given<br />
by<br />
s i (x 0 → · · · → x i−1 → x i → x i+1 → · · · → x n )<br />
= x 0 → · · · → x i−1 → x i<br />
1 xi<br />
→xi → x i+1 → · · · → x n .<br />
One can verify that these maps satisfy the simplicial identities.<br />
2. We consider a special situation <strong>of</strong> the first example. It<br />
shall fill the gap which may have occurred during transition<br />
from the geometric definition <strong>of</strong> st<strong>and</strong>ard n-simplicies to<br />
the abstract category-theoretic reformulation. We fix a nonnegative<br />
integer m. The totally ordered set m is indeed a<br />
category 0 → 1 → · · · → m. Thus we can consider its nerve<br />
Nm. By the preceding example we have Nm n = Ob(m n ).<br />
In other words the combinatorial m-simplex m corresponds<br />
to the functor Hom △ (−, m) : △ → Set. This correspondence<br />
actually extends to a (covariant) functor, given by
46 2 Simplicial methods<br />
ι(m) = Hom △ (−, m),<br />
By Yondeda Lemma we have<br />
ι : △ → SimpSet = Set △ .<br />
Hom SimpSet (Hom △ (−, m), Hom △ (−, n)) ∼ = Hom △ (m, n).<br />
It means ι is fully faithful.<br />
In the next section we shall talk about the geometric realization<br />
<strong>of</strong> simplicial sets. Then we will see that the simplicial<br />
set Hom △ (−, m) precisely gives rise to △ m .<br />
3. Let A be an associative ring <strong>and</strong> A-Mod the category <strong>of</strong> all<br />
A-modules. A simplicial object in A-Mod is called a simplicial<br />
A-module. From Set to A-Mod, there exists a natural<br />
covariant functor, given by constructing free A-modules.<br />
Suppose X : △ → Set is a simplicial set. Then we naturally<br />
obtain a simplicial A-module<br />
A[X] : △ → Set → A-Mod.<br />
Given a simplicial A-module Y , one can construct a (nonnegatively<br />
graded) chain complex <strong>of</strong> A-modules by defining<br />
C n (Y ) = Y n <strong>and</strong> ∂ n = ∑ n<br />
i=0 (−1)i d i : C n (Y ) → C n−1 (Y ). If<br />
Y comes from a simplicial set X, that is, Y = A[X], then<br />
we write the chain complex as C ∗ (X, A) instead <strong>of</strong> C ∗ (Y )<br />
or C ∗ (A[X]) for various good reasons.<br />
The following characteristic statement is enlightening. Suppose<br />
X is a simplicial set. Then we define the corresponding<br />
simplex category to be ι/X, where ι : △ → SimpSet is the<br />
functor in Example 2.2.2 (2).<br />
Theorem 2.2.3. Let X be a simplicial set. Then we have<br />
an isomorphism<br />
X ∼ = lim −→ι/X<br />
P,
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 47<br />
where P : ι/X → SimpSet is given by P(n, Φ) = Hom △ (−, n).<br />
Pro<strong>of</strong>. From the contravariant version <strong>of</strong> Theorem 1.1.28 we<br />
know for the functor X : △ op → Set we can define a category<br />
D <strong>and</strong> a functor ˜X : D → Set<br />
△ op = SimpSet such that<br />
X ∼ = lim −→D ˜X.<br />
We show the category D is exactly ι/X <strong>and</strong> subsequently ˜X<br />
can be identified with P.<br />
The objects in the overcategory ι/X are <strong>of</strong> the form (n, Φ),<br />
where Φ : ι(n) = Hom △ (−, n) → X is a simplicial map. By<br />
the Yoneda Lemma Hom SimpSet (Hom △ (−, n), X) ∼ = X(n) =<br />
X n . Thus Φ can be regarded as an element <strong>of</strong> the set X(n).<br />
This provides a bijection between Ob(ι/X) <strong>and</strong> Ob D. From<br />
here one can continue to finish the pro<strong>of</strong>. We leave the details<br />
for the reader.<br />
⊓⊔<br />
Definition 2.2.4. Let C be a small category <strong>and</strong> NC its<br />
nerve. Suppose R is a commutative ring with identity. Then<br />
the simplicial R-module R[NC] gives rise to a complex <strong>of</strong> R-<br />
modules, written as C ∗ (C, R). The n-th homology <strong>of</strong> C ∗ (C, R),<br />
denoted by Hn(C, R), is called the n-th homology <strong>of</strong> C with<br />
coefficients in R. The n-th cohomology <strong>of</strong> the cochain complex<br />
C ∗ (C, R) := C ∗ (C, R) ∧ = Hom R (C ∗ (C, R), R),<br />
denoted by H n (C, R), is called the n-th cohomology <strong>of</strong> C with<br />
coefficients in R.<br />
By direct calculation, one can see that H0(C, R) ∼ = H 0 (C, R)<br />
is a free R-module with rank equal to the number <strong>of</strong> connected<br />
components <strong>of</strong> C. For any chain complex C ∗ (C, R), define as<br />
above, we can insert the base ring R at degree -1 <strong>and</strong> obtain<br />
the so-called reduced chain complex ˜C∗ (C, R) = C ∗ (C, R) →
48 2 Simplicial methods<br />
R → 0. Then the homology <strong>of</strong> the reduced chain complex is<br />
called the reduced homology <strong>of</strong> C, written as ˜H∗ (C, R). We<br />
see ˜H−1 (C, R) = 0 <strong>and</strong> ˜H0 (C, R) is a free R-module with rank<br />
equal to rk RH0(C, R) − 1. A small category C is called R-<br />
acyclic if ˜H∗ (C, R) vanishes.<br />
Now we address the issue <strong>of</strong> degeneracy maps. Suppose X is<br />
a simplicial object in an abelian category T . Then we have a<br />
complex <strong>of</strong> objects in T , denoted by X ∗ . It has a subcomplex<br />
X ′ ∗ such that X ′ n = ∑ i s i(X n−1 ) if n > 0 <strong>and</strong> X ′ 0 = 0. Then<br />
we continue to define a quotient complex X † ∗ = X ∗ /X ′ ∗. We<br />
can also define the normalized complex <strong>of</strong> X, N ∗ (X), by<br />
N n (X) =<br />
n−1<br />
⋂<br />
i=0<br />
Ker(d i )<br />
<strong>and</strong> ∂ n = (−1) n d n . This is a subcomplex <strong>of</strong> X ∗ .<br />
Theorem 2.2.5.1. We have X ∗ = N ∗ (X)⊕X ∗ ′ <strong>and</strong> N ∗ (X) ∼ =<br />
X ∗. † Furthermore the quotient map X ∗ ↠ X ∗<br />
† induces a<br />
chain homotopy equivalence <strong>and</strong> X ∗ ′ is contractible.<br />
2. (Dold-Kan Correspondence) Suppose T is an abelian<br />
category <strong>and</strong> Ch ≥0 (T ) is the category <strong>of</strong> non-negatively<br />
graded chain complexes in T . Then the normalized chain<br />
complex functor functor<br />
is a category equivalence.<br />
N : SimpT → Ch ≥0 (T ),<br />
Pro<strong>of</strong>. For the first part, see [53, Chapter VIII, Section] <strong>and</strong><br />
[84, Section 8.3], <strong>and</strong> for the second part, the reader is referred<br />
to [84, Section 8.4].<br />
⊓⊔<br />
Because <strong>of</strong> the above theorem, we also call X † ∗ the normalized<br />
complex <strong>of</strong> X ∗ .
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 49<br />
When T = A-Mod, <strong>and</strong> X is a simplicial A-module, then<br />
X ′ n, for any n > 0, consists <strong>of</strong> degenerate elements <strong>of</strong> the form<br />
s i (x ′ ) for some x ′ ∈ X n−1 . When X = R[NC], a simplicial R-<br />
module defined over a small category C <strong>and</strong> a commutative<br />
ring R, the degenerate elements in R[NC] n are all the linear<br />
combinations <strong>of</strong> the degenerate elements in NC n which are<br />
exactly those n-chains containing an identity morphism. In<br />
practice, when we compute (co-)homology <strong>of</strong> a category, we<br />
only need to use the normalized complex C † ∗(C, R).<br />
Example 2.2.6.1. Let C be the following category with two<br />
objects, two identity <strong>and</strong> non-identity morphisms<br />
x<br />
α<br />
β<br />
To compute H∗(C, Z), we only have to write down the normalized<br />
chain complex 0 → C † 1 (C, Z) → C† 0 (C, Z) → 0<br />
y<br />
0 → Z{α, β} → Z{x, y} → 0.<br />
(By comparison, C ∗ (C, Z) is infinite.) The non-trivial differential<br />
is given by α ↦→ y − x <strong>and</strong> β ↦→ y − x. Then<br />
the only non-trivial homology groups are H0(C, Z) ∼ = Z <strong>and</strong><br />
H1(C, Z) ∼ = Z. The calculation <strong>of</strong> cohomology groups is left<br />
to the reader.<br />
2. The second category D is slightly different from the first.<br />
One can easily verify that the normalized complex C † ∗(D, Z)<br />
is infinite, <strong>of</strong> dimension two at each degree. By direct computation<br />
H0(D, Z) ∼ = Z is the only non-trivial homology<br />
group.<br />
α<br />
<br />
x y<br />
α −1
50 2 Simplicial methods<br />
However there is an easy way to see it, if one notices that<br />
D ≃ • <strong>and</strong> knows that a category equivalence induces isomorphism<br />
on homology (see Proposition 2.2.19).<br />
Definition 2.2.7. Let D <strong>and</strong> C be two small categories.<br />
Then the join <strong>of</strong> D with C, denoted by D ∗ C, is a category<br />
whose objects are Ob D ∪ Ob C, <strong>and</strong> whose morphisms<br />
are Mor D ∪ Mor C, plus exactly one extra morphism γ a,x ∈<br />
Hom D∗C (a, x) introduced for every pair <strong>of</strong> objects a ∈ Ob D<br />
<strong>and</strong> x ∈ Ob C. The composition laws in D ∗ C are determined<br />
by the composition laws in C <strong>and</strong> D, plus the equalities<br />
αγ a,x = γ a,y , γ a,x β = γ b,x for any α ∈ Hom C (x, y) <strong>and</strong><br />
β ∈ Hom D (b, a).<br />
By definition D ∗ C <strong>and</strong> C ∗ D are two different categories.<br />
One can easily construct N(D ∗ C) from ND <strong>and</strong> NC.<br />
Proposition 2.2.8. Suppose both D <strong>and</strong> C are connected<br />
small categories. If R = k is a field, then<br />
˜Hn(D ∗ C, k) ∼ =<br />
⊕ ˜Hi(D, k) ⊗ ˜Hj (C, k)<br />
i+j=n−1<br />
if n ≥ 0. Particularly H∗(D ∗ C, k) ∼ = H∗(C ∗ D, k).<br />
Pro<strong>of</strong>. For n ≥ 0, C n (D ∗ C, k) = k[N(D ∗ C)] n has a basis<br />
consisting <strong>of</strong> the following elements<br />
γx i ,y i+1<br />
x 0 → · · · → x i −→ yi+1 → · · · → y n<br />
where x 1 , · · · , x i ∈ Ob D <strong>and</strong> y i+1 , · · · , y n ∈ Ob C, for 0 ≤<br />
i ≤ n. Here γ xi ,y i+1<br />
is the unique morphism in Hom D∗C (x i , y i+1 ).<br />
Its differential is
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 51<br />
∑ i−1<br />
j=0 (−1)j γx i ,y i+1<br />
x 0 → · · · → ˆx j → · · · → x i −→ yi+1 → · · · → y n<br />
+ (−1) i γx i−1 ,y i+1<br />
x 0 → · · · → x i−1 −→ yi+1 → · · · → y n<br />
+ (−1) i+1 γx i ,y i+2<br />
x 0 → · · · → x i −→ yi+2 → · · · → y n<br />
+ ∑ n<br />
l=i+2 (−1)l γx i ,y i+1<br />
x 0 → · · · → x i −→ yi+1 → · · · → ŷ l → · · · → y n .<br />
Suppose ˜C∗ (D ∗ C, k), ˜C∗ (D, k) <strong>and</strong> ˜C∗ (C, k) are the reduced<br />
chain complexes with k inserted in degree -1. Then we can<br />
define a degree -1 chain map<br />
˜C ∗ (D ∗ C, k) → ˜C∗ (D, k) ⊗ ˜C∗ (C, k)<br />
such that, at degree −1 it is identity <strong>and</strong> at degree n ≥ 0,<br />
˜C n (D ∗ C, k) → [˜C∗ (D, k) ⊗ ˜C∗ (C, k)] n−1 is given by<br />
γx i ,y i+1<br />
x 0 → · · · → x i −→ yi+1 → · · · → y n ↦→ [x 0 → · · · → x i ]⊗[y i+1 → · · ·<br />
along with<br />
<strong>and</strong><br />
x 0 → · · · → x n ↦→ [x 0 → · · · → x n ] ⊗ 1,<br />
y 0 → · · · → y n ↦→ 1 ⊗ [x 0 → · · · → x n ].<br />
Here 1 ∈ k = ˜C−1 (D, k) = ˜C−1 (C, k). This chain map is an<br />
isomorphism <strong>and</strong> thus by Künneth formula we get<br />
˜Hn(D ∗ C, k) ∼ = ⊕ i+j=n−1 Hi(˜C∗ (D, k)) ⊗ Hj(˜C∗ (C, k))<br />
∼ =<br />
⊕i+j=n−1 ˜Hi(D, k) ⊗ ˜Hj (C, k).<br />
Example 2.2.6 (2) tells us that it is important to know how to<br />
compare various categories or more generally simplicial sets.<br />
Definition 2.2.9. A natural transformation Φ : X → Y between<br />
simplicial objects in T is called a simplicial map. The<br />
simplicial objects in T form a category SimpT .<br />
⊓⊔
52 2 Simplicial methods<br />
By definition a simplicial map Φ consists <strong>of</strong> a sequence <strong>of</strong><br />
morphisms in T , Φ n : X n → Y n , commuting with the relevant<br />
face <strong>and</strong> degeneracy maps.<br />
Lemma 2.2.10. Let R be a commutative ring with identity.<br />
A simplicial map Φ : X → Y between simplicial sets induces<br />
a chain map CΦ : C ∗ (X, R) → C ∗ (Y, R).<br />
Pro<strong>of</strong>. Because every Φ n : X n → Y n commutes with d i , it commutes<br />
with the differentials <strong>of</strong> these chain complexes which<br />
are alternating sums <strong>of</strong> d i ’s.<br />
⊓⊔<br />
Lemma 2.2.11. Suppose D <strong>and</strong> C are two small categories<br />
<strong>and</strong> Φ : ND → NC is a simplicial map. Then Φ determines<br />
a functor u : D → C which in turn gives rise to Φ as Nu.<br />
Pro<strong>of</strong>. The map Φ 0 : ND 0 → NC 0 gives an assignment<br />
Ob D → Ob C while Φ 1 : ND 1 → NC 1 gives an assignment<br />
Mor D → Mor C. We show that these define a functor<br />
u : D → C such that u(d) = Φ 0 (d) if d ∈ Ob D <strong>and</strong><br />
u(f) = Φ 1 (f) if f ∈ Mor D.<br />
First <strong>of</strong> all, Φ 0 <strong>and</strong> Φ 1 are compatible in the sense that if<br />
c f →d ∈ ND 1 then u(f) = Φ 1 (f) is a morphism from u(c) =<br />
Φ 0 (c) to u(d) = Φ 0 (d). This follows from the commutative<br />
diagram by choosing i = 0 or 1<br />
Φ<br />
ND<br />
1 <br />
1 NC 1<br />
d i d i<br />
<br />
<br />
<br />
ND 0 Φ0<br />
NC 0<br />
Second <strong>of</strong> all, we have to demonstrate u(fg) = u(f)u(g) for<br />
any two composable morphisms in Mor D. This time we just<br />
use a similar commutative diagram involving d 1 , Φ 1 <strong>and</strong> Φ 2 .<br />
At last we can show u(1 d ) = 1 u(d) by invoking a degeneracy<br />
map
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 53<br />
ND 1<br />
Φ 1 <br />
NC 1<br />
s 0<br />
<br />
s 0<br />
ND<br />
0 Φ0<br />
NC 0<br />
Since Φ is completely determined by Φ 0 <strong>and</strong> Φ 1 , one can prove<br />
that Nu is exactly Φ.<br />
⊓⊔<br />
Example 2.2.12. Suppose u : D → C is a functor between<br />
small categories <strong>and</strong> R is a commutative ring with identity.<br />
Then it induces a simplicial map Nu : ND → NC. Furthermore<br />
it induces a chain map Cu : C ∗ (D, R) → C ∗ (C, R).<br />
Corollary 2.2.13. A functor u : D → C induces a map u ∗ :<br />
H∗(D, R) → H∗(C, R) as well as u ∗ : H ∗ (C, R) → H ∗ (D, R).<br />
Pro<strong>of</strong>. The map u ∗ is induced by Cu while u ∗ is induced by<br />
Hom R (Cu, R).<br />
⊓⊔<br />
We shall illustrate the previous comparison results by an<br />
example. Note that if there are two functors v : E → D<br />
<strong>and</strong> u : D → C, we can easily verify that (uv) ∗ = u ∗ v ∗ <strong>and</strong><br />
(uv) ∗ = v ∗ u ∗ .<br />
Example 2.2.14. Let us consider the following category C with<br />
four non-identity morphisms<br />
µ <br />
z<br />
Its opposite category is pictured as<br />
x<br />
α<br />
β<br />
γ<br />
y
54 2 Simplicial methods<br />
y op α op <br />
β op <br />
γ x op<br />
op <br />
z op µ op<br />
In order not to make confusion in the opposite category we<br />
write x op for x etc. By direct calculation we can find that<br />
H0(C, Z) ∼ = H0(C op , Z) ∼ = Z <strong>and</strong> H1(C, Z) ∼ = H1(C op , Z) ∼ =<br />
Z ⊕ Z. These are the only non-trivial homology groups.<br />
We can list all (covariant) functors from C → C op , since<br />
there are not many. Suppose u : C → C op is a functor. We<br />
write u(C) to be the image <strong>of</strong> C, a subcategory <strong>of</strong> C op .<br />
1. If u(x) = x op , then both u(y) <strong>and</strong> u(z) have to be x op . In this<br />
case u(C) is a trivial category <strong>and</strong> has the only non-trivial<br />
homology at degree zero, which is Z.<br />
2. If u(x) = y op , then there are several possibilities. In any case<br />
we must have u(y) = u(z). Firstly, we may have u(C) =<br />
{y op }. Then Hi(u(C), Z) ∼ = Z if i = 0 or zero otherwise.<br />
Secondly we may have u(C) equals<br />
or<br />
<br />
y op α op x op<br />
β op<br />
y op <br />
x op .<br />
In the former situation we will have H0(u(C), Z) ∼ = H1(u(C), Z) ∼ =<br />
Z <strong>and</strong> zero otherwise. In the latter situation we have only<br />
non-trivial homology H0(u(C), Z) ∼ = Z.<br />
3. If u(x) = z op , then it is similar to 2.<br />
Since u can be decomposed into a sequence C ↠ u(C) ↩→<br />
C op , u ∗ is also the composite <strong>of</strong> H∗(C, Z) → H∗(u(C), Z) →<br />
H∗(C op , Z). Under the circumstance we underst<strong>and</strong> u ∗ com-
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 55<br />
pletely. A crucial fact is that, our previous calculations assert<br />
that there exists no such functor u : C → C op that induces the<br />
isomorphism <strong>of</strong> graded abelian groups H∗(C, Z) ∼ = H∗(C op , Z).<br />
Simplicial maps are used to compare two simplicial sets (<strong>and</strong><br />
resulting complexes). Now we introduce a way to compare two<br />
simplicial maps.<br />
Definition 2.2.15. For any two simplicial sets X, Y , we can<br />
define the Cartesian product X × Y by (X × Y ) n = X n × Y n<br />
with face <strong>and</strong> degeneracy maps d i = (d X i , dY i ), s i = (s X i , sY i ).<br />
Example 2.2.16. If D <strong>and</strong> C are two small categories, then<br />
N(D × C) ∼ = ND × NC.<br />
For brevity we denote by {0, · · · , 0, 1, · · · , 1} the element<br />
0 = · · · = 0 < 1 = · · · = 1 <strong>of</strong> the set N1 n with exactly i<br />
copies <strong>of</strong> 1 for 0 ≤ i ≤ n + 1.<br />
Definition 2.2.17. Let Φ, Ψ : X → Y be two simplicial<br />
maps between simplicial sets. We say Φ is simplicially<br />
homotopic to Ψ if there exists a natural transformation<br />
: X × N1 → Y such that n | Xn ×{(0,··· ,0)} = Φ n <strong>and</strong><br />
n | Xn ×{(1,··· ,1)} = Ψ n for all n ≥ 0. We call a simplicial<br />
homotopy from Φ to Ψ, written as : Φ→Ψ ≃ or simply Φ ≃ Ψ.<br />
We say two simplicial sets X <strong>and</strong> Y are homotopic, written<br />
as X ≃ Y , if there exist natural transformations Φ : X → Y<br />
<strong>and</strong> Ψ : Y → X such that ΨΦ ≃ Id X <strong>and</strong> ΦΨ ≃ Id Y .<br />
There are two canonical simplicial maps ι 0 , ι 1 : N0 → N1<br />
by sending {0, · · · , 0} to {0, · · · , 0} or {1, · · · , 1}, respectively.<br />
Based on this observation, being a simplicial homotopy<br />
is equivalent to having the following commutative diagram
56 2 Simplicial methods<br />
X ∼ = X × N0<br />
Id X ×ι 0<br />
<br />
X × N1<br />
Id X ×ι 1<br />
X × N0 ∼ = X<br />
Φ<br />
<br />
Ψ<br />
Y .<br />
Since (X × N1) n consists <strong>of</strong> n + 2 copies <strong>of</strong> X n in the<br />
form <strong>of</strong> X n × {0, · · · , 1, · · · , 1}, combinatorially, the above<br />
definition is equivalent to saying that there exist maps ′ i =<br />
s i n | Xn ×{0,··· ,1,··· ,1} : X n → Y n+1 for 0 ≤ i ≤ n such that<br />
d 0 ′ 0 = Φ n<br />
d n+1 ′ n = Ψ n<br />
d i ′ j = ′ j−1 d i i < j<br />
d j+1 ′ j+1 = d j+1 ′ j<br />
d i ′ j = ′ j d i−1 i > j + 1<br />
s i ′ j = ′ j+1 s i i ≤ j<br />
s i ′ j = ′ j s i−1 i > j.<br />
Remember in the definition <strong>of</strong> ′ i, {0, · · · , 1, · · · , 1} denotes<br />
the element 0 = · · · = 0 < 1 = · · · = 1 in N1 with i copies<br />
<strong>of</strong> 1.<br />
Definition 2.2.18. If the induced simplicial map Npt : NC →<br />
N• <strong>of</strong> the canonical functor pt : C → • gives rise to a homotopy<br />
NC ≃ N•, then we say C is contractible.<br />
Note that there always exist various functors • → C. If<br />
C is contractible, then any functor ι : • → C will induce<br />
N• ≃ NC.<br />
Proposition 2.2.19. If Φ : u → u ′ is a natural transformation<br />
between two functors u, u ′ : D → C, then Nu is<br />
homotopic to Nu ′ . Consequently Cu <strong>and</strong> Cu ′ are chain homotopic.<br />
In particular, if ND ≃ NC, then C ∗ (D, R) <strong>and</strong> C ∗ (C, R)<br />
are chain homotopy equivalent. Hence H∗(D, R) ∼ = H∗(C, R)<br />
<strong>and</strong> H ∗ (D, R) ∼ = H ∗ (C, R).
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 57<br />
Pro<strong>of</strong>. We can define a functor ˜Φ : D × 1 → C by ˜Φ(a, 0) =<br />
u(a), ˜Φ(a, 1) = u ′ (a) <strong>and</strong> ˜Φ(α, 1{0} ) = u(α), ˜Φ(α, 1{1} ) =<br />
u ′ (α) <strong>and</strong> ˜Φ(1a ,
58 2 Simplicial methods<br />
when T possesses all limits. This is the historical context for<br />
the Kan extensions.<br />
2.2.2 Classifying spaces<br />
It is <strong>of</strong>ten very useful to have some knowledge about classifying<br />
spaces when computing the (co-)homology <strong>of</strong> a small<br />
category. In this section we recall several important results as<br />
facts, in order to obtain a balanced view towards key results<br />
in the preceding section.<br />
Definition 2.2.22. The geometric realization |X|, <strong>of</strong> a simplicial<br />
set X, is the quotient <strong>of</strong><br />
⋃<br />
X n × △ n<br />
n≥0<br />
by the equivalence relation given by (d i x, y) ∼ (x, d i y) <strong>and</strong><br />
(s i x, y) ∼ (x, s i y).<br />
Definition 2.2.23. The geometric realization <strong>of</strong> NC, the<br />
nerve <strong>of</strong> a small category C, is called the classifying space<br />
<strong>of</strong> C, customarily denoted by BC.<br />
In the literature the classifying space <strong>of</strong> a small category is<br />
<strong>of</strong>ten written as |C|. However in these notes we try to avoid<br />
this notation since when C is a finite group, it has been used<br />
to denote the order <strong>of</strong> the group.<br />
In the following examples, the first two are not hard to verify<br />
while the other two requires further knowledge from algebraic<br />
topology so we point out places where the reader may find<br />
pro<strong>of</strong>s.<br />
Example 2.2.24.1. The classifying space <strong>of</strong> n is the st<strong>and</strong>ard<br />
n-simplex △ n .<br />
2. The classifying space <strong>of</strong> the following category is S 1 . Recall<br />
its integral homology <strong>and</strong> compare with Example 2.2.6 (1).
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 59<br />
x<br />
3. The classifying space <strong>of</strong> Z 2 is RP ∞ . One can see this by using<br />
Milnor’s construction EZ 2 . Then up to homotopy BZ 2 ≃<br />
EZ 2 /Z 2 .<br />
4. Let D <strong>and</strong> C be two small categories. There is a natural way<br />
to define the join <strong>of</strong> ND <strong>and</strong> NC. Then N(D∗C) ∼ = ND∗NC<br />
<strong>and</strong> consequently B(D ∗ C) ≃ BD ∗ BC, see [26].<br />
The following theorem asserts that the classifying spaces are<br />
“good”.<br />
Theorem 2.2.25. The space |X| is a CW-complex having<br />
one n-cell for each non-degenerate n-simplex <strong>of</strong> X.<br />
It explains the structures <strong>of</strong> classifying spaces in Example<br />
2.2.24.<br />
Proposition 2.2.26. The singular homology <strong>and</strong> cohomology<br />
<strong>of</strong> BC with coefficients in R, namely H∗(BC, R) <strong>and</strong><br />
H ∗ (BC, R), are identified with H∗(C, R) <strong>and</strong> H ∗ (C, R).<br />
Pro<strong>of</strong>. The normalized complex C † ∗(C, R) is a cellular complex<br />
for computing the singular homology <strong>of</strong> H∗(BC, R).<br />
The essential connection between simplicial sets <strong>and</strong> topological<br />
spaces is given by the following result.<br />
Theorem 2.2.27 (Kan). Let X be a simplicial set <strong>and</strong> Y<br />
a topological space. Then we have an adjunction<br />
Hom T op (|X|, Y ) ∼ = Hom SimpSet (X, SY ).<br />
Moreover the adjunct preserves homotopies.<br />
Pro<strong>of</strong>. Let f : |X| → Y be a continuous map. Then we define<br />
a simplicial map Φ f by [(Φ f ) n (x n )](a n ) = f(x n , a n ) for n ≥ 0,<br />
x n ∈ X n <strong>and</strong> a n ∈ △ n . Conversely if Φ : X → SY is a<br />
α<br />
β<br />
y
60 2 Simplicial methods<br />
simplicial map, we define f Φ : |X| → Y by f Φ (x n , a n ) =<br />
[Φ n (x n )](a n ) for n ≥ 0, x n ∈ X n <strong>and</strong> a n ∈ △ n . These two<br />
assignments give the adjunction.<br />
⊓⊔<br />
As an example, every CW-complex is homotopy equivalent<br />
to the classifying space <strong>of</strong> the poset <strong>of</strong> its singular simplices.<br />
Directly from Theorem 2.2.3 we have an alternative characterization<br />
<strong>of</strong> the geometric realization <strong>of</strong> a simplicial set X<br />
as<br />
|X| ∼ = lim −→ι/X<br />
|P|,<br />
where |P| sends each object (n, Φ) to △ n , because a left adjoint<br />
functor preserves direct limits. Given this homeomorphism<br />
we can prove Theorem 2.2.27 alternatively as follows.<br />
Hom T op (|X|, Y ) ∼ = Hom T op (lim −→ι/X<br />
|P|, Y )<br />
∼ = lim ←−ι/X<br />
Hom T op (|P|, Y )<br />
∼ = lim ←−ι/X<br />
Hom SimpSet (P, SY )<br />
∼ = HomSimpSet (lim −→ι/X<br />
P, SY )<br />
∼ = HomSimpSet (X, SY ).<br />
The third isomorphism does not depend on Theorem 2.2.27<br />
<strong>and</strong> it is true because<br />
Hom T op (△ n , Y ) = SY n<br />
∼ = HomSimpSet (Hom △ (−, n), SY ).<br />
Theorem 2.2.28. Let X, Y be two simplicial sets. Then<br />
there exists a natural homeomorphism<br />
|X × Y | ∼ = |X| × |Y |,<br />
if |X| × |Y | is a CW-complex.<br />
The space |X| × |Y | is a CW-complex if both X <strong>and</strong> Y are<br />
countable (that is, both ∪ n≥0 X n <strong>and</strong> ∪ n≥0 Y n are countable)
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 61<br />
or if either |X| or |Y | is locally finite (that is, every point is<br />
an inner point <strong>of</strong> a finite subcomplex).<br />
We collect some useful statements from last section which<br />
are adapted to classifying spaces. The upshot is that a homotopy<br />
between two simplicial maps or sets does give rise to a<br />
homotopy between continuous maps or spaces, in the usual<br />
topological sense. A small category C is contractible if BC has<br />
the same homotopy type <strong>of</strong> a point. By Theorem 2.2.27 it is<br />
equivalent to saying that NC ≃ N•.<br />
Corollary 2.2.29. A functor u : D → C induces a continuous<br />
map Bu : BD → BC.<br />
1. If u ′ : D → C is another functor <strong>and</strong> : ND × N1 →<br />
NC is a homotopy between Nu <strong>and</strong> Nu ′ , then we obtain a<br />
homotopy B : BD × △ 1 → BC between Bu <strong>and</strong> Bu ′ .<br />
2. A natural transformation Φ between u, u ′ : D → C induces<br />
a homotopy B Φ : BD × △ 1 → BC between Bu <strong>and</strong> Bu ′ .<br />
3. If C has an initial or a terminal object, then BC is contractible.<br />
The topological map Bu : BD → BC induces the maps u ∗<br />
<strong>and</strong> u ∗ as in Corollary 2.2.13.<br />
Remark 2.2.30. We must emphasize that continuous maps between<br />
classifying spaces are not always realized by functors. In<br />
other words, not all topological maps BD → BC come from<br />
a simplicial map ND → NC by Lemma 2.2.11. For instance,<br />
there is a natural homeomorphism between BC <strong>and</strong> BC op , but<br />
one cannot construct a functor between C <strong>and</strong> C op realizing<br />
this homeomorphism, except in very special situations (e.g. C<br />
is a group). See Example 2.2.14.<br />
We shall come back to comparing classifying spaces via functors<br />
by introducing Quillen’s work.
62 2 Simplicial methods<br />
2.2.3 Cup product <strong>and</strong> cohomology ring<br />
In this section, we introduce cross product (or external product),<br />
cup product (or internal product) <strong>and</strong> the resulting multiplicative<br />
structure on simplicial cohomology.<br />
Theorem 2.2.31 (Eilenberg-Zilber). Let X, Y be two<br />
simplicial sets. There is a natural chain homotopy equivalence<br />
ξ : C ∗ (X × Y, R) → C ∗ (X, R) ⊗ C ∗ (Y, R).<br />
Pro<strong>of</strong>. The chain map, called the Alex<strong>and</strong>er-Whitney map,<br />
can be explicitly constructed as follows<br />
n∑<br />
ξ(a, b) = d i+1 · · · d n (a) ⊗ d i 0(b),<br />
i=0<br />
where (a, b) ∈ C n (X × Y, R) = R[X n ] ⊗ R[Y n ].<br />
The inverse <strong>of</strong> ξ (up to chain equivalence), called the Eilenber-<br />
Zilber map<br />
is defined by<br />
η(c ⊗ d) =<br />
η : C ∗ (X, R) ⊗ C ∗ (Y, R) → C ∗ (X × Y, R),<br />
∑<br />
(p,q)-shuffles<br />
σ<br />
(−1) |σ| (s σ(1) · · · s σ(p) c, s σ(p+1) · · · s σ(n) d)<br />
for c ∈ C p (X, R) <strong>and</strong> d ∈ C q (Y, R) with p + q = n. Here a<br />
(p, q)-shuffle σ is a permutation <strong>of</strong> n letters such that<br />
σ(1) < · · · < σ(p) <strong>and</strong> σ(p + 1) < · · · < σ(n).<br />
⊓⊔<br />
Remark 2.2.32. The Alex<strong>and</strong>er-Whitney <strong>and</strong> Eilenberg-Zilber<br />
maps lead to chain homotopy equivalences between normalized<br />
chain complexes, in light <strong>of</strong> Theorem 2.2.5.
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 63<br />
Let C be a small category <strong>and</strong> R a commutative ring with<br />
identity. For C ∗ (−, R) = C ∗ (−, R) ∧ , we have a cross product<br />
induced by the Alex<strong>and</strong>er-Whitney map<br />
× : C i (X, R)⊗C j (Y, R) → Hom R (C i (X, R)⊗C j (Y, R), R) ≃ →C i+j (X×<br />
with the left map given by<br />
(f × g)(a, b) = f(d i+1 · · · d n (a))g(d i 0(b)),<br />
if f ∈ C i (X, R), g ∈ C j (Y, R), a ∈ C n (X, R) <strong>and</strong> b ∈<br />
C n (Y, R). One can check that ∂ ∗ (f ×g) = ∂ ∗ (f)×g+(−1) i f ×<br />
∂ ∗ (g). Thus it induces a cross product on cohomology<br />
× : H i (X, R) ⊗ H j (Y, R) → H i+j (X × Y, R)<br />
When R is a field, the map × is an isomorphism by Künneth<br />
formula.<br />
Proposition 2.2.33. Let X, Y be two simplicial sets <strong>and</strong><br />
τ : X ×Y → Y ×X the twist map, defined by τ(a, b) = (b, a)<br />
for any a ∈ X n <strong>and</strong> b ∈ Y n . Then τ induces an isomorphism<br />
τ ∗ : H ∗ (X × Y, R) → H ∗ (Y × X, R).<br />
Suppose f ∈ H i (X, R) <strong>and</strong> g ∈ H j (Y, R). Then τ ∗ (f ×g) =<br />
(−1) ij g × f.<br />
Pro<strong>of</strong>. Since τ is an isomorphism <strong>of</strong> simplicial sets, it certainly<br />
induces an isomorphism on (co-)chain complexes <strong>and</strong><br />
(co-)homology. Now we calculate the value <strong>of</strong> τ ∗ .<br />
Define τ ∗<br />
′ = ξτ ∗ η : C ∗ (Y, R) ⊗ C ∗ (X, R) → C ∗ (X, R) ⊗<br />
C ∗ (Y, R), as shown in the following commutative diagram<br />
C ∗ (Y × X, R)<br />
η ≃<br />
τ ∗ <br />
C ∗ (X × Y, R)<br />
C ∗ (Y, R) ⊗ C ∗ (X, R)<br />
<br />
τ<br />
′ C ∗ (X, R) ⊗ C ∗ (Y, R).<br />
∗<br />
≃<br />
<br />
ξ
64 2 Simplicial methods<br />
Then by direct calculation, it follows τ ∗ ′ is chain homotopy<br />
equivalent to the canonical chain equivalence τ ∗ ′′ : C ∗ (Y, R) ⊗<br />
C ∗ (X, R) → C ∗ (X, R) ⊗ C ∗ (Y, R) determined by b ⊗ a ↦→<br />
(−1) ij a ⊗ b for any a ∈ X i <strong>and</strong> b ∈ Y j . Hence τ ∗ : H ∗ (X ×<br />
Y, R) → H ∗ (Y × X, R) is given by the chain equivalence<br />
induced by τ ∗<br />
′′<br />
Hom R (C ∗ (X, R)⊗C ∗ (Y, R), R) → Hom R (C ∗ (Y, R)⊗C ∗ (X, R), R)<br />
On the other h<strong>and</strong>, there is a canonical chain equivalence<br />
C ∗ (X, R) ⊗ C ∗ (Y, R) → C ∗ (Y, R) ⊗ C ∗ (X, R)<br />
defined in the same fashion as τ ′′<br />
∗ . Thus there is a commutative<br />
diagram <strong>of</strong> cochain complexes<br />
C ∗ (X, R) ⊗ C ∗ (Y, R)<br />
∼ =<br />
C ∗ (Y, R) ⊗ C ∗ (X, R)<br />
Hom R (C ∗ (X, R) ⊗ C ∗ (Y, R), R) ∼=<br />
<br />
Hom R (C ∗ (Y, R) ⊗ C ∗ (X, R), R.<br />
Since the vertical maps give rise to the cross products by<br />
sending f ⊗g to f ×g for any f ∈ C i (X, R) <strong>and</strong> g ∈ C j (Y, R),<br />
while the horizontal maps map f ⊗g <strong>and</strong> f ×g to (−1) ij g ⊗f<br />
<strong>and</strong> (−1) ij g × f, respectively, our formula for τ ∗ follows from<br />
it.<br />
⊓⊔<br />
Definition 2.2.34. There exists a diagonal functor ∆ : C →<br />
C × C such that ∆(x) = (x, x) <strong>and</strong> ∆(α) = (α, α) for any<br />
x ∈ Ob C <strong>and</strong> α ∈ Mor C, respectively.<br />
The diagonal functor induces a natural chain map, the simplicial<br />
diagonal map, also denoted by<br />
∆ : C ∗ (C, R) → C ∗ (C × C, R).<br />
Composing with the Alex<strong>and</strong>er-Whitney map<br />
C ∗ (C × C, R) → C ∗ (C, R) ⊗ C ∗ (C, R),
2.2 Nerves, classifying spaces <strong>and</strong> cohomology 65<br />
we get the diagonal map,<br />
∆ : C ∗ (C, R) → C ∗ (C, R) ⊗ C ∗ (C, R).<br />
More explicitly for any x 0 → x 1 → · · · → x n , a base element<br />
<strong>of</strong> C n (C, R),<br />
n∑<br />
∆(x 0 → · · · → x n ) = (x 0 → · · · → x i )⊗(x i → · · · → x n ).<br />
i=0<br />
By dualizing the diagonal map, we obtain the cup product on<br />
C ∗ (C, R) = C ∗ (C, R) ∧ as<br />
∪ : C i (C, R) ⊗ C j (C, R) ×<br />
−→C i+j (C × C, R) ∆<br />
−→C i+j (C, R).<br />
Suppose f ∈ C i (C, R) <strong>and</strong> g ∈ C j (C, R) with i + j = n. Then<br />
(f ∪ g)(x 0 → · · · → x n ) = ∑ n<br />
k=0 f(x 0 → · · · → x k )g(x k → · · · → x n )<br />
= f(x 0 → · · · → x i )g(x i → · · · → x n ).<br />
From direct calculation we get ∂ ∗ (f ∪ g) = ∂ ∗ (f) ∪ g +<br />
(−1) i f ∪ ∂ ∗ (g) for any f ∈ C i (C, R) <strong>and</strong> g ∈ C j (C, R). Consequently<br />
the cup product passes to cohomology<br />
∪ : H i (C, R) ⊗ H j (C, R) ×<br />
−→H ∗ (C × C, R) ∆<br />
−→H i+j (C, R),<br />
which makes the graded R-module H ∗ (C, R) = ⊕ i≥0 H i (C, R)<br />
a graded ring. This is the ordinary cohomology ring <strong>of</strong> C with<br />
coefficients in R. We may readily verify the following equality.<br />
Proposition 2.2.35. f ∪ g = (−1) ij g ∪ f for any f ∈<br />
H i (C, R) <strong>and</strong> g ∈ H j (C, R).<br />
Pro<strong>of</strong>. It follows from Proposition 2.2.33 because the cup<br />
product is induced by a cross product.<br />
⊓⊔<br />
It means that the ordinary cohomology ring <strong>of</strong> a small category<br />
is graded commutative. Since the cup product is exactly<br />
the same as the one for the topological cohomology ring
66 2 Simplicial methods<br />
H ∗ (BC, R) = ⊕ i≥0 H i (BC, R). We actually have a ring isomorphism<br />
H ∗ (C, R) ∼ = H ∗ (BC, R).<br />
Remark 2.2.36. When C is a (discrete) group, H ∗ (C, R) is the<br />
usual group cohomology ring.<br />
Corollary 2.2.37. Let u : D → C be a functor. Then u ∗ :<br />
H ∗ (C, R) → H ∗ (D, R) is a ring homomorphism.<br />
This ring homomorphism is usually called the restriction.<br />
Proposition 2.2.38. Let D <strong>and</strong> C be small categories. Suppose<br />
k is a field. Then the cup product <strong>of</strong> any two positive<br />
degree elements in H ∗ (D ∗ C, k) is constantly zero.<br />
Pro<strong>of</strong>. In Proposition 2.2.8 we computed homology groups <strong>of</strong><br />
the join <strong>of</strong> two small categories. Accordingly there are formulas<br />
for cochain complexes <strong>and</strong> cohomology as well, i.e. for<br />
each n > 0<br />
˜C n (D ∗ C, k) ∼ = [˜C∗ (D, k) ⊗ ˜C∗ (C, k)] n−1 ,<br />
<strong>and</strong><br />
H n (D ∗ C, k) =<br />
⊕<br />
i+j=n−1<br />
˜H i (D, k) ⊗ ˜Hj (C, k).<br />
It implies that H n (D ∗ C, k) is spanned over a set <strong>of</strong> cohomology<br />
classes represented by functions <strong>of</strong> the form η α such<br />
that η α (β) = 1 if β = α <strong>and</strong> zero otherwise, <strong>and</strong> such that<br />
α ∈ N n (D ∗ C) consists <strong>of</strong> exactly one morphism γ x,y where<br />
x ∈ Ob D <strong>and</strong> y ∈ Ob C, as introduced in Definition 2.2.7.<br />
By direct computation, the cup product <strong>of</strong> any two such functions<br />
is constantly zero. Hence we are done.<br />
⊓⊔<br />
There is a topological interpretation to the above result.<br />
In fact the cohomology ring <strong>of</strong> a join <strong>of</strong> two spaces always<br />
have trivial cup product. See Example 2.2.24 (4). There is a<br />
topological interpretation. Suppose X <strong>and</strong> Y are two spaces.
2.3 Quillen’s work on classifying spaces 67<br />
Then their join X∗Y is a suspension Σ(X×Y/X∨Y ). For any<br />
suspended space, its cohomology ring has trivial cup product.<br />
The structure <strong>of</strong> X ∗ Y also explains Proposition 2.2.8. In<br />
Example 2.2.6 (1), the cup product <strong>of</strong> any two elements <strong>of</strong><br />
positive degree in H ∗ (C, R) is zero.<br />
2.3 Quillen’s work on classifying spaces<br />
In order to assure a functor u : D → C inducing a simplicial<br />
homotopy equivalence ND ≃ NC, one must have a functor on<br />
the opposite direction. However one may very <strong>of</strong>ten obtain a<br />
(topological) homotopy equivalence Bu : BD → BC without<br />
the existence <strong>of</strong> such v : C → D that Bv is also a homotopy<br />
equivalence. Such homotopies certainly are not simplicial <strong>and</strong><br />
will be investigated now.<br />
2.3.1 Quillen’s Theorem A<br />
Quillen’s Theorem A (also called Quillen’s Fibre Theorem)<br />
provides sufficient conditions on a functor between two small<br />
categories which guarantee the functor inducing a homotopy<br />
equivalence between classifying spaces.<br />
Theorem 2.3.1 (Quillen’s Theorem A). Let u : D →<br />
C be a functor between two small categories. If either all<br />
the overcategories or all the undercategories are contractible.<br />
Then Bu is a homotopy equivalence.<br />
Pro<strong>of</strong>. The pro<strong>of</strong> is given in Section 2.4.3.<br />
Under certain circumstances, one may replace the over- or<br />
undercategories with some simpler constructions.<br />
Definition 2.3.2. Let u : D → C be a functor between small<br />
categories. For each x ∈ Ob C, we define the fibre <strong>of</strong> u over x<br />
⊓⊔
68 2 Simplicial methods<br />
to be the category u −1 (x) whose objects are preimages <strong>of</strong> x<br />
<strong>and</strong> whose morphisms are preimages <strong>of</strong> 1 x .<br />
Apart from intuition, the fibres are usually not good for<br />
comparing the homotopy types <strong>of</strong> two classifying spaces. For<br />
instance if G is a group <strong>and</strong> H is a subgroup, then the inclusion<br />
functor has fibre at • equal to the trivial category. But<br />
there is no reason to say that BH would be homotopy equivalent<br />
to BG, especially when we think about the extreme case<br />
where H is the trivial subgroup <strong>of</strong> G.<br />
Definition 2.3.3. Let u : D → C be a functor between small<br />
categories. We call D prefibred over C if the natural functor i x :<br />
u −1 (x) → x\u has a right adjoint h x for each x ∈ Ob C. We<br />
call D prec<strong>of</strong>ibred over C if the natural functor i x : u −1 (x) →<br />
u/x has a left adjoint h x for each x ∈ Ob C.<br />
Corollary 2.3.4. Suppose u : D → C such that D is either<br />
prefibred or prec<strong>of</strong>ibred over C. If the fibre u −1 (x) over every<br />
x ∈ Ob C is contractible, then Bu is a homotopy equivalence.<br />
Pro<strong>of</strong>. Since there are adjoint functors between u −1 (x) <strong>and</strong><br />
x\u (or u/x) for every x ∈ Ob C. The condition implies the<br />
contractibility <strong>of</strong> all the undercategories (or overcategories).<br />
Then we apply Quillen’s Theorem A.<br />
⊓⊔<br />
We will see some prefibred <strong>and</strong> prec<strong>of</strong>ibred examples in the<br />
next section. Note that, for G a group <strong>and</strong> H a subgroup,<br />
i : H ↩→ G is a simple example that H is neither prefibred<br />
nor prec<strong>of</strong>ibred over G, unless H = G.<br />
Suppose u : D → C makes D prec<strong>of</strong>ibred over C. Then we<br />
have<br />
Hom u −1 (x)(h x (a, α), b) ∼ = Hom u/x ((a, α), i x (b))<br />
for any b ∈ Ob u −1 (x) <strong>and</strong> (a, α) ∈ Ob u/x. Let us try<br />
to underst<strong>and</strong> h x . The set Hom u/x ((a, α), b) consists <strong>of</strong> ex-
2.3 Quillen’s work on classifying spaces 69<br />
actly those f ∈ Hom D (a, b) such that u(f) = α : u(a) →<br />
x = u(b). The identity 1 hx (a,α) corresponds to a morphism<br />
in Hom u/x ((a, α), i x h x (a, α)), which is some morphism ˜f ∈<br />
Hom D (a, h x (a, α)) such that u( ˜f) = α. Then each f ∈<br />
Hom D (a, b), satisfying u(f) = α, must uniquely factor through<br />
˜f. If f = g ˜f for g ∈ HomD (h x (a, α), b) such that u(g) = 1 x ,<br />
then g corresponds to f ∈ Hom u/x ((a, α), i x (b)) in the isomorphism.<br />
We have alternative descriptions <strong>of</strong> h x <strong>and</strong> h x .<br />
Lemma 2.3.5. Suppose u : D → C is as above <strong>and</strong> γ : x →<br />
y is a morphism in C.<br />
1. If D is prefibred over C, then it induces a functor<br />
γ ⋆ : u −1 (y) → y\u → x\u → u −1 (x).<br />
Furthermore the right adjoint <strong>of</strong> u −1 (x) → x\u is given by<br />
(α, a) ↦→ α ⋆ (a).<br />
2. If D is prec<strong>of</strong>ibred over C, then it induces a functor<br />
γ ⋆ : u −1 (x) → u/x → u/y → u −1 (y).<br />
Furthermore the left adjoint <strong>of</strong> u −1 (y) → u/y is given by<br />
(b, β) ↦→ β ⋆ (b).<br />
Pro<strong>of</strong>. We shall only prove (2) because the pro<strong>of</strong> for (1) is<br />
similar.<br />
Let (b, β) ∈ Ob u/y. Then<br />
Hence β ⋆ (b) = h y (b, β).<br />
β ⋆ : b ↦→ (b, 1 y ) ↦→ (b, β) ↦→ h y (b, β).<br />
Definition 2.3.6. Let u : D → C be a functor between small<br />
categories. We call D fibred over C if it is prefibred over C <strong>and</strong><br />
moreover (βα) ⋆ = α ⋆ β ⋆ for any composable α, β ∈ Mor C.<br />
⊓⊔
70 2 Simplicial methods<br />
We call D c<strong>of</strong>ibred over C if it is prec<strong>of</strong>ibred over C <strong>and</strong><br />
moreover (βα) ⋆ = β ⋆ α ⋆ for any composable α, β ∈ Mor C.<br />
2.3.2 Constructions over categories <strong>and</strong> relevant functors<br />
We describe several important constructions here, which also<br />
serve an examples to illustrate Quillen’s Theorem A.<br />
Definition 2.3.7. Suppose C is a small category. The (first)<br />
category <strong>of</strong> factorizations in C, denoted by F (C), has Ob F (C) =<br />
Mor C. An object admits a morphism to another object if <strong>and</strong><br />
only if as morphisms in C, the first object factors through the<br />
second.<br />
Iterating the construction, for non-negative integer n, we<br />
can define F n (C) = F (F n−1 (C)) as the n-th category <strong>of</strong> factorizations<br />
in C. Here F 0 (C) is defined to be C.<br />
We shall see that in order to underst<strong>and</strong> C very <strong>of</strong>ten we<br />
have to utilize F (C).<br />
In order not to cause confusions, when a morphism α ∈<br />
Mor C is considered as an object in F (C), we write it as [α].<br />
In the above definition, a morphism in F (C) is a pair (α ′ , α ′′ ) :<br />
[α] → [β] <strong>and</strong> is customarily pictured by<br />
y<br />
α ′ <br />
α<br />
x<br />
y ′ x ′ α ′′<br />
β<br />
Example 2.3.8.1. When C is a group, F (C) has all group elements<br />
as objects. The category F (C) is a groupoid whose<br />
skeleton is isomorphic to the group itself.<br />
2. When C is a poset, then F (C) is another poset. For 1, we<br />
have F (1) as follows
2.3 Quillen’s work on classifying spaces 71<br />
(α,1 0 )<br />
(1 1 ,1 0 )<br />
<br />
[1 ← 0]<br />
(1 1 ,α)<br />
[0 ← 0]<br />
[1 ← 1]<br />
<br />
(1 0 ,1 0 )<br />
(1 1 ,1 1 )<br />
For convenience we denote the non-identity morphism 1 ← 0<br />
by α.<br />
One can see that there are two natural covariant functors<br />
t : F (C) → C <strong>and</strong> s : F (C) → C op , where t st<strong>and</strong>s for target<br />
while s means source.<br />
Proposition 2.3.9. Suppose D ≃ C. Then F (D) ≃ F (C).<br />
Pro<strong>of</strong>. Let u : D → C <strong>and</strong> v : C → D be such that vu ∼ = Id D<br />
<strong>and</strong> uv ∼ = Id C . We construct two functors between F (C) <strong>and</strong><br />
F (D) which induce an equivalence.<br />
We define F (u) : F (D) → F (C) by F (u)([α]) = [u(α)],<br />
<strong>and</strong> F (u)(α ′ , α ′′ ) = (u(α ′ ), u(α ′′ )). One can verify this gives a<br />
functor. Similarly there exists a functor F (v) : F (C) → F (D).<br />
Since F (v)F (u)([α]) = [vu(α)] ∼ = [α], we have F (v)F (u) ∼ =<br />
Id F (D) . Similarly we get F (u)F (v) ∼ = Id F (C) . Thus F (D) ≃<br />
F (C).<br />
⊓⊔<br />
Proposition 2.3.10. Consider the canonical functors C←F t (C) →C s o<br />
Then<br />
1. F (C) is c<strong>of</strong>ibred over both C <strong>and</strong> C op . It implies, for any<br />
x ∈ Ob C, the natural functor i x : t −1 (x) → t/x has a<br />
left adjoint, <strong>and</strong>, for any y ∈ Ob C op , the natural functor<br />
i y : t −1 (y) → s/y has a left adjoint;<br />
2. for any x ∈ Ob C, the category t −1 (x) ∼ = (Id C /x) op has an<br />
initial object;
72 2 Simplicial methods<br />
3. for any y ∈ Ob C op , the category s −1 (y) ∼ = y\Id C has an<br />
initial object.<br />
Consequently both t <strong>and</strong> s induce homotopy equivalences.<br />
Pro<strong>of</strong>. The category F (C) is prec<strong>of</strong>ibred over C because we<br />
can define h x ([α], α ′ ) = [α ′ α] for any x ∈ Ob C <strong>and</strong> ([α], α ′ ) ∈<br />
Ob t/x such that<br />
Hom t −1 (x)(h x ([α], α ′ ), [β]) ∼ = Hom t/x (([α], α ′ ), i x (β)).<br />
We can compute t −1 (x) <strong>of</strong> t : F (C) → C for every x ∈ Ob C<br />
<strong>and</strong> find t −1 (x) ∼ = (Id C /x) op . It has an initial object [1 x ]<br />
<strong>and</strong> thus is contractible. This means that t/x is always contractible<br />
<strong>and</strong> thus t induces an equivalence by Quillen’s Theorem<br />
A. Moreover from the description <strong>of</strong> t −1 (x) we can see<br />
that F (C) is c<strong>of</strong>ibred over C.<br />
We can similarly prove that s : F (C) → C op induces an<br />
equivalence as well, with s −1 (y) ∼ = y\Id C .<br />
⊓⊔<br />
A hidden fact in the pro<strong>of</strong> is that whenever there is a category<br />
C, we can produce many over- <strong>and</strong> undercategories Id C /x<br />
<strong>and</strong> x\Id C , all <strong>of</strong> which are contractible. This will be used for<br />
an important construction later on.<br />
Remark 2.3.11. Here the homotopy equivalence Bt : BF (C) →<br />
BC usually is not a simplicial homotopy equivalence as we do<br />
not expect to have a functor from C to F (C) giving a homotopy<br />
equivalence BC → BF (C). If such a functor were to<br />
exist, then we would obtain a functor inducing the homotopy<br />
equivalence BC → BC op , <strong>and</strong> hence inducing isomorphism on<br />
homology H∗(C, R) → H∗(C op , R), which rarely happens. See<br />
Example 2.2.14.<br />
Definition 2.3.12. The functor ∇ = (t, s) : F (C) → C e =<br />
C × C op is called the skew diagonal functor. The category C e<br />
is called the enveloping category <strong>of</strong> C.
2.3 Quillen’s work on classifying spaces 73<br />
The above construction is a special case <strong>of</strong> a more general<br />
definition F (u) for any u : D → C, naturally equipped with a<br />
functor F (u) → D × C op . Indeed F (Id C ) = F (C), see Section<br />
2.4.3.<br />
Example 2.3.13. When C is a poset, both F (C) <strong>and</strong> C e are<br />
posets <strong>and</strong> ∇ : F (C) → C e is an embedding. For the poset<br />
C = 1, we have F (1) ↩→ 1 e as follows<br />
[1 ← 0] (1, 0)<br />
[0 ← 0]<br />
[1 ← 1]<br />
↩→ (0, 0)<br />
(1, 1)<br />
(0, 1)<br />
Proposition 2.3.14. The category F (C) is c<strong>of</strong>ibred over C e .<br />
It implies that, for any (y, x) ∈ Ob C e , the functor i (y,x) :<br />
∇ −1 (y, x) → ∇/(y, x) has a left adjoint. Thus N∇ −1 (y, x) ≃<br />
N∇/(y, x).<br />
Furthermore ∇ −1 (y, x) ∼ = Hom C (x, y).<br />
Pro<strong>of</strong>. We first prove that F (C) is prec<strong>of</strong>ibred over C e . To<br />
this end, we need to show that for every (y, x) ∈ Ob C e , the<br />
functor i (y,x) : ∇ −1 (y, x) → ∇/(y, x) has a left adjoint. It is<br />
easy to identify ∇ −1 (y, x) with the discrete set Hom C (x, y).<br />
Then we define h (y,x) ([α], (α ′ , α ′′ )) = [α ′ αα ′′ ] ∈ Hom C (x, y)<br />
which leads to an isomorphism<br />
Hom ∇ −1 (y,x)([α ′ αα ′′ ], [β]) ∼ = Hom ∇/(y,x) (([α], (α ′ , α ′′ )), ([β], (1 y , 1 x ))).<br />
Note that in any case either morphism set has at most one<br />
element.<br />
If (γ ′ , γ ′′ ) : (y, x) → (y ′ , x ′ ) is a morphism, then (γ ′ , γ ′′ ) ⋆ is<br />
given by<br />
[α] ↦→ ([α], (1 y , 1 x )) ↦→ ([α], (γ ′ , γ ′′ )) ↦→ [γ ′ αγ ′′ ],
74 2 Simplicial methods<br />
for any α ∈ Hom C (x, y). Hence one can verify that F (C) is<br />
c<strong>of</strong>ibred over C e .<br />
⊓⊔<br />
We note that some <strong>of</strong> the overcategories or undercategories<br />
or fibres can be empty. As a reminder, the homology <strong>of</strong> ∅ is<br />
constantly zero.<br />
Remark 2.3.15. In the previous two propositions F (C) are<br />
c<strong>of</strong>ibred over C, C op as well as C e . It is not fibred over these<br />
categories. For instance in Example 2.3.13, i x : ∇ −1 (1, 0) =<br />
[1 ← 0] → ∇/(1, 0) ∼ = F (1) has only a left adjoint <strong>and</strong> no<br />
right adjoint because [1 ← 0] is a terminal object in F (1). In<br />
these notes, it is very important to compute limits over various<br />
over <strong>and</strong> undercategories. We <strong>of</strong>ten do it by establishing<br />
adjoint functors between these categories <strong>and</strong> some simpler<br />
categories. Then by Proposition 1.2.4 we will know whether<br />
we can simplify the computations <strong>of</strong> either direct or inverse<br />
limits (usually not both!). We shall see many explicitly calculations<br />
from Chapter 3.<br />
2.4 Further categorical <strong>and</strong> simplicial constructions<br />
In this section we will introduce the definition <strong>of</strong> a Grothendieck<br />
construction which is important when we deal with finite<br />
categories constructed from groups. Grothendieck constructions<br />
are a certain kind <strong>of</strong> homotopy colimits. By Thomason’s<br />
Homotopy Colimit Theorem, every homotopy colimit<br />
is a Grothendieck construction. In order to provide the interested<br />
reader a necessary underst<strong>and</strong>ing <strong>of</strong> these constructions,<br />
we will define bisimplicial sets, <strong>and</strong> since bisimplicial sets are<br />
introduced, there is no reason not to give pro<strong>of</strong>s <strong>of</strong> Quillen’s<br />
<strong>and</strong> Thomason’s theorems. Although this chapter is the most<br />
suitable place to present these materials, it is possible to postpone<br />
reading them until they are needed. Indeed, only un-
2.4 Further categorical <strong>and</strong> simplicial constructions 75<br />
derst<strong>and</strong>ing the Grothendieck constructions <strong>and</strong> knowing the<br />
statements are sufficient for our purposes in these notes.<br />
2.4.1 Grothendieck constructions<br />
Let C be a small category <strong>and</strong> F : C → Cat a functor. Then<br />
we can construct a small category, called the Grothendieck<br />
construction Gr C F, as follows.<br />
The objects are pairs <strong>of</strong> the form (x, a) with x ∈ Ob C <strong>and</strong><br />
a ∈ Ob F(x). Let (x, a) <strong>and</strong> (y, b) be two objects. A morphism<br />
(x, a) → (y, b) is a pair (α, f) such that α ∈ Hom C (x, y) <strong>and</strong><br />
f ∈ Hom F(b) (F(α)(a), b). It can be pictured as<br />
x<br />
α<br />
α<br />
y A ′ <br />
B ′<br />
C ′<br />
D ′<br />
y A ′ <br />
B ′<br />
x<br />
Note that there always exists a functor π : Gr C F → C.<br />
Example 2.4.1. Let u : D → C be a functor between small<br />
categories. Then the over- <strong>and</strong> undercategories can be realized<br />
as Grothendieck constructions. Given an object x ∈ Ob C, we<br />
can define two functors u = Hom C (u(−), x) : D → Set <strong>and</strong><br />
u = Hom C (x, u(−)) : D → Set. One can verify that Gr D u<br />
<strong>and</strong> Gr D u are isomorphic to u/x <strong>and</strong> x\u, respectively.<br />
We record one more important Grothendieck construction,<br />
although it is not needed in this chapter.<br />
Definition 2.4.2. A small category C is an EI category if all<br />
endomorphisms are isomorphisms.<br />
C ′<br />
D ′
76 2 Simplicial methods<br />
Typical examples <strong>of</strong> EI categories are groups <strong>and</strong> posets.<br />
For each EI category C, one can introduce a partial order on<br />
the set <strong>of</strong> isomorphism classes <strong>of</strong> objects by [x] ≤ [y] if <strong>and</strong><br />
only if Hom C (x, y) ≠ ∅. Let us denote by [C] the poset <strong>of</strong><br />
isomorphism classes <strong>of</strong> objects in C. Obviously there exists a<br />
natural functor p : C → [C]. Since [C] is a poset, we know the<br />
barycentric subdivision <strong>of</strong> [C], written as S([C]), is another<br />
poset whose objects are chains <strong>of</strong> non-isomorphism objects<br />
[x] n = [x 0 ] → [x 1 ] → · · · → [x n ] <strong>and</strong> there is a morphism<br />
[x] n → [y] m if <strong>and</strong> only if [y] m is a subchain <strong>of</strong> [x] n . We define<br />
a functor<br />
˜p : S([C]) → Cat<br />
by asking ˜p([x] n ) to be a category whose objects are the y n =<br />
y 0 → y 1 → · · · → y n in NC n such that p(y i ) = [x i ] for<br />
all 0 ≤ i ≤ n, <strong>and</strong> whose morphisms are (n + 1)-tuples <strong>of</strong><br />
morphisms in C making the following diagram commutative<br />
α<br />
y 1 0 y 1<br />
α 2 <br />
· · · αn <br />
y n<br />
γ 0 <br />
γ 1 <br />
γ n <br />
y ′ 0 β 1<br />
<br />
y ′ 1 β 2<br />
<br />
· · · βn<br />
<br />
y ′ n<br />
Since C is EI, all γ i are isomorphism. This means that ˜p([x] n )<br />
is always a groupoid. One can readily verify that for any<br />
x n → y m in Mor(sd[C]), there is a well defined functor<br />
˜p(x n ) → ˜p(y m ) given by first removing objects in x n which<br />
are not isomorphic to any object in y m , <strong>and</strong> then composing<br />
appropriate morphisms in what is left <strong>of</strong> x n .<br />
Definition 2.4.3 (Slominska). Let C be an EI category.<br />
Then its subdivision, written as S(C), is defined to be Gr S([C])˜p.<br />
Alternatively S(C) is a category whose objects are chains <strong>of</strong><br />
non-isomorphisms x n = x 0 → x 1 → · · · → x n in C, <strong>and</strong> a<br />
morphism from x n to y m = y 0 → y 1 → · · · → y m is an (m +
2.4 Further categorical <strong>and</strong> simplicial constructions 77<br />
1)-tuple <strong>of</strong> isomorphisms (γ 0 , · · · , γ m ), making the following<br />
diagram commutative<br />
<br />
<br />
<br />
<br />
<br />
α 1<br />
x n1<br />
α 2<br />
· · · αn x nm<br />
y 0<br />
γ 1<br />
β1<br />
y 1<br />
γ n<br />
β2<br />
· · · βn<br />
y m<br />
x n0<br />
γ 0<br />
<br />
in which every x ni ∈ {x i } n i=0 <strong>and</strong> x ni<br />
∼ = yi in C. From this<br />
description there exists a functor s : S(C) → C by x n ↦→ x 0 .<br />
Thus there are maps s ∗ : H∗(S(C), R) → H∗(C, R) <strong>and</strong> s ∗ :<br />
H ∗ (C, R) → H ∗ (S(C), R).<br />
Proposition 2.4.4 (Slominska). The map s induces a homotopy<br />
equivalence BS(C) ≃ BC. Hence we have isomorphisms<br />
s ∗ : H∗(S(C), R) → H∗(C, R) <strong>and</strong> s ∗ : H ∗ (C, R) →<br />
H ∗ (S(C), R).<br />
Pro<strong>of</strong>. We shall prove all undercategories are contractible. In<br />
order to do so, we show S(C) is prefibred over C. Then one<br />
can easily verify that for each x ∈ Ob C, s −1 (x) has x as a<br />
terminal object <strong>and</strong> hence is contractible.<br />
To show S(C) is prefibred over C we only have to find a<br />
right adjoint to the inclusion i x : s −1 (x) → x\s. Suppose<br />
(α, x 0 → x 1 → · · · → x n ) is an object in x\s. Then we define<br />
a functor x\s → s −1 (x) by<br />
(x 0<br />
α 1<br />
→x1 → · · · → x n , α) ↦→ (x α 1α<br />
→x 1 → · · · → x n ).<br />
Moreover if (γ 0 , γ 1 , · · · , γ m ) is a morphism from (α, x 0 →<br />
x 1 → · · · → x n ) to (α ′ , x ′ 0 → x ′ 1 → · · · → x ′ m), then its image<br />
is defined as (1 x , γ 1 , · · · , γ n ). Now one can check this functor<br />
is right adjoint to i x . Hence we are done.<br />
⊓⊔<br />
These isomorphisms will be generalized in Chapter 4.
78 2 Simplicial methods<br />
There are two or three occasions in this book that we need<br />
to underst<strong>and</strong> the classifying spaces <strong>of</strong> some categories. Thus<br />
we provide two examples on subdivisions <strong>of</strong> categories.<br />
Example 2.4.5.1. Let C be the following EI category<br />
G<br />
x<br />
α <br />
y H .<br />
Then its underlying poset [C] is x → y, <strong>and</strong> S([C]) is the<br />
poset [x] ← [x → y] → [y]. One can easily find that ˜p[x]<br />
<strong>and</strong> ˜p[y] are<br />
G x , y H<br />
respectively. The groupoid ˜p[x → y] is<br />
G×H {x α <br />
y} ,<br />
in which we denote by α the only object in ˜p[x → y]. The two<br />
functors ˜p[x → y] → ˜p[x] ˜p[x → y] → ˜p[y] are the obvious<br />
projections. Thus the category S(C) has objects <strong>of</strong> the forms<br />
([x], x), ([y], y) <strong>and</strong> ([x → y], x α →y). All the morphisms in<br />
S(C) are <strong>of</strong> the following forms<br />
•([x → y] → x, g), ∀g ∈ Mor(˜p[x]) = Aut C (x) = G;<br />
•([x → y] → y, h), ∀h ∈ Mor(˜p[y]) = Aut C (y) = H;<br />
•(1 [x→y] , (g ′ , h ′ )), ∀(g ′ , h ′ ) ∈ Mor(˜p[x → y]) = Aut C (x) ×<br />
Aut C (y) = G × H.<br />
As one can see the subdivision S(C) is more complicated<br />
than C itself so it seems not helpful in terms <strong>of</strong> underst<strong>and</strong>ing<br />
BC through BS(C). However, in the next section, we shall<br />
see it does help if we combine with Thomason’s Homotopy<br />
Colimit Theorem.<br />
For future reference, we record two more finite EI categories,<br />
which share the same underlying poset with C. The reader<br />
may try to figure out their subdivisions. We shall give only<br />
partial descriptions <strong>of</strong> these constructions.
2.4 Further categorical <strong>and</strong> simplicial constructions 79<br />
2. Let D be the following category<br />
h<br />
x<br />
1 x<br />
α<br />
β<br />
y {1 y }<br />
,<br />
where h 2 = 1 x <strong>and</strong> βh = α. Then we also have poset [D] =<br />
x → y, <strong>and</strong> S([D]) = [x] ← [x → y] → [y]. We can find ˜p[x]<br />
<strong>and</strong> ˜p[y] are<br />
h<br />
1 x<br />
x , y {1 y }<br />
respectively. The third groupoid ˜p[x → y] has two isomorphic<br />
objects x α →y <strong>and</strong> x β →y. Its skeleton is the trivial category.<br />
3. Let E be the following EI category<br />
h<br />
g<br />
x<br />
1 x<br />
α<br />
β<br />
gh<br />
y {1 y }<br />
,<br />
where g 2 = h 2 = 1 x , gh = hg, αh = α, αg = β, βh =<br />
β, βg = β. Then we again have poset [E] = x → y, <strong>and</strong><br />
S([E]) = [x] ← [x → y] → [y]. By direct calculations, ˜p[x]<br />
<strong>and</strong> ˜p[y] are<br />
h<br />
g<br />
x<br />
1 x<br />
<br />
, y {1 y } <br />
gh<br />
respectively. The groupoid ˜p[x → y] has two isomorphic<br />
objects x→y α <strong>and</strong> x→y. β However in this case, its skeleton<br />
is isomorphic to C 2 = {1 x , h}, which is the subgroup <strong>of</strong><br />
Aut C (x) × Aut C (y) that stabilizes the set {α, β}.<br />
We will come back to these examples at the end <strong>of</strong> the next<br />
section.
80 2 Simplicial methods<br />
2.4.2 Bisimplicial sets <strong>and</strong> homotopy colimits<br />
Definition 2.4.6. A bisimplicial set is a contravariant functor<br />
△ × △ to Set.<br />
Because<br />
Hom Cat (△ op ×△ op , SimpSet) ∼ = Hom Cat (△ op , Hom Cat (△ op , SimpSet<br />
clearly we may regard a bisimplicial set as a simplicial object<br />
in the category <strong>of</strong> simplicial sets. Thus a bisimplicial set is<br />
indeed a simplicial simplicial set.<br />
Alternatively, a bisimplicial set is a bigraded sequence <strong>of</strong> sets<br />
X p,q for p, q ≥ 0, together with horizontal face <strong>and</strong> degeneracy<br />
maps d h : X p,q → X p−1,q <strong>and</strong> s h : X p,q → X p+1,q as well as<br />
vertical face <strong>and</strong> degeneracy maps d v : X p,q → X p,q−1 <strong>and</strong> s v :<br />
X p,q → X p,q+1 . These maps must satisfy simplicial identities<br />
in either direction, <strong>and</strong> in addition every horizontal map must<br />
commutes with every vertical map. The total complex <strong>of</strong> a<br />
bisimplicial set, Tot(X), is the simplicial set<br />
Tot(X) n := ⊕ n≥0<br />
X n,∗ × △ n / ∼,<br />
in which the equivalence is given by (d h i x, y) ∼ (x, di y) <strong>and</strong><br />
(s h i x, y) ∼ (x, si y).<br />
We can also consider bisimplicial objects in an abelian category<br />
A-Mod for some associative ring. Then by Dold-Kan<br />
Correspondence, the category <strong>of</strong> bisimplicial A-modules is<br />
equivalent to the category <strong>of</strong> first quadrant double complexes<br />
<strong>of</strong> A-modules.<br />
Fix a commutative ring R. Each bisimplicial set gives rise<br />
to a first quadrant double complex R[X] p,q with differentials<br />
∂ h p,q = ∑ i (−1)i d h i <strong>and</strong> ∂ v p,q = (−1) p ∑ i (−1)i d v i . Consequently
2.4 Further categorical <strong>and</strong> simplicial constructions 81<br />
the category <strong>of</strong> bisimplicial R-modules is equivalent to the category<br />
<strong>of</strong> first quadrant double chain complexes <strong>of</strong> R-modules.<br />
Definition 2.4.7. The diagonal, diagX, <strong>of</strong> a bisimplical set<br />
X is the simplicial set obtained via restriction along the diagonal<br />
functor △ → △ × △.<br />
Theorem 2.4.8 (Bousfield-Kan ). Let X, Y be two bisimplicial<br />
sets.<br />
1. There is a natural simplicial isomorphism Tot(X) ∼ = diagX.<br />
2. Given a bisimplicial map Φ p,q : X p,q → Y p,q , if for all q,<br />
X ∗,q → Y ∗,q is a weak homotopy equivalence, then so is<br />
diagΦ : diagX → diagY .<br />
Pro<strong>of</strong>. For the first part, we define a map Ψ, induced by the<br />
maps X p,∗ × △ p → diagX p given by<br />
(x, d p y) ↦→ d h px ∈ X p,p ,<br />
for any (x, d p y) ∈ X p,q × △ p n. One may readily check that Ψ<br />
is a simplicial isomorphism.<br />
As for Part (2), we will not provide a pro<strong>of</strong> here but it is<br />
not surprising from Part (1). See [9], [Tornehave], [Reedy] <strong>and</strong><br />
[8, 16].<br />
Definition 2.4.9. Suppose G : C → SimpSet is a functor.<br />
⊕Then the simplicial replacement <strong>of</strong> G is the bisimplicial set<br />
∗<br />
G which in dimension n consists <strong>of</strong> the coproduct<br />
( ⊕ G) ⊕<br />
n :=<br />
G(x 0 ).<br />
∗<br />
x 0 →···→x n ∈NC n<br />
The horizontal face map d i maps G(s(σ)) to G(s(d i σ)) by the<br />
identity map if i > 0 <strong>and</strong> by α 0 if i = 0. The horizontal degeneracy<br />
map s i maps the simplicial set G(s(σ)) to G(s(s i σ))<br />
by the identity map.
82 2 Simplicial methods<br />
Definition 2.4.10. Let G : C → SimpSet be a functor.<br />
The homotopy colimit <strong>of</strong> G, hocolim C G, is diag( ⊕ ∗ G).<br />
For example one can easily see that hocolim C Npt = NC.<br />
Theorem 2.4.11 (Thomason’s Homotopy Colimit Theorem).<br />
Suppose C is a small category <strong>and</strong> F : C → Cat is<br />
a functor. Let Gr C F be the Grothendieck construction on F.<br />
Then there is a natural weak equivalence<br />
NGr C F ≃ hocolim C NF.<br />
The pro<strong>of</strong> is in the next section. It is clear from the pro<strong>of</strong> <strong>of</strong><br />
Thomason’s theorem, <strong>and</strong> Definitions 2.2.2 <strong>and</strong> 2.4.9 that<br />
BGr C F ≃ hocolim C BF.<br />
Remark 2.4.12. The homotopy colimit is a replacement in<br />
T op <strong>of</strong> the colimit we introduced earlier. In the category T op,<br />
the colimit <strong>of</strong> a diagram <strong>of</strong> spaces does exit. However it does<br />
not behave very well <strong>and</strong> thus is not good to work with in algebraic<br />
topology. One can find a well-known example in [DH].<br />
There is also a dual notion <strong>of</strong> homotopy limit but we shall<br />
not touch it in this book.<br />
The following examples provide the only three homotopy<br />
colimits that the reader needs to know for our purposes. In<br />
fact, they are only used in Section 4.2.2, for computing certain<br />
ordinary (co)homology.<br />
Example 2.4.13. When C is the poset P : a ← b → c, for any<br />
F : P → Cat the homotopy colimit hocolim P BF is called a<br />
homotopy pushout because it fits into the right lower corner<br />
<strong>of</strong> the following diagram
2.4 Further categorical <strong>and</strong> simplicial constructions 83<br />
BF(b→a)<br />
<br />
BF(b) BF(b→c) <br />
BF(a)<br />
BF(c)<br />
<br />
hocolim P BF<br />
such that it is commutative up to homotopy <strong>of</strong> continuous<br />
maps, <strong>and</strong> hocolim C BF enjoys a certain universal property,<br />
analogous to that <strong>of</strong> a colimit.<br />
Example 2.4.14. In Example 2.4.5, the classifying spaces <strong>of</strong><br />
the subdivisions <strong>of</strong> all three categories can be realized as homotopy<br />
pushouts, according to Example 2.4.13. For instance,<br />
in Example 2.4.5 (3) we have BE ≃ B(C 2 ×C 2 )/BC 2 because<br />
we have the following homotopy pushout diagram<br />
inclusion<br />
B˜p([x → y]) = BC<br />
<br />
2 B(C 2 × C 2 ) = B˜p([x])<br />
Bpt<br />
<br />
B˜p([y]) = B•<br />
<br />
B(C 2 × C 2 )/BC 2 ≃ hocolim P B˜p ≃ BS(E<br />
Similarly, in Example 2.4.5 (1), we get BC ≃ BG ∗ BH (a<br />
join <strong>of</strong> spaces), <strong>and</strong> in Example 2.4.5 (2) BD ≃ BC 2 . At<br />
least in these small examples, subdivisions are useful to obtain<br />
the homotopy type <strong>of</strong> classifying spaces <strong>of</strong> categories. This is<br />
one <strong>of</strong> the the reasons why we introduced the Grothendieck<br />
construction earlier.<br />
2.4.3 Pro<strong>of</strong>s <strong>of</strong> Quillen’s Theorem A <strong>and</strong> Thomason’s theorem<br />
The pro<strong>of</strong>s are taken directly from the original papers <strong>of</strong><br />
Quillen [61] <strong>and</strong> Thomason [77].<br />
In order to prove Quillen’s Theorem A, we shall first generalize<br />
the construction <strong>of</strong> category <strong>of</strong> factorizations.<br />
Definition 2.4.15. Suppose u : D → C is a functor between<br />
two small categories. We define F (u) to be a category
84 2 Simplicial methods<br />
whose objects are (a, α, x) with a ∈ Ob D, x ∈ Ob C <strong>and</strong><br />
α ∈ Hom C (x, u(a)). A morphism from (a, α, x) to (b, β, y)<br />
is a pair <strong>of</strong> morphisms (f, γ) with f ∈ Hom D (a, b) <strong>and</strong><br />
γ ∈ HomC(y, x) such that β = u(f)αγ as in the diagram<br />
u(a)<br />
u(f)<br />
<br />
u(b)<br />
Note that there exists two natural functors P t : F (u) →<br />
D op <strong>and</strong> P s : F (u) → C, as well as P t × P s : F (u) →<br />
D × C op . For any category C, F (Id C ) is exactly the category<br />
<strong>of</strong> factorizations in C, F (C), such that P s = s <strong>and</strong> P t = t.<br />
Let us notice that any element (a 0 , α 0 , x 0 ) → (a 1 , α 1 , x 1 ) →<br />
· · · → (a n , α n , x n ) ∈ NF (u) n can be pictured as a commutative<br />
diagram<br />
x 0<br />
α 0 <br />
x 1<br />
α 1 <br />
α <br />
β<br />
x<br />
y<br />
γ<br />
· · ·<br />
x n<br />
α n <br />
u(a 0 )<br />
<br />
u(a 1 )<br />
<br />
· · ·<br />
<br />
u(a n )<br />
Pro<strong>of</strong> (<strong>of</strong> Quillen’s Theorem A). We will only prove the case<br />
where all undercategories associated wtih u are contractible.<br />
The other case can be shown similarly.<br />
Let F (u) be as above. We define a functor N[Id C /u(−)] ∗ :<br />
⊕D → SimpSet <strong>and</strong> consider its simplicial replacement X =<br />
∗ N[Id C/u(−)] ∗ the bisimplicial set with X p,q consists <strong>of</strong><br />
pairs <strong>of</strong> sequences <strong>of</strong> morphisms in C <strong>and</strong> D, respectively,<br />
(x p → · · · → x 0 → u(a 0 ), a 0 → a 1 → · · · → a q ).<br />
The horizontal <strong>and</strong> vertical face <strong>and</strong> degeneracy maps are<br />
given in the obvious way. We have a simplicial map X ∗q →<br />
ND q as well as a simplicial map X p∗ → NCp op . Since the
2.4 Further categorical <strong>and</strong> simplicial constructions 85<br />
diagonal <strong>of</strong> X is the nerve <strong>of</strong> F (u), these two functors provide<br />
two more simplicial maps<br />
⊕<br />
N(Id C /u(a 0 )) op<br />
q → ⊕<br />
N• q = ND q<br />
a 0 →···→a q a 0 →···→a q<br />
<strong>and</strong> ⊕<br />
N(x 0 \u) p →<br />
⊕<br />
N• p = NCp op .<br />
x p →···→x 0 x p →···→x 0<br />
The geometric realizations <strong>of</strong> these two maps are homotopy<br />
equivalences because both Id D /u(a 0 ) <strong>and</strong> x 0 \u are contractible<br />
(the latter by assumption). Hence by Theorem 2.4.7<br />
(2) we obtain two homotopy equivalences<br />
BD<br />
BP t<br />
BP<br />
BF (u)<br />
s<br />
BC op .<br />
By examining the following diagram<br />
C op<br />
P s<br />
C op<br />
P s =s<br />
F (u)<br />
u ′<br />
F (C)<br />
P t<br />
D<br />
u<br />
P t =t C<br />
where u ′ is defined by u ′ (a, α, x) = (u(a), α, x). It forces Bu ′<br />
<strong>and</strong> thus Bu to be homotopy equivalences.<br />
⊓⊔<br />
The pro<strong>of</strong> <strong>of</strong> Thomason’s homotopy colimit theorem is in<br />
spirit similar to that <strong>of</strong> Quillen’s Theorem A. We remind the<br />
reader that NF (u) ≃ diagX = hocolim D N[Id C /u(−)] in the<br />
above pro<strong>of</strong>, <strong>and</strong> one verifies that Gr D Id C /u(−) ∼ = F (u).<br />
Pro<strong>of</strong> (<strong>of</strong> Thomason’s theorem). We shall establish a natural<br />
map<br />
η : hocolim C NF → NGr C F<br />
<strong>and</strong> construct a new functor ˜F : C → Cat so that we have<br />
natural equivalences
86 2 Simplicial methods<br />
λ 1<br />
λ<br />
hocolim C NF<br />
<br />
hocolim C N ˜F 2 <br />
NGr C F<br />
satisfying ηλ 1 ≃ λ 2 .<br />
Step 1: Similar to the pro<strong>of</strong> <strong>of</strong> Quillen’s Theorem A, hocolim C NF<br />
is the diagonal <strong>of</strong> ⊕ ∗<br />
NF such that for any integer n ≥ 0,<br />
( ⊕ ∗ NF) n consists <strong>of</strong> n-simplicies<br />
α<br />
(x<br />
1 α<br />
0→ · · ·<br />
n f 1 f n<br />
→xn , a 0→ · · · →an )<br />
where x 0 → · · · → x n ∈ NC n <strong>and</strong> a 0 → · · · → a n ∈ NF(x 0 ) n .<br />
We define η by<br />
α<br />
η n (x<br />
1 α<br />
0→ · · ·<br />
n f 1 f n<br />
→xn , a 0→ · · · →an )<br />
= (x 0 , a 0 ) (α 1,F(α 1 )(f 1 ))<br />
−→ (x 1 , F(α 1 )(a 1 )) (α 2,F(α 2 α 1 )(f 2 ))<br />
−→ · · ·<br />
· · · (α n,F(α n···α 1 )(f n ))<br />
−→ (x n , F(α n · · · α 1 )(a n ))<br />
Step 2: Let us construct λ 1 . We define ˜F : C → Cat to be<br />
the functor<br />
˜F(x) = π/x<br />
for π : Gr C F → C. Recall that the objects in ˜F(x) are <strong>of</strong> the<br />
form ((y, a), α) such that y ∈ Ob C, a ∈ Ob F(y) <strong>and</strong> α : y →<br />
x, <strong>and</strong> moreover any α : y → x induces ˜F(α) : ˜F(y) → ˜F(x).<br />
There exists a canonical functor<br />
˜F(x) → F(x)<br />
for any x ∈ Ob C, given by ((y, a), α) ↦→ F(α)(a). This functor<br />
has a right adjoint F(x) → ˜F(x) by a ↦→ ((x, a), 1x ). Consequently<br />
N ˜F(x) → NF(x) is a simplicial homotopy equivalence.<br />
Since all functors constructed previously, ˜F(x) → F(x),<br />
assemble to a natural transformation
2.4 Further categorical <strong>and</strong> simplicial constructions 87<br />
˜F → F<br />
<strong>of</strong> functors C → Cat, we obtain a simplicial homotopy equivalence<br />
N ˜F → NF which gives rise to a homotopy equivalence,<br />
by Theorem 2.4.7 (2),<br />
λ 1 : hocolim C N ˜F<br />
≃<br />
→hocolimC NF.<br />
Step 3: Now we show there is a simplicial homotopy equivalence<br />
λ 2 : hocolim C N ˜F → GrC F.<br />
Similar to Step 1, hocolim C N ˜F is the diagonal <strong>of</strong><br />
⊕∗ N ˜F such<br />
that for any integer n ≥ 0, ( ⊕ ∗ N ˜F)n consists <strong>of</strong> n-simplicies<br />
α<br />
(x<br />
1 α<br />
0→ · · ·<br />
n (h 1 ,l 1 )<br />
→xn , c 0 → · · · (h n,l n )<br />
→ c n )<br />
where x 0 → · · · → x n ∈ NC n <strong>and</strong> c 0 → · · · → c n ∈<br />
N ˜F(x0 ) n = [N(π/x 0 )] n . Note that each c i is <strong>of</strong> the form<br />
((y i , b i ), β i ) such that y i ∈ Ob C, b i ∈ Ob F(y i ) <strong>and</strong> β i : y i →<br />
x 0 . Moreover, h i+1 : y i → y i+1 satisfies β i+1 h i+1 = β i , <strong>and</strong><br />
l i+1 : F(h i+1 )(b i ) → b i+1 , for every i ≥ 0. Note that there are<br />
morphisms α i · · · α 1 β i : y i → x i . We define λ 2 by<br />
α<br />
λ 2 (x<br />
1 α<br />
0→ · · ·<br />
n<br />
→xn , ((y 0 , b 0 ), β 0 ) (h 1,l 1 )<br />
→ · · · (h n,l n )<br />
→ ((y n , b n ), β n ))<br />
= (y 0 , b 0 ) (h 1,l 1 )<br />
→ · · · (h n,l n )<br />
→ (y n , b n ).<br />
Let us consider NGr C F as a bisimplicial set with (p, q)-<br />
simplicies NGr C F p,q = NGr C F q . This means it is constant<br />
in horizontal direction. Clearly NGr C F ∗ = diagNGr C F ∗,∗ .<br />
The simplicial map λ 2 is the diagonalization <strong>of</strong> an obvious<br />
bisimlicial map λ : ⊕ ∗ Gr C ˜F → NGr C F ∗,∗ .<br />
Thus, by Theorem 2.4.7 (1), we only have to show for each q,<br />
λ ∗,q : ⊕ ∗ N ˜F∗,q → NGr C F ∗,q is a homotopy equivalence. But<br />
λ ∗,q is the coproduct over all q-simplicies (x 0 , c 0 ) → · · · →
88 2 Simplicial methods<br />
(x q , c q ) <strong>of</strong> NGr C F <strong>of</strong> the map ⊕ p x q\Id C → ⊕ p<br />
•. Then by<br />
Theorem 2.4.7 (2) λ ∗,q , <strong>and</strong> so λ 2 , are homotopy equivalences.<br />
Step 4: Finally we establish a simplicial hopmotopy<br />
: hocolim C N ˜F × N1 → NGrC F<br />
from ηλ 1 to λ 2 . We define it by<br />
(i+1) zeros<br />
α<br />
(x<br />
1 α<br />
0→ · · ·<br />
n<br />
{ }} {<br />
→xn , ((y 0 , b 0 ), β 0 ) → · · · → ((y n , b n ), β n ))×( 0, · · · , 0, 1, · · ·<br />
= (y 0 , b 0 ) (h 1,l 1 )<br />
→ · · · (h i,l i )<br />
→ (y i , b i ) (α i+1···α 1 β i ,F(α i+1···α 1 β i+1 )(l i+1 ))<br />
−→<br />
(x i+1 , F(α i+1 · · · α 1 β i+1 )(b i+1 )) (α i+2,F(α i+2···α 1 β i+2 )(l i+2 ))<br />
−→<br />
(x i+2 , F(α i+2 · · · α 1 β i+2 )(b i+2 )) → · · · → (x n , F(α n · · · α 1 β n )(b n )).<br />
One may verify that all simplicial identities are met. Hence<br />
we are done.<br />
⊓⊔
Chapter 3<br />
Category algebras <strong>and</strong> their representations<br />
Abstract The concept <strong>of</strong> a category algebra is introduced<br />
here. The relation ship between the module category <strong>of</strong> a category<br />
algebra <strong>and</strong> an appropriate functor category is demonstrated<br />
at the very beginning. We shall characterize the intrinsic<br />
properties <strong>of</strong> category algebras. Especially we show<br />
how one may compare a category algebra with a cocommutative<br />
bialgebra so such algebras enjoy similar homological<br />
properties. We are particularly interested in finite EI category<br />
algebras because we may classify their indecomposable<br />
projective <strong>and</strong> simple modules. When dealing with category<br />
algebras, the usual homological tools, such as induction <strong>and</strong><br />
coinduction, are replaced by Kan extensions. Examples are<br />
supplied to show how one may compute Kan extensions <strong>of</strong><br />
various modules.<br />
Throughout this chapter, the base ring R is always a commutative<br />
ring with identity. A module will normally be a left<br />
module, unless otherwise specified. If S is a set, then RS<br />
st<strong>and</strong>s for the free R-module generated by S.<br />
3.1 Basic concepts <strong>and</strong> examples<br />
3.1.1 Category algebras<br />
The category algebra is a natural way to linearize a category.
90 3 Category algebras <strong>and</strong> their representations<br />
Definition 3.1.1. Let C be a small category <strong>and</strong> R a commutative<br />
ring. The category algebra RC is a free R-module<br />
whose basis is the set <strong>of</strong> morphisms <strong>of</strong> C. We define a product<br />
on the basis elements <strong>of</strong> RC by<br />
{<br />
α ◦ β, if α <strong>and</strong> β can be composed in C;<br />
α ∗ β =<br />
0 , otherwise<br />
<strong>and</strong> then extend this product linearly to all elements <strong>of</strong> RC.<br />
With this product, RC becomes an associative R-algebra.<br />
If Ob C is finite, it is easy to see that ∑ x∈Ob C 1 x is the identity<br />
<strong>of</strong> RC where 1 x is the identity <strong>of</strong> Aut C (x). If a category<br />
C is finite, then Ob C is finite <strong>and</strong> RC is <strong>of</strong> finite R-rank.<br />
Another way to linearize a category C over a ring R is to<br />
construct a new category C R whose objects are the same as C<br />
while the morphisms between any two objects x, y ∈ C R are<br />
Hom CR (x, y) := RHom C (x, y). This category C R is additive<br />
<strong>and</strong> one can obtain the category algebra RC by forgetting the<br />
category structure in an obvious way.<br />
Example 3.1.2. Suppose C is the following finite category<br />
1 x <br />
x<br />
with g 2 = 1 y <strong>and</strong> α = gα. Then the category algebra RC is<br />
<strong>of</strong> rank 4 with identity 1 RC = 1 x + 1 y . It contains two group<br />
algebras R{1 x } <strong>and</strong> R{1 y , g}.<br />
We say C is connected if C as a (directed) graph is connected.<br />
Every category C can be written as the disjoint union<br />
<strong>of</strong> connected components C = ⊎ i∈J C i , where each C i is a connected<br />
full subcategory <strong>and</strong> J is an index set. As a consequence<br />
the category algebra RC becomes a direct product <strong>of</strong><br />
α<br />
1 y<br />
<br />
<br />
y<br />
g
3.1 Basic concepts <strong>and</strong> examples 91<br />
ideals RC i , i ∈ J. Thus in order to study the properties <strong>of</strong> RC<br />
it suffices to study the properties <strong>of</strong> each RC i . For simplicity<br />
<strong>and</strong> some technical reasons we <strong>of</strong>ten make the connectedness<br />
assumption.<br />
3.1.2 <strong>Representations</strong> <strong>of</strong> categories <strong>and</strong> Mitchell’s Theorem<br />
We shall show that a fundamental property <strong>of</strong> the category<br />
algebra RC is that it provides a mechanism for investigating<br />
R-representations <strong>of</strong> C, which we define now.<br />
Definition 3.1.3. An (R-)representation <strong>of</strong> a category C is<br />
a (covariant) functor M : C → R-mod.<br />
When the base ring is understood, we <strong>of</strong>ten abbreviate an<br />
R-representation as a representation <strong>of</strong> C. All representations<br />
<strong>of</strong> C form a functor category (R-mod) C , which is an abelian<br />
category with enough projectives <strong>and</strong> injectives so we can<br />
talk about subfunctors <strong>and</strong> quotient functors. The following<br />
fundamental theorem <strong>of</strong> category representations tells us an<br />
alternative description <strong>of</strong> the functor category.<br />
Theorem 3.1.4 (Mitchell). For any small category C with<br />
finitely many objects, there exist functors ι : (R-mod) C →<br />
RC-mod <strong>and</strong> σ : RC-mod → (R-mod) C such that<br />
1. σ ◦ ι ∼ = Id (R-mod)<br />
C, <strong>and</strong><br />
2. ι is fully faithful, <strong>and</strong> moreover if Ob C is finite then ι◦σ ∼ =<br />
Id RC-mod .<br />
Thus if Ob C is finite, the R-representations <strong>of</strong> C can be<br />
identified with the unital RC-modules.<br />
Pro<strong>of</strong>. Assume F : C → R-mod is a representation <strong>of</strong> C.<br />
We construct a free R-module M F = ⊕ x∈Ob C<br />
F (x). For any<br />
m ∈ F (x) <strong>and</strong> morphism α ∈ Mor C, we ask α·m = F (α)(m),<br />
if t(α) = x, or α · m = 0 if t(α) ≠ x. By extending this
92 3 Category algebras <strong>and</strong> their representations<br />
operation linearly we obtain an RC-module structure on M F .<br />
This construction can be easily verified to define a functor<br />
ι : (R-mod) C → RC-mod.<br />
Conversely, if M is an RC-module, we may define a functor<br />
F M by F M (x) = 1 x · M. Since, if α ∈ Hom C (x, y) <strong>and</strong> m ∈<br />
1 x · M, α · m = (1 y ◦ α) · m = 1 y · (α · m) ∈ 1 y · M, we see<br />
that F M : C → R-mod is well defined. It induces a functor<br />
σ : RC-mod → (R-mod) C .<br />
We may readily check these two functors satisfy 1 <strong>and</strong> 2. ⊓⊔<br />
Similar statements can be made between right RC-modules<br />
<strong>and</strong> contravariant functors from C to R-mod.<br />
Remark 3.1.5. Throughout these notes we will be particularly<br />
interested in RC-modules lying in ι{(R-mod) C }. One reason<br />
is that many important modules are indeed <strong>of</strong> this form. Another<br />
reason is that, on top <strong>of</strong> module-theoretic methods, we<br />
may apply simplicial or homotopy-theoretic tools. Hence we<br />
want to focus on categories with finitely many objects, although<br />
many results make sense for arbitrary small categories.<br />
Moreover since there is a well developed representation theory<br />
<strong>of</strong> finite-dimensional algebras, from now on we will restrict<br />
our attention to the finite categories. Consequently for practical<br />
reasons we will not distinguish between RC-mod <strong>and</strong><br />
(R-mod) C , <strong>and</strong> normally refer to an object in these categories<br />
as an RC-module.<br />
Among all RC-modules, there is a distinguished one that we<br />
introduce below. It plays a key role throughout these notes.<br />
Definition 3.1.6. For any category C, the constant functor<br />
or trivial module R : C → R-mod, is defined by R(x) = R<br />
for all x ∈ Ob C <strong>and</strong> R(α) = Id R for all α ∈ Mor C.
3.1 Basic concepts <strong>and</strong> examples 93<br />
We provide several examples <strong>of</strong> simplicially constructed<br />
modules where both the module <strong>and</strong> functor aspects are described.<br />
Example 3.1.7.1. The constant functor R corresponds to the<br />
module R Ob C, on which RC acts via α · x = y, if α ∈<br />
Hom C (x, y), or zero otherwise.<br />
2. The regular module RC corresponds to a functor such that<br />
RC(x) = 1 x · RC = RHom C (−, x).<br />
3. The opposite algebra (RC) op is isomorphic to the algebra <strong>of</strong><br />
the opposite category C op , <strong>and</strong> thus a set <strong>of</strong> base elements<br />
are Mor C op . Consider the enveloping category <strong>of</strong> C, namely<br />
C e = C × C op . Then we have<br />
RC e ∼ = (RC) e = (RC) ⊗ R (RC) op ,<br />
the enveloping algebra <strong>of</strong> RC. Consequently RC is an RC e -<br />
module via (α, β op ) · γ = αγβ, for any α, β, γ ∈ Mor C. As<br />
a functor RC : C e → R-mod, we have<br />
RC(y, x) = RHom C (x, y)<br />
if (x, y) ∈ Ob C e .<br />
Occasionally we will need to discuss right RC-modules. A<br />
dual version <strong>of</strong> Mitchell’s theorem says that the right RCmodules<br />
corresponds to the contra-variant functors/representations<br />
<strong>of</strong> C. There exists a natural functor (−) ∧ = Hom R (−, R) :<br />
RC-mod → mod-RC. We give an explicit description <strong>of</strong> the<br />
dual module M ∧ = Hom R (M, R) (a right RC-module) <strong>of</strong><br />
M ∈ RC-mod. In case R is a field, ∧ induces an antiisomorphism<br />
between RC-mod <strong>and</strong> mod-RC.<br />
Lemma 3.1.8. Suppose M ∈mod-RC. Then its dual M ∧ ∈<br />
RC-mod has values M ∧ (x) = M(x) ∧ = Hom R (M(x), R),<br />
<strong>and</strong> each α ∈ Mor C acts via M(α) ∧ = Hom R (M(α)(−), R).
94 3 Category algebras <strong>and</strong> their representations<br />
Pro<strong>of</strong>. As a free R-module, M ∧ = ⊕ x∈Ob C M(x)∧ . Each α ∈<br />
Hom C (x, y) acts as M(α) : M(y) → M(x) <strong>and</strong> thus induces<br />
a map M(α) ∧ : M(x) ∧ → M(y) ∧ . One can readily check that<br />
M ∧ (x) is exactly M(x) ∧ so we know the structure <strong>of</strong> M ∧ as<br />
a right RC-module.<br />
⊓⊔<br />
The next statement follows from direct calculation.<br />
Corollary 3.1.9. We have R ∧ = R. Here the first R is a<br />
left module <strong>and</strong> the second is a right module.<br />
3.1.3 Three examples<br />
Let us consider three motivating examples.<br />
Example 3.1.10. Let G be a (discrete) group. Then the group<br />
can be regarded as a category with only one object •, whose<br />
morphisms are the elements <strong>of</strong> G. The group algebra RG is<br />
the same as the category algebra RG. A left RG-module M<br />
can be regarded as the representation <strong>of</strong> G given by a certain<br />
functor Φ : G → R-mod, sending • to M <strong>and</strong> Aut G (•) = G<br />
into Aut R (M). The trivial RG-module R is exactly the trivial<br />
module <strong>of</strong> RG.<br />
Note that in this case G ∼ = G op <strong>and</strong> thus RG ∼ = RG op <strong>and</strong><br />
RG e ∼ = RG ⊗ RG ∼ = R(G × G).<br />
A quiver q = (Γ 0 , Γ 1 ) is a directed graph having Γ 0 <strong>and</strong><br />
Γ 1 as the set <strong>of</strong> vertices <strong>and</strong> the set <strong>of</strong> arrows, respectively.<br />
The path algebra <strong>of</strong> q can be thought as a category algebra<br />
as follows. Any directed graph G = (Γ 0 , Γ 1 ) may be used to<br />
generate a category C G on the same set Γ 0 <strong>of</strong> objects, where<br />
the morphisms <strong>of</strong> this category are the strings <strong>of</strong> composable<br />
arrows <strong>of</strong> G. It is called the free category generated by G.<br />
Example 3.1.11. The path algebra <strong>of</strong> a quiver q is the category<br />
algebra RC q <strong>of</strong> the free category C q .
3.1 Basic concepts <strong>and</strong> examples 95<br />
It is necessary to point out that, given a category, its category<br />
algebra is usually different from its path algebra, if we<br />
consider the category as a quiver at the same time. Nevertheless,<br />
there is a relationship between these two algebras, as we<br />
now explain.<br />
Proposition 3.1.12. Let q be a category. We may regard q<br />
as a quiver <strong>and</strong> form the free category C q over q. There is a<br />
natural functor u : C q → q, which extends to a surjective homomorphism,<br />
still denoted by u : RC q → Rq, from the path<br />
algebra <strong>of</strong> q to the category algebra <strong>of</strong> q, such that its kernel<br />
I is generated by { α 1<br />
← α 2<br />
← − α ←<br />
1α 2<br />
}, where α1 <strong>and</strong> α 2 are arrows<br />
<strong>of</strong> q. This epimorphism induces a natural isomorphism <strong>of</strong><br />
R-algebras RC q /I <strong>and</strong> Rq.<br />
Pro<strong>of</strong>. The functor u is defined as follows. For each x ∈ Ob C q ,<br />
φ(x) = x. For each α ∈ Mor C q , u(α) = the composite <strong>of</strong><br />
the maps in the string α. It can be extended linearly to an<br />
epimorphism u : RC q → Rq, having the kernel I. ⊓⊔<br />
The last example <strong>of</strong> a category algebra is the incidence algebra<br />
<strong>of</strong> a locally finite poset (partially ordered set). A (closed)<br />
interval <strong>of</strong> P is a subposet [x, y] ⊂ P which consists <strong>of</strong> all<br />
objects z such that x ≤ z ≤ y for a given pair <strong>of</strong> objects<br />
x, y in P. The incidence algebra I(P, R) is an R-algebra <strong>of</strong> all<br />
functions f : Int(P) → R, where Int(P) is the set <strong>of</strong> intervals<br />
<strong>of</strong> P. The multiplication (also called convolution) if defined<br />
by<br />
(fg)([x, y]) = ∑<br />
f([x, z])g([z, y]).<br />
x≤z≤y<br />
It has an identity δ such that<br />
{<br />
1, if x = y;<br />
δ([x, y]) =<br />
0 , otherwise.
96 3 Category algebras <strong>and</strong> their representations<br />
On the other h<strong>and</strong> since whenever x ≤ y in a poset we can<br />
replace ≤ with an arrow x → y, a poset is always a category.<br />
Example 3.1.13. Let P be a finite poset. Then the incidence<br />
algebra I(P, R) <strong>of</strong> P is isomorphic to the category algebra<br />
RP.<br />
3.2 A closed symmetric monoidal category<br />
Suppose C is finite. We describe the intrinsic structure <strong>of</strong> RC<br />
which make it comparable with a cocommutative bialgebra.<br />
Later on we shall see RC-mod possesses similar homological<br />
properties as the module category <strong>of</strong> a cocommutative bialgebra.<br />
3.2.1 Tensor structure <strong>and</strong> an intrinsic characterization <strong>of</strong> category algebras<br />
Suppose G is a group. The category RG-mod is a symmetric<br />
monoidal category equipped with a tensor product − ⊗ R −<br />
<strong>and</strong> a tensor identity R. The reason why RG-mod enjoys such<br />
good properties is that RG is a cocommutative Hopf algebra.<br />
For general properties <strong>of</strong> Hopf algebras, we refer the reader<br />
to []. For an arbitrary unital associative R-algebra there are<br />
maps<br />
1. There is a map µ : A ⊗ R A → A, called the multiplication.<br />
2. The following diagram is commutative (associativity)<br />
A ⊗ A ⊗ A µ⊗Id<br />
Id⊗µ <br />
A ⊗ A µ<br />
A ⊗ A<br />
µ<br />
3. There is a map ι : R → A, called the unit, <strong>and</strong> the following<br />
diagram is commutative<br />
<br />
A
3.2 A closed symmetric monoidal category 97<br />
R ⊗ A ι⊗Id <br />
A ⊗ A<br />
∼ =<br />
µ<br />
Id⊗ι<br />
<br />
∼ =<br />
A ⊗ R<br />
A<br />
If A = RG, it is also a counital coalgebra, which means, on<br />
top <strong>of</strong> the previous maps, we have the following extra maps,<br />
reversing maps in 1-3.<br />
4. There is a comultiplication, RG → RG ⊗ R RG.<br />
6. The following diagram is commutative (coassociativity)<br />
∆<br />
RG<br />
<br />
RG ⊗ RG<br />
∆ <br />
RG ⊗ RG<br />
∆⊗Id<br />
Id⊗∆<br />
<br />
RG ⊗ RG ⊗ RG,<br />
5. The group algebra has an augmentation map ɛ : RG → R,<br />
also called the counit.<br />
R ⊗ RG<br />
∼ =<br />
ɛ⊗Id<br />
RG<br />
∆ <br />
∼ =<br />
RG ⊗ RGId⊗ɛ RG ⊗ R<br />
Any R-module satisfying conditions 4-6 is called a counital<br />
coassociative coalgebra. Any R-algebra satisfying 4-6 is<br />
named a bialgebra.<br />
A group algebra is usually not commutative, but<br />
7. there is a twist map τ : RG ⊗ R RG → RG ⊗ R RG, given<br />
by τ(a ⊗ b) = b ⊗ a, such that the following diagram is<br />
commutative<br />
∆<br />
RG ⊗ RG<br />
RG<br />
τ<br />
∆<br />
<br />
RG ⊗ RG.
98 3 Category algebras <strong>and</strong> their representations<br />
An R-coalgebra or bialgebra satisfying 7 is called cocommutative.<br />
8. Moreover there is an antipode η : RG → RG in the sense<br />
that if δ(a) = ∑ i b i ⊗ c i then ∑ i b iη(c i ) = ∑ i η(b i)c i =<br />
ɛ(a)e.<br />
If a bialgebra has an antipode, then it is called a Hopf algebra.<br />
Remark 3.2.1. For a group algebra, η is explicitly given by<br />
g → g −1 . The comultiplication is given by g ↦→ g ⊗ g.<br />
A cocommutative bialgebra A has the significant property<br />
that A-mod is a symmetric monoidal category with tensor<br />
identity R. More precisely it means that for any two A-<br />
modules M <strong>and</strong> N, M ⊗ R N is still an A-module, M ⊗ R N ∼ =<br />
N ⊗ R M, <strong>and</strong> M ⊗ R R ∼ = M. In other words, A-mod inherits<br />
various constructions on R-mod. Note that the tensor identity<br />
R plays the role <strong>of</strong> both the unit <strong>and</strong> co-unit <strong>of</strong> A. If A<br />
is Hopf, then even Hom R (M, N) becomes an A-module such<br />
that<br />
Hom A (L ⊗ M, N) ∼ = Hom A (L, Hom R (M, N)).<br />
In the literature, Hom R (M, N) is called a function object or<br />
the internal hom. Furthermore when R = k is a field, A<br />
is cocommutative Hopf <strong>and</strong> M, N are finite-dimensional, we<br />
have an isomorphism <strong>of</strong> A-modules<br />
M ∧ ⊗ N ∼ = Hom R (M, N).<br />
Since R ∈ A-mod, M ∧ is a left A-module under the circumstance,<br />
satisfying (M ∧ ) ∧ ∼ = M.<br />
In what follows, we begin with a description about how R-<br />
mod gives rise to a symmetric monoidal category structure<br />
on RC-mod. Then we characterize the structure <strong>of</strong> RC using
3.2 A closed symmetric monoidal category 99<br />
some structure maps comparable to those <strong>of</strong> a cocommutative<br />
bialgebra. In this way we demonstrate why a category algebra<br />
<strong>and</strong> a cocommutative bialgebra, as well as their module categories,<br />
are similar yet different. It is the intrinsic structure <strong>of</strong> a<br />
category algebra that makes it a natural <strong>and</strong> interesting subject<br />
<strong>of</strong> investigation. We note that R-mod itself is the module<br />
category <strong>of</strong> the R-category algebra <strong>of</strong> the trivial category •,<br />
based on Mitchell’s theorem.<br />
Fix a finite category C. The so-called internal product on<br />
(R-mod) C ≃ RC-mod, in which the tensor product is denoted<br />
by ˆ⊗ R , is defined by (M ˆ⊗ R N)(x) = M(x) ⊗ R N(x)<br />
for any M, N ∈ RC-mod ≃ (R-mod) C <strong>and</strong> x ∈ Ob C. The<br />
module structure <strong>of</strong> M ˆ⊗N can be viewed as given by the comultiplication<br />
∆ : RC → RC ⊗ R RC, induced by the canonical<br />
diagonal functor ∆ : C → C × C whose action on each<br />
α ∈ Mor C is ∆(α) = α ⊗ α. One can easily verify that the<br />
trivial module R is the tensor identity with respect to ˆ⊗ R . For<br />
the sake <strong>of</strong> simplicity, we shall write ⊗ for ⊗ R , <strong>and</strong> ˆ⊗ for ˆ⊗ R ,<br />
throughout this book. We note that in the literature (see for<br />
example [25]), the symbol ⊗ is <strong>of</strong>ten used instead <strong>of</strong> ˆ⊗ for the<br />
internal tensor product <strong>of</strong> functors. The new notation ˆ⊗ is introduced<br />
because we need to distinguish between M ⊗ N <strong>and</strong><br />
M ˆ⊗N. In fact, as R-modules, the inclusion M ˆ⊗N ⊂ M ⊗N,<br />
for any RC-modules M <strong>and</strong> N, is <strong>of</strong>ten strict.<br />
As we mentioned above, the diagonal functor ∆ : C → C ×C<br />
induces a co-multiplication on the category algebra RC. The<br />
co-multiplication ∆ : RC → RC ⊗ RC almost gives us a coalgebra<br />
structure on RC except that there is not a suitable<br />
choice <strong>of</strong> a map RC → R which would serve as the co-unitary<br />
map. Recall from Example 3.1.7 (1) that the trivial module<br />
R can be realized by R Ob C. The next result states a natural<br />
augmentation map from RC to R Ob C.
100 3 Category algebras <strong>and</strong> their representations<br />
Lemma 3.2.2. There exists a surjective linear map<br />
defined on base elements by<br />
where t(α) is the target <strong>of</strong> α.<br />
RC ɛ ↠R Ob C = R,<br />
ɛ(α) = t(α),<br />
Pro<strong>of</strong>. The surjective map induces a RC-module structure on<br />
R Ob C as follows. If x ∈ Ob C <strong>and</strong> β ∈ Hom C (x ′ , y), then<br />
βx = y when x = x ′ , <strong>and</strong> βx = 0 otherwise.<br />
⊓⊔<br />
If we return to ∆, we realize that the image <strong>of</strong> it really<br />
lies in RC ˆ⊗RC, a subspace <strong>of</strong> RC ⊗ RC, <strong>and</strong> moreover<br />
∆ : RC → RC ˆ⊗RC becomes a RC-map, since RC ˆ⊗RC, unlike<br />
RC ⊗ RC, is a well-defined RC-module. The above observations<br />
hint that, in order to get a sound “co-algebra” structure,<br />
one needs to use ˆ⊗ other than ⊗ to define the structure<br />
maps. This motivates us to write down the following maps<br />
which resemble almost all <strong>of</strong> the structure maps for a cocommutative<br />
bialgebra. Abusing terminology, we adopt the same<br />
names for the structure maps <strong>of</strong> a category algebra, such as<br />
co-multiplication <strong>and</strong> co-unit, as their counterparts for a bialgebra.<br />
In a coalgebra, the augmentation map <strong>and</strong> co-unit are<br />
the same, so by analogy the map ɛ in the preceding lemma<br />
will occasionally be called the co-unit. We emphasize that the<br />
unit, given by the natural inclusion map R ∼ ι<br />
= R · 1 RC →RC,<br />
is different from the co-unit.<br />
Proposition 3.2.3. Let RC be the category algebra <strong>of</strong> a finite<br />
category C. Then we have the following R-linear maps:<br />
a co-multiplication ∆ : RC → RC⊗RC, defined by ∆( ∑ ∑<br />
α λ αα) =<br />
α λ αα ⊗ α, a co-unit ɛ : RC → R defined as above, <strong>and</strong> a<br />
twist map τ : RC ⊗RC → RC ⊗RC defined on base elements
3.2 A closed symmetric monoidal category 101<br />
by τ(α ⊗ α ′ ) = α ′ ⊗ α, such that the following diagrams are<br />
commutative: 1. co-associativity<br />
∆<br />
RC<br />
<br />
RC ⊗ RC<br />
∆ <br />
RC ⊗ RC<br />
2. co-unitary property<br />
R ˆ⊗RC<br />
∼ =<br />
ɛ⊗Id<br />
<strong>and</strong> 3. co-commutativity<br />
∆<br />
RC ⊗ RC<br />
∆⊗Id<br />
Id⊗∆ RC ⊗ RC ⊗ RC,<br />
RC<br />
∆ <br />
∼ =<br />
<br />
RC ˆ⊗RC<br />
Id⊗ɛ RC ˆ⊗R<br />
RC<br />
τ<br />
∆<br />
<br />
RC ⊗ RC.<br />
If we denote by µ the multiplication, then we have furthermore<br />
three commutative diagrams: 4. Multiplication <strong>and</strong> comultiplication<br />
RC ⊗ RC<br />
∆⊗∆ <br />
RC ⊗ RC ⊗ RC ⊗ RC<br />
µ<br />
RC<br />
∆ <br />
RC ⊗ RC<br />
Id⊗τ⊗Id<br />
5. Unit <strong>and</strong> co-multiplication:<br />
RC<br />
<strong>and</strong> 6. Unit <strong>and</strong> co-unit<br />
µ<br />
R ⊗ R<br />
∆<br />
ι⊗ι<br />
<br />
µ⊗µ<br />
<br />
RC ⊗ RC ⊗ RC ⊗ RC,<br />
<br />
RC ˆ⊗RC,
102 3 Category algebras <strong>and</strong> their representations<br />
RC<br />
ι<br />
R<br />
ɛ<br />
where the R-linear map η : R → R is defined by 1 ↦→<br />
∑<br />
x∈Ob C x,<br />
We note that the only missing diagram is the compatibility<br />
<strong>of</strong> multiplication <strong>and</strong> co-unit. The reason is that one usually<br />
cannot give R a meaningful algebra structure so that<br />
ɛ : RC → R becomes an algebra homomorphism. The existence<br />
<strong>of</strong> the above tensor structure ˆ⊗ is well known. Notably<br />
it has been used to define the internal product in functor<br />
homology theory, see for example [25].<br />
Remark 3.2.4. If we remove the finiteness condition on C, then<br />
there is no identity in the algebra kC. Nevertheless many <strong>of</strong><br />
the constructions are still valid, although in this case, we are<br />
forced to use the full subcategory V ect C k<br />
because kC-mod does<br />
not have a tensor structure.<br />
Remark 3.2.5. If C = P happens to be a poset, then there<br />
is another co-multiplication that one can find in [72]. Let<br />
α ∈ Mor P, one has ¯∆ : kP → kP ⊗ kP such that<br />
¯∆(α) = ∑ {β,γ|βγ=α}<br />
β ⊗ γ. Nevertheless it is possible to give<br />
an incidence algebra a Hopf algebra structure. The augmentation<br />
map ¯ɛ : kP → k is given by ¯ɛ(α) = 1 if α is an identity,<br />
or ¯ɛ(α) = 0 otherwise. The antipode kP → kP can also be<br />
explicitly constructed. Suppose α ∈ Mor P is a morphism in<br />
the poset. Then a factorization <strong>of</strong> α is a way <strong>of</strong> writing α as<br />
a product α 1 α 2 · · · α n for some non-identity α i ∈ Mor P. The<br />
length l(α) <strong>of</strong> α is the maximal n for such a factorization to<br />
exist. Then the antipode is given by α ↦→ ∑ β (−1)l(β) β, in<br />
which β runs over all possible factors <strong>of</strong> α.<br />
η<br />
<br />
R,
3.2 A closed symmetric monoidal category 103<br />
Indeed one can do this for the category algebras <strong>of</strong> the socalled<br />
(finite) Möbius categories. A main feature <strong>of</strong> a Möbius<br />
category is that each object only has one automorphism.<br />
It seems that RC can only be a (cocommutative) Hopf algebra<br />
if C is <strong>of</strong> the two extreme structures: either a group<br />
(all morphisms are invertible), or a Möbius category (identity<br />
morphisms are the only invertible morphisms) .<br />
Remark 3.2.6. Let A be an R-algebra. Although A-mod is not<br />
monoidal in general, the module category A e -mod is always<br />
equipped with a tensor product ⊗ A such that A is the tensor<br />
identity.<br />
When A = RC is a category algebra, (RC) e ∼ = RC e becomes<br />
the category algebra <strong>of</strong> the category C e := C × C op , <strong>and</strong> hence<br />
there are two distinct monoidal structures on RC e -mod. The<br />
monoidal structure given by ⊗ RC is more interesting to us, <strong>and</strong><br />
we shall deal with it in Chapter 5 on Hochschild cohomology.<br />
3.2.2 The internal hom<br />
Usually one can not define an antipode for a category algebra.<br />
This causes a problem when one attempts to define<br />
Hom k (M, N) as a sensible kC-module (generalizing the group<br />
case). We shall give a remedy below. Another relevant fact is<br />
that the product <strong>of</strong> two projective kC-modules is in general<br />
not projective, see Example 3.. These make many homological<br />
properties, such as the cohomology theory, <strong>of</strong> a finitedimensional<br />
category algebra different from those <strong>of</strong> a finitedimensional<br />
cocommutative Hopf algebra.<br />
Proposition 3.2.7 (Swenson). Suppose C is a finite category<br />
(or at least Ob C is finite). Let M, N ∈ RC-mod. Then<br />
we can define an internal hom Hom(M, N) ∈ RC-mod.
104 3 Category algebras <strong>and</strong> their representations<br />
Pro<strong>of</strong>. We want to define Hom(M, N) such that there is an<br />
isomorphism<br />
Hom RC (L ˆ⊗M, N) ∼ = Hom RC (L, Hom(M, N)).<br />
Let RC = RC · 1 x . Assume the above isomorphism. Then the<br />
right h<strong>and</strong> side is Hom RC (RC·1 x , Hom(M, N)) ∼ = Hom(M, N)(x),<br />
<strong>and</strong> the left h<strong>and</strong> side is Hom RC (RC · 1 x ˆ⊗M, N). Hence we<br />
may define Hom(M, N) by<br />
Hom(M, N)(x) = Hom RC (RC · 1 x ˆ⊗M, N).<br />
When C = G is a group, it is straightforward from Swenson’s<br />
definition that Hom(M, N) ∼ = Hom R (M, N) as RG-modules.<br />
Alternatively consider the diagonal functor ∆ : C → C × C.<br />
Because R(C ×C) ∼ = RC ⊗RC, for L, M, N ∈ RC-mod, L⊗M<br />
is a R(C × C)-module, <strong>and</strong><br />
Hom RC (L ˆ⊗M, N) = Hom RC (Res ∆ (L ⊗ M), N)<br />
∼ = HomR(C×C) (L ⊗ M, RK ∆ N)<br />
∼ = HomRC⊗RC (L ⊗ M, RK ∆ N)<br />
∼ = HomRC (L, Hom RC (M, RK ∆ N)).<br />
Here in Hom RC (M, RK ∆ N), RK ∆ N ∈ RC ⊗ RC-mod is<br />
considered as a (R · 1 RC ) ⊗ RC-module <strong>and</strong> hence a RCmodule.<br />
The RC ⊗ RC-, or rather the RC ⊗ (R · 1 RC )-, module<br />
structure on RK ∆ N provides a RC-module structure on<br />
Hom RC (M, RK ∆ N). Then we may verify that Hom(M, N) =<br />
Hom RC (M, RK ∆ N).<br />
With the internal hom, one might attempt to define a dual<br />
module <strong>of</strong> M by Hom(M, R) in order to generalize the group<br />
case. The following example explain why this is not a good<br />
idea.<br />
⊓⊔
3.3 Functors between module categories 105<br />
Example 3.2.8. Let k be a field <strong>of</strong> characteristic two <strong>and</strong> C<br />
the following category<br />
{1 x } <br />
x<br />
α<br />
β<br />
y<br />
{1 y ,g}<br />
with g 2 = 1 y , gα = α <strong>and</strong> gβ = β.<br />
We consider two modules S x,k <strong>and</strong> S y,k such that S x,k (x) =<br />
k, S x,k (y) = 0, S y,k (x) = 0 <strong>and</strong> S y,k (y) = k. Then Hom(S x,k , k)<br />
can be easily shown to be isomorphic to S x,k . However Hom(S y,k , k) ∼ =<br />
S 2 x,k ⊕ S y,k because kC1 x ˆ⊗S y,k<br />
∼ = S<br />
2<br />
y,k<br />
<strong>and</strong> kC1 y ˆ⊗S y,k<br />
∼ = kC1y .<br />
Thus Hom(Hom(S y,k , k), k) ≁ = Sy,k .<br />
3.3 Functors between module categories<br />
We investigate functors for comparing two module categories<br />
over category algebras.<br />
3.3.1 Restriction on algebras <strong>and</strong> modules<br />
We prove some basic properties, many <strong>of</strong> which follow directly<br />
from simple reasoning.<br />
Definition 3.3.1. Suppose u : D → C is a (covariant) functor.<br />
We define Res u : RC-mod → RD-mod to be the restriction<br />
along u. Given a module M ∈ RC-mod, we have<br />
Res u M = M ◦ u ∈ RD-mod.<br />
Lemma 3.3.2. Let u : D → C be a functor. Then Res u R =<br />
R.<br />
Since a functor u : D → C also extends linearly to a natural<br />
map <strong>of</strong> R-modules ū : RD → RC, it is reasonable to ask if<br />
ū is always an algebraic homomorphism because if it is, the<br />
so-called change-<strong>of</strong>-base-ring or the representation-theoretic<br />
restriction, ↓ RC<br />
RD : RC-mod → RD-mod, should coincide with
106 3 Category algebras <strong>and</strong> their representations<br />
Res u . The answer to the question is no, <strong>and</strong> here is a simple<br />
example. Let D = 1 <strong>and</strong> C = •. There is a unique functor pt :<br />
D → C. The induced map ¯pt : RD → RC is not an algebra<br />
homomorphism since the product <strong>of</strong> the two isomorphisms in<br />
D is zero while the product <strong>of</strong> their images is not.<br />
Proposition 3.3.3. A functor u : D → C extends linearly<br />
to an algebra homomorphism ū : RD → RC if <strong>and</strong> only<br />
if u is injective on Ob D. When this happens, the induced<br />
functor followed by 1 RD , 1 RD ◦ ↓ RC<br />
RD : RC-mod → RD-mod, is<br />
exactly Res u .<br />
Pro<strong>of</strong>. We know u(βα) = u(β)u(α) for any pair <strong>of</strong> composable<br />
morphisms α, β in D. The injectivity <strong>of</strong> u implies two<br />
morphisms α, β ∈ Mor(D) are composable if <strong>and</strong> only if<br />
u(α), u(β) ∈ Mor(C) are composable.<br />
If u is injective on Ob D, then we define a map ū : RD →<br />
RC as the linear extension <strong>of</strong> functor u, i.e., ū( ∑ ∑<br />
i r iα i ) =<br />
i r iū(α i ) for any r i ∈ R, α i ∈ Mor(D). This ū is indeed<br />
an algebra homomorphism because our previous observation<br />
<strong>of</strong> u implies ū(( ∑ j r jβ j )( ∑ i r iα i )) = ū( ∑ j r jβ j )ū( ∑ i r iα i ) is<br />
always true.<br />
On the other h<strong>and</strong> if the linear extension ū : RD → RC is<br />
an algebra homomorphism then we must have ū(0) = 0 <strong>and</strong><br />
then ū(1 x )ū(1 y ) = ū(1 x · 1 y ) = 0 unless x = y. This suggests<br />
that u is injective on Ob C ′ .<br />
When ū : RD → RC is an algebra homomorphism, we<br />
show 1 RD ◦ ↓ RC<br />
RD = Res u : RC-mod → RD-mod. Let M be an<br />
RC-module <strong>and</strong> γ ∈ Mor(D). Then it corresponds to some<br />
F M ∈ (R-mod) C . The RD-modules Res u M <strong>and</strong> 1 RD·(M ↓ RC<br />
RD )<br />
are isomorphic because Res u M = M Resu (F M ), γ · M = u(γ)M<br />
for any γ ∈ Mor D <strong>and</strong> 1 RD kills the elements <strong>of</strong> M ↓ RC<br />
RD that<br />
are not supported on any objects <strong>of</strong> D.<br />
⊓⊔
3.3 Functors between module categories 107<br />
Proposition 3.3.4. Let D <strong>and</strong> C be equivalent small categories.<br />
Then (R-mod) C ≃ (R-mod) D , an equivalence which<br />
sends the constant functor to the constant functor. If both<br />
Ob C <strong>and</strong> Ob D are finite then RC <strong>and</strong> RD are Morita equivalent.<br />
Pro<strong>of</strong>. We show that the two functor categories (R-mod) C <strong>and</strong><br />
(R-mod) D are equivalent. Then it implies the module categories<br />
RC-mod <strong>and</strong> RD-mod are equivalent, hence RC <strong>and</strong><br />
RD are Morita equivalent. In fact if u : D → C <strong>and</strong> v : C → D<br />
are equivalences, we have Res u Res v<br />
∼ = IdRD := Id RD-mod : (Rmod)<br />
D → (R-mod) D because <strong>of</strong> the following diagram<br />
M(vu(x)) = (Res u Res v M)(x) ∼ =<br />
(Id RD M)(x) = M(x)<br />
M(vu(α))=(Res u Res v M)(α)<br />
M(vu(y)) = (Res u Res v M)(y) ∼=<br />
(Id RD M)(α)=M(α)<br />
<br />
(Id RD M)(y) = M(y)<br />
where M ∈ (R-mod) D , α : x → y ∈ Mor D <strong>and</strong> Id RD is the<br />
identity functor. Similarly we can show Res v Res u<br />
∼ = IdRC : (Rmod)<br />
C → (R-mod) C . Clearly the constant functor restricts to<br />
the constant functor always.<br />
⊓⊔<br />
Remark 3.3.5. Recall from Chapter 2 that if D ≃ C then<br />
ND ≃ NC (<strong>and</strong> BD ≃ BC). The above result is an algebraic<br />
consequence <strong>of</strong> a category equivalence. On one h<strong>and</strong> the existence<br />
<strong>of</strong> a Morita equivalence implies that the two (category)<br />
algebras are the same, in terms <strong>of</strong> homological properties.<br />
On the other h<strong>and</strong> there are non-equivalent finite categories<br />
whose algebras are Morita equivalent, see Example 3.1.16 (5).<br />
By contrary a simplicial/topological equivalence does not<br />
provide a Morita equivalence. For instance the posets 0 <strong>and</strong><br />
1 are simplicial homotopy equivalent, because there are natural<br />
adjoint functors between them. However if k is a field<br />
then k0 <strong>and</strong> k1 are not equivalent as algebras. The reason
108 3 Category algebras <strong>and</strong> their representations<br />
is that k0 ∼ = k has only one simple module k while k1 has<br />
two simple modules because its identity can be written as a<br />
sum <strong>of</strong> orthogonal primitive idempotents 1 0 + 1 1 . Since their<br />
category algebras have different numbers <strong>of</strong> non-isomorphic<br />
simple modules, k0-mod <strong>and</strong> k1-mod cannot be equivalent<br />
as categories.<br />
3.3.2 Kan extensions <strong>of</strong> modules<br />
Since not every functor u : D → C has an adjoint, we have to<br />
look for other functors for comparing RD-mod <strong>and</strong> RC-mod.<br />
Clearly the restriction Res u possesses two adjoint functors,<br />
the left <strong>and</strong> right Kan extensions LK u <strong>and</strong> RK u . Here will<br />
be our first attempt to study Kan extensions <strong>of</strong> modules. We<br />
note that, because Res u is exact, LK u preserves projectives<br />
while RK u preserves injectives.<br />
In light <strong>of</strong> Proposition 3.2.3, if one is careful enough, the<br />
restriction can be written using the usual module-theoretic<br />
notation ↓ C D :=↓RC RD , <strong>and</strong> then in the literature one may see that<br />
its left adjoint <strong>and</strong> right adjoint are denoted by ↑ C D := RC ⊗ RD<br />
−, the induction, <strong>and</strong> ⇑ C D := Hom RD(RC, −), the coinduction.<br />
They are the Kan extensions in different forms. Since there<br />
is plenty <strong>of</strong> description <strong>of</strong> the induction <strong>and</strong> coinduction for<br />
algebra representations, in these notes we shall mainly discuss<br />
the functor-theoretic aspect <strong>of</strong> them through Kan extensions.<br />
A special situation <strong>of</strong> Kan extensions, the group case, can<br />
be found in Example 1.2.11. One can see that using moduletheoretic<br />
methods it is not easy to obtain most <strong>of</strong> the following<br />
results for general category algebras.<br />
Suppose M ∈ RD-mod <strong>and</strong> u : D → C is a functor. Then<br />
we obtain an RC-module LK u M. As a functor, for each x ∈<br />
Ob C, (LK u M)(x) is given by a direct limit, <strong>and</strong> if α : x → y
3.3 Functors between module categories 109<br />
is a morphism, it induces a map (LK u M)(x) → (LK u M)(y)<br />
by the universal property <strong>of</strong> −→<br />
lim. This specifies the RC-action<br />
on LK u M.<br />
Suppose P is a poset <strong>and</strong> Q is a subposet. We call Q an<br />
ideal <strong>of</strong> P if for a pair <strong>of</strong> object x ∈ Ob Q <strong>and</strong> y ∈ Ob P<br />
such that x ≤ y in P then y belongs to Q. Before we start<br />
the first calculation we mention that if D is a full subcategory<br />
<strong>of</strong> a small category C then any functor over D can be naively<br />
considered as a functor over C by asking its value to be zero<br />
on any objects that do not belong to D. In other words, any<br />
RD-module is naturally an RC-module under the assumption.<br />
Proposition 3.3.6. Suppose P is a poset <strong>and</strong> Q is an ideal<br />
together with the inclusion i : Q ↩→ P. Then for any M ∈<br />
RQ-mod, the RP-module LK i M is exactly M, regarded as<br />
an RP-module.<br />
Pro<strong>of</strong>. One can find that (LK i M)(x) ∼ = lim<br />
−→Q≤x<br />
M, where Q ≤x<br />
is a subposet <strong>of</strong> Q consisting <strong>of</strong> objects smaller or equal to x.<br />
By assumption this poset is either empty or has a terminal<br />
object x. Consequently (LK i M)(x) is either zero if x ∉ Ob Q<br />
or M(x) otherwise. Hence we are done.<br />
Alternatively we can compute RP ⊗ RQ M. For any x ∈<br />
Ob P, we have<br />
1 x·(RP⊗ RQ M) = ∑ y≤x<br />
RHom P (y, x)⊗M = ∑ y≤x<br />
RHom P (y, x)⊗1 y·M.<br />
This is not zero if <strong>and</strong> only if 1 y · M ≠ 0 which means y ∈<br />
Ob Q. By assumption, it forces x belongs to Q. Thus 1 x ·<br />
(RP⊗ RQ M) ≠ 0 if <strong>and</strong> only if x ∈ Ob Q. When this happens,<br />
1 x · (RP ⊗ RQ M) = 1 x · M. Thus we obtain the same result.<br />
⊓⊔
110 3 Category algebras <strong>and</strong> their representations<br />
The concept <strong>of</strong> an idea can be defined for a certain class <strong>of</strong><br />
categories, which are called EI-categories <strong>and</strong> will be introduced<br />
shortly. The above result stays true in this generality.<br />
We will also give a dual concept <strong>of</strong> coideal later on.<br />
Proposition 3.3.7. Suppose u : D → C satisfies the condition<br />
that for every x ∈ Ob C u/x is connected. Then<br />
LK u R = R.<br />
Especially for t : F (C) → C, we have LK t R ∼ = R as an<br />
RC-module.<br />
Pro<strong>of</strong>. When u/x is connected for an x ∈ Ob C, we have<br />
lim R = R. Then by the universal property <strong>of</strong> direct limit<br />
−→u/x<br />
we see any α : x → y must induce the identity morphism on<br />
R. Thus LK u R = R. ⊓⊔<br />
For any M ∈ RF (C)-mod we can compute LK t M explicitly.<br />
If x ∈ Ob C, then −→t/x<br />
lim M ∼ = lim −→t<br />
M by Propositions<br />
2.3.10 <strong>and</strong> 1.2.4. The category t −1 (x) has an initial ob-<br />
−1 (x)<br />
ject [1 x ] <strong>and</strong> thus in general we cannot simplify the direct<br />
limit. We want to construct an example based on this description<br />
showing that LK t (M ˆ⊗N) ≁ = LKt (M) ˆ⊗LK t (N) for<br />
M, N ∈ RF (C)-mod.<br />
Example 3.3.8. Consider the following poset P <strong>and</strong> its factorization<br />
category F (P) (a poset too)<br />
2<br />
α<br />
1 [α] [β]<br />
β<br />
3<br />
, [1 2 ]<br />
We define an RF (P)-module M by<br />
0<br />
R<br />
0<br />
R<br />
<br />
0<br />
[1 1 ]<br />
[1 3 ]
3.3 Functors between module categories 111<br />
Since we know explicitly t −1 (1), t −1 (2) <strong>and</strong> t −1 (3), we can<br />
compute directly that LK t M(1) = R 2 , LK t M(2) = LK t M(3) =<br />
0. Thus the RP-module LK t M ˆ⊗LK t M is pictured by<br />
0<br />
R 4<br />
On the other h<strong>and</strong>, since M ˆ⊗M = M, LK t (M ˆ⊗M) ≁ =<br />
LK t M ˆ⊗LK t M.<br />
Proposition 3.3.9. Let C be a small category <strong>and</strong> ∇ :<br />
F (C) → C e the skew diagonal functor. Then LK ∇ R ∼ = RC<br />
as an RC e -module.<br />
Pro<strong>of</strong>. We compute (LK ∇ R)(y, x) = lim −→∇/(y,x)<br />
R for any<br />
(y, x) ∈ Ob C e . By Proposition 2.3.13 the functor i (y,x) :<br />
∇ −1 (y, x) → ∇/(y, x) has a left adjoint. Hence Proposition<br />
1.2.4 implies that −→∇/(y,x)<br />
lim R ∼ = lim −→∇<br />
R. However<br />
−1 (y,x)<br />
∇ −1 (y, x) ∼ = Hom C (x, y) so we have −→∇<br />
lim R = RC(y, x),<br />
−1 (y,x)<br />
where RC is considered as a functor C e → R-mod. We are<br />
done.<br />
⊓⊔<br />
0<br />
For a finite group algebra RG <strong>and</strong> M, N ∈ RG-mod, we<br />
have a canonical isomorphism Hom RG (M, N) ∼ = Hom RG (R, Hom R (M,<br />
where Hom R (−, −) is the internal hom for RG-mod. The<br />
above proposition allows us to generalizes this isomorphism,<br />
but without using the previously defined internal hom for RCmod.<br />
Corollary 3.3.10. Let M, N ∈ RC-mod. Then Hom R (M, N) ∈<br />
RC e -mod <strong>and</strong><br />
Hom RC (M, N) ∼ = Hom RF (C) (R, Res ∇ Hom R (M, N))<br />
∼ = HomRC (R, RK t Res ∇ Hom R (M, N)).
112 3 Category algebras <strong>and</strong> their representations<br />
Pro<strong>of</strong>. We can define an RC e -module structure on Hom R (M, N)<br />
by<br />
[(α, β op ) · f](m) = αf(βm),<br />
for any f ∈ Hom R (M, N) <strong>and</strong> m ∈ M. Then<br />
Hom RC (M, N) ∼ = Hom RC e(RC, Hom R (M, N))<br />
∼ = HomRC e(LK ∇ R, Hom R (M, N))<br />
∼ = HomRF (C) (R, Res ∇ Hom R (M, N))<br />
∼ = HomRF (C) (Res t R, Res ∇ Hom R (M, N))<br />
∼ = HomRC (R, RK t Res ∇ Hom R (M, N)).<br />
Since F (G) is equivalent to G as categories, under Morita<br />
equivalence between RF (G) <strong>and</strong> RG, RK t Res ∇ Hom R (M, N) ∈<br />
RG-mod is exactly the usual RG-module Hom R (M, N). In<br />
Chapter 5, we shall give a higher version <strong>of</strong> this corollary<br />
in terms <strong>of</strong> Ext, based on Hom RC (M, N) = Ext 0 RC(M, N) =<br />
Ext 0 RC e(RC, Hom R(M, N)). The reader should compare the<br />
above isomorphism with Hom RC (M, N) ∼ = Hom RC (R, Hom(M, N)).<br />
We will explain later on that the latter usually does not admit<br />
an higher version using Ext.<br />
Proposition 3.3.11. Let p : C × C op → C be the projection<br />
to the first component. Then LK p RC = R as an RC-module.<br />
Pro<strong>of</strong>. For each x ∈ Ob C one can easily verify that p/x ∼ =<br />
Id C /x × C op . Since Id C /x has an terminal object (x, 1 x ),<br />
the inclusion {(x, 1 x )} → Id C /x has a left adjoint. Thus<br />
the natural functor C op ∼ = • × C op → Id C /x × C op ∼ = p/x<br />
also has a left adjoint. By Proposition 1.2.4 (LK p RC)(x) =<br />
lim RC ∼ = lim<br />
−→p/x −→C opRC. As a functor from C op to R-mod<br />
(RC)(x) = RHom C (x, −) <strong>and</strong> thus we can define a morphism<br />
(RC)(x) → R by ∑ i r iα i ↦→ ∑ i r i so that R fits into the<br />
defining diagram for direct limit. In fact if M fits into the<br />
⊓⊔
3.3 Functors between module categories 113<br />
limit defining diagram<br />
RHom C (y, −)<br />
RHom C (x, −)<br />
θ y<br />
θ x<br />
M ,<br />
then θ y (α) = θ x (β) for any β ∈ Hom C (y, −) <strong>and</strong> α ∈<br />
Hom C (x, −). Thus (LK p RC)(x) = R <strong>and</strong> we are done. ⊓⊔<br />
The last proposition can be generalized to a functor isomorphism<br />
LK p<br />
∼ = − ⊗RC R, <strong>and</strong> we shall come back to it in<br />
Chapter 4.<br />
Note that if we have two functors u : D → C <strong>and</strong> v :<br />
E → D then Res uv = Res v Res u <strong>and</strong> as a consequence<br />
LK uv = LK v LK u . It is enlightening if we summarize our<br />
three calculations in one diagram for t = p∇.<br />
R<br />
RF (C)-mod<br />
LK ∇<br />
RC e -mod<br />
RC<br />
LK t<br />
LK p<br />
∼ =−⊗RC R<br />
RC-mod .<br />
R<br />
Based on explicit formulas for computing LK ∇ <strong>and</strong> LK p ,<br />
one should be able to find examples, similar to Example 3.3.8,<br />
that they do not commute with corresponding tensor products.<br />
Definition 3.3.12. Let C be a small category <strong>and</strong> x an object.<br />
Then we define a full subcategory C x which consists <strong>of</strong><br />
one object x <strong>and</strong> all <strong>of</strong> its endomorphisms. We also define C [x]<br />
to be the full subcategory <strong>of</strong> C containing all objects that are<br />
isomorphic to x.
114 3 Category algebras <strong>and</strong> their representations<br />
Proposition 3.3.13. Let C be a small category <strong>and</strong> x ∈<br />
Ob C. Consider the inclusion ι : C x ↩→ C. Then LK ι [REnd C (x)] ∼ =<br />
RHom C (x, −).<br />
Pro<strong>of</strong>. For every y ∈ Ob C, the overcategory ι/y has objects<br />
{(x, α)} where α runs over the set Hom C (x, y). A morphism<br />
(x, α) → (x, β) is an endomorphism g ∈ End C (x) such that<br />
α = βg. Hence we can fit RHom C (x, y) into the commutative<br />
diagram<br />
[REnd C (x)](x, α)<br />
α◦−<br />
β◦−<br />
[REnd C (x)](x, β)<br />
RHom C (x, y) ,<br />
in which [REnd C (x)](x, α) = [REnd C (x)](x, β) = REnd C (x),<br />
<strong>and</strong> α ◦ −, β ◦ − are compositions. If M is another R-module<br />
that can replace RHom C (x, y) <strong>and</strong> make a new commutative<br />
diagram, then, by examining the images <strong>of</strong> End C (x), one can<br />
easily establish a canonical map RHom C (x, y) → M such<br />
that the whole diagram is commutative. This means that<br />
RHom C (x, y) ∼ = lim −→ι/y<br />
[REnd C (x)] <strong>and</strong> thus LK ι [REnd C (x)] ∼ =<br />
RHom C (x, −).<br />
⊓⊔<br />
3.3.3 Dual modules <strong>and</strong> Kan extensions<br />
In group representations, the two Kan extensions are isomorphic.<br />
Many important properties <strong>of</strong> group representations rely<br />
on this fact. Unfortunately it is not the case for category representations.<br />
To obtain a balanced underst<strong>and</strong>ing <strong>of</strong> the homological<br />
properties <strong>of</strong> category algebras, we describe some<br />
compatibility between the two Kan extensions. Here we assume<br />
the base ring is a field R = k.<br />
Let M ∈ mod-kC be a right module. Its dual M ∧ =<br />
Hom k (M, k) becomes a left kC-module. The functor (−) ∧ =
3.3 Functors between module categories 115<br />
Hom k (−, k) is an anti-isomorphism between the two module<br />
categories. Suppose G is a group <strong>and</strong> H is a subgroup. In<br />
group representation theory, there are two well-known isomorphisms<br />
(M ↓ G H )∧ ∼ = M ∧ ↓ G H , if M ∈ kG-mod, <strong>and</strong><br />
(N ↑ G H )∧ ∼ = N ∧ ↑ G H , if N ∈ kH-mod. We shall extend these<br />
to category representations.<br />
Lemma 3.3.14. Let τ : D → C be a functor between finite<br />
categories. Then we have Res τ M ∧ ∼ = (Res τ M) ∧ , for any<br />
M ∈mod-kC.<br />
Pro<strong>of</strong>. The isomorphism follows from Lemma 3.1.8.<br />
Lemma 3.3.15. With the same assumption as above, we<br />
have RK τ N ∧ ∼ = (LK τ N) ∧ , for any N ∈ mod-kD.<br />
Pro<strong>of</strong>.<br />
RK τ N ∧ ∼ = Hom kC -mod(kC, RK τ N ∧ )<br />
∼ = HomkD-mod(Res τ kC, N ∧ )<br />
∼ = Hommod-kD(N, (Res τ kC) ∧ )<br />
∼ = Hommod-kD(N, Res τ kC ∧ )<br />
∼ = Hommod-kC(LK τ N, kC ∧ )<br />
∼ = HomkC -mod(kC, (LK τ N) ∧ )<br />
∼ = (LKτ N) ∧ .<br />
The last isomorphism can be immediately combined with<br />
computations from the preceding section to get the right Kan<br />
extensions <strong>of</strong> certain modules.<br />
Since this duality exchanges projective <strong>and</strong> injective modules,<br />
whenever the base ring is a field or a complete local ring<br />
if we can characterize the projective modules then the structures<br />
<strong>of</strong> injective modules are understood through duality.<br />
⊓⊔<br />
⊓⊔
116 3 Category algebras <strong>and</strong> their representations<br />
3.4 EI categories, projectives <strong>and</strong> simples<br />
In this section, we investigate the representation theory <strong>of</strong> EIcategories.<br />
It will help us in computing various (co)homology<br />
groups. For technical reasons we always assume in this section<br />
the base ring R is a field or a complete discrete valuation ring,<br />
in order to have the unique decomposition property for every<br />
RC-module.<br />
Some <strong>of</strong> the general theory <strong>of</strong> EI-categories is given by tom<br />
Dieck in [15] (I.11), much <strong>of</strong> which was due to Lück, see also<br />
[51].<br />
3.4.1 EI condition <strong>and</strong> its implications<br />
Recall from Definition 2.4.2 that EI-category is a small category<br />
C in which all endomorphisms are isomorphisms.<br />
One <strong>of</strong> the important features <strong>of</strong> EI-categories is described<br />
as follows. Given an EI-category C, there is a preorder defined<br />
on Ob C, that is, y ≤ x if <strong>and</strong> only if Hom C (y, x) ≠ ∅. Let [y]<br />
be the isomorphism class <strong>of</strong> an object y ∈ Ob C. This preorder<br />
induces a partial order on the set Is C <strong>of</strong> isomorphism classes<br />
<strong>of</strong> Ob C (specified by [y] ≤ [x] if <strong>and</strong> only if Hom C (y, x) ≠<br />
∅), which plays an important role in studying representations<br />
<strong>and</strong> cohomology <strong>of</strong> EI-categories. Because <strong>of</strong> the existence <strong>of</strong><br />
an order for the isomorphism classes <strong>of</strong> objects in any EIcategory,<br />
EI-categories are sometimes referred to as ordered<br />
categories by some authors, see [57], [42].<br />
Definition 3.4.1. For any EI category C <strong>and</strong> any object<br />
x ∈ Ob C, we can define a full subcategory D ≤x ⊂ D consisting<br />
<strong>of</strong> all y ∈ Ob D such that [y] ≤ [x], or equivalently<br />
Hom C (y, x) ≠ ∅. Similarly we can define other full subcategories<br />
<strong>of</strong> D: D x .
3.4 EI categories, projectives <strong>and</strong> simples 117<br />
An object in an EI category C is called maximal if C >x = ∅,<br />
<strong>and</strong> is minimal if C
118 3 Category algebras <strong>and</strong> their representations<br />
<strong>and</strong> otherwise Mˆx (y) = 0. For each minimal x we have a short<br />
exact sequence<br />
0 → Mˆx → M → M/Mˆx → 0<br />
such that M/Mˆx is atomic. In either case, we can repeat the<br />
process for Mˆx . Since C is finite, we will eventually obtain a<br />
filtration <strong>of</strong> M with every factor atomic. We can further refine<br />
this filtration until every factor is simple.<br />
For each RC-module, even if C is not finite, the above analysis<br />
can visualize its structure, which will help us in future<br />
investigations. For example, every simple RC-module has to<br />
be atomic.<br />
3.4.2 Some representation theory<br />
Here we recall some st<strong>and</strong>ard facts in representation theory.<br />
Suppose A is a finite-dimensional algebra over a field k. An A-<br />
module M is indecomposable if M is not a direct sum <strong>of</strong> two<br />
non-trivial submodules. An element e ∈ A is an idempotent if<br />
e ≠ 0 <strong>and</strong> e 2 = e. For example 1 is always an idempotent <strong>and</strong><br />
if e ≠ 1 is an idempotent then so is 1 − e. Two idempotents<br />
e 1 <strong>and</strong> e 2 are orthogonal if e 1 e 2 = e 2 e 1 = 0. An idempotent<br />
e is primitive if one cannot write e = e 1 + e 2 such that both<br />
e 1 <strong>and</strong> e 2 are orthogonal idempotents.<br />
Suppose e ∈ A is a non-identity idempotent. Then e <strong>and</strong><br />
1−e are a pair <strong>of</strong> orthogonal idempotents with 1 = e+(1−e).<br />
It results in a decomposition<br />
A = Ae ⊕ A(1 − e).<br />
The module Ae is projective <strong>and</strong> one can check that it is<br />
indecomposable (not a direct sum <strong>of</strong> two non-trivial modules)<br />
if <strong>and</strong> only if e is primitive.
3.4 EI categories, projectives <strong>and</strong> simples 119<br />
Since A is finite-dimensional, 1 can be decomposed as a sum<br />
<strong>of</strong> finitely many pairwise orthogonal primitive idempotents<br />
(called a primitive decomposition)<br />
1 = e 1 + e 2 + · · · + e n .<br />
Consequently the regular module A becomes a direct sum<br />
A = Ae 1 ⊕ Ae 2 ⊕ · · · ⊕ Ae n .<br />
<strong>and</strong> we obtain a list <strong>of</strong> indecomposable projective A-modules,<br />
Ae i . Up to isomorphism this is a complete list <strong>of</strong> indecomposable<br />
projective A-modules.<br />
In fact if 1 = e ′ 1 + e ′ 2 + · · · + e ′ m is another sum <strong>of</strong> primitive<br />
idempotents in A, then m = n <strong>and</strong> there exists an invertible<br />
element a ∈ A such that 1 = a1a −1 = ae ′ 1a −1 +ae ′ 2a −1 +· · ·+<br />
ae ′ na −1 such that up to an reordering one has e i = ae ′ i a−1<br />
for all possible i. Following from this fact, the automorphism<br />
<strong>of</strong> A, induced by a, maps each direct summ<strong>and</strong> Ae ′ i <strong>of</strong> A =<br />
Ae ′ 1⊕Ae ′ 2⊕· · ·⊕Ae ′ n isomorphically to Ae i . Thus the number<br />
<strong>of</strong> idempotents in a primitive decomposition is equal to the<br />
number <strong>of</strong> indecomposable direct summ<strong>and</strong> <strong>of</strong> A.<br />
Example 3.4.4. Suppose C is a finite category <strong>and</strong> R is a commutative<br />
ring with identity. Then 1 RC is a sum <strong>of</strong> orthogonal<br />
idempotents ∑ x∈Ob C 1 x. Hence we have a decomposition <strong>of</strong><br />
the regular module<br />
RC = ⊕<br />
RC · 1 x = ⊕<br />
RHom C (x, −),<br />
x∈Ob C<br />
x∈Ob C<br />
<strong>and</strong> each summ<strong>and</strong> RHom C (x, −) is a projective RC-module.<br />
However since an idempotent 1 x may be decomposable in RC,<br />
the above decomposition can be refined, in a way depending<br />
on the choice <strong>of</strong> R. If this happens, the corresponding projec-
120 3 Category algebras <strong>and</strong> their representations<br />
tive module RHom C (x, −) is a direct sum <strong>of</strong> other projective<br />
modules. See Example 3.4.7.<br />
In order to describe the simple A-modules, we have to introduce<br />
a few more terminologies. Let J(A) be the Jacobson<br />
radical <strong>of</strong> A. It is the intersection <strong>of</strong> all left maximal ideals <strong>of</strong><br />
A. In particular it is nilpotent in the sense that there exists an<br />
integer n > 0 such that J(A) n = 0. By contrast an element<br />
a ∈ A is nilpotent if a n = 0 for some positive integer n. If an<br />
ideal satisfies the condition that every element is nilpotent,<br />
then this ideal is contained in the radical.<br />
Example 3.4.5. If C is an EI-category, the any non-isomorphism,<br />
as an element in the category algebra RC is nilpotent. Since<br />
they form an ideal <strong>of</strong> RC, it implies that all non-isomorphisms<br />
are contained in J(RC). A important fact is that every element<br />
in J(A) is nilpotent, but the converse if not true (see<br />
Example 3.4.7 (4)).<br />
For each A-module, the radical <strong>of</strong> M, RadM, is the intersection<br />
<strong>of</strong> all maximal submodules <strong>of</strong> M. For example the<br />
regular module has its radical RadA = J(A) because left<br />
ideal <strong>of</strong> A is exactly the same as a left submodule <strong>of</strong> A. In<br />
general RadM = J(A)M. It is easy to see that M/RadM has<br />
a trivial radical. Any A-module with trivial radical is called<br />
semi-simple. A semi-simple module is called simple if it is indecomposable.<br />
Equivalently an A-module is simple if it does<br />
not contain any non-trivial submodule. For example A/RadA<br />
is semi-simple. Every simple A-module occurs as a direct summ<strong>and</strong><br />
in this semi-simple module up to isomorphism.<br />
The quotient A/RadA is itself an algebra with identity ¯1,<br />
the image <strong>of</strong> 1 ∈ A. A pairwise-orthogonal primitive decomposition<br />
1 = e 1 + e 2 + · · · + e n gives rise to ¯1 ∈ A/RadA,<br />
¯1 = ē 1 + ē 2 + · · · + ē n , which is again a sum <strong>of</strong> pairwiseorthogonal<br />
primitive idempotents in A/RadA. This simple
3.4 EI categories, projectives <strong>and</strong> simples 121<br />
observation actually establishes a one-to-one correspondence<br />
between the sets <strong>of</strong> isomorphism classes <strong>of</strong> indecomposable<br />
projective A-modules <strong>and</strong> <strong>of</strong> simple A-modules.<br />
Proposition 3.4.6. Every indecomposable projective A-module,<br />
up to isomorphism, is <strong>of</strong> the form Ae, for some primitive<br />
idempotent e ∈ A. Moreover, Ae/Rad(Ae) is a simple A-<br />
module <strong>and</strong> every simple A-module arises in this way.<br />
Moreover the number <strong>of</strong> idempotents in a primitive decomposition<br />
<strong>of</strong> 1 ∈ A, the number <strong>of</strong> indecomposable summ<strong>and</strong>s<br />
<strong>of</strong> A <strong>and</strong> the number <strong>of</strong> indecomposable summ<strong>and</strong>s<br />
<strong>of</strong> A/Rad(A) equal to each other.<br />
There exists a large collection <strong>of</strong> good references on representation<br />
theory <strong>of</strong> associative algebras. However for those who<br />
do not plan to go over the whole theory, just bear in mind the<br />
basic constructions <strong>and</strong> important facts that we record here.<br />
Then through upcoming examples one can see how they work.<br />
We shall use them to classify projective <strong>and</strong> simple modules<br />
<strong>of</strong> certain finite category algebras. It will be sufficient for us<br />
to develop (co)homology theory <strong>of</strong> categories <strong>and</strong> modules.<br />
Example 3.4.7.1. Let G = {g ∣ g 2 = 1 • } be the cyclic group<br />
<strong>of</strong> order 2, regarded as a category with one object. If k = C<br />
is the field <strong>of</strong> complex numbers, the identity 1 • can be written<br />
as 1 •+g<br />
, a decomposition into a sum <strong>of</strong> orthogonal<br />
primitive idempotents. The regular module is a direct sum<br />
<strong>of</strong> two one dimensional modules CG ∼ = C(1 • +g)⊕C(1 • −g).<br />
Thus both C(1 • + g) <strong>and</strong> C(1 • − g) are projective. They are<br />
simple as well because they cannot have non-trivial submodules.<br />
It means CG is semi-simple with trivial radical. The<br />
module C(1 • +g) is the trivial module <strong>and</strong> C(1 • −g) is called<br />
the sign representation.<br />
However when k is a field <strong>of</strong> characteristic 2, 1 • is primitive.<br />
Hence kG is indecomposable. The regular module has ex-<br />
2<br />
+ 1 •−g<br />
2
122 3 Category algebras <strong>and</strong> their representations<br />
actly one non-trivial submodule k(1 • + g), which has to be<br />
the radical Rad(kG). Then kG/Rad(kG) is one-dimensional<br />
<strong>and</strong> is simple. It is the only simple kG-module, the trivial<br />
module.<br />
2. The poset 1 = 0 → 1 is a category with two objects 0 <strong>and</strong> 1.<br />
For any field k, the identity 1 k1 = 1 0 +1 1 in the category algebra<br />
k1. The two identity morphisms 1 0 <strong>and</strong> 1 1 are primitive<br />
orthogonal idempotents so k1 = k{1 0 , α} ⊕ k1 1 . The first<br />
indecomposable summ<strong>and</strong> has exactly one non-trivial submodule<br />
k{α}, the radical Rad(k{1 0 , α}) <strong>of</strong> k{1 0 , α}. Then<br />
it gives rise to a one-dimensional simple module S 0 . As a<br />
functor, S 0 (0) = k <strong>and</strong> S 0 (1) = 0. The second summ<strong>and</strong> is<br />
<strong>of</strong> dimension one so it is already simple. If we denote it by<br />
S 1 . As a functor S 1 (0) = 0 <strong>and</strong> S 1 (1) = k.<br />
3. Now we examine the category C <strong>of</strong> Example 3.1.2 that is<br />
neither a group nor a poset.<br />
1 x <br />
x<br />
α<br />
1 y<br />
<br />
<br />
y<br />
g<br />
with g 2 = 1 y <strong>and</strong> α = gα. We always have 1 kC = 1 x + 1 y so<br />
kC = k{1 x , α} ⊕ k{1 y , g}. Similar to 2, the first summ<strong>and</strong>,<br />
named P x , is indecomposable <strong>and</strong> has radical k{α}. The<br />
quotient <strong>of</strong> P x by its radical is a one-dimensional simple<br />
module S x (analogues to S 0 as above). According to 1, the<br />
second direct summ<strong>and</strong> is decomposable if k = C. Whence<br />
we have CG ∼ = C(1 • + g) ⊕ C(1 • − g). It means when CC<br />
has three indecomposable projective modules <strong>and</strong> the same<br />
number <strong>of</strong> simple modules.
3.4 EI categories, projectives <strong>and</strong> simples 123<br />
The category algebra CC<br />
Indecomposable projective modules Simples module<br />
P x,1 = C{1 x , α} S x,1 = C{1 x }<br />
P y,1 = C{1 y + g} S y,1 = C{1 y + g}<br />
P y,−1 = C{1 y − g} S y,−1 = C{1 y − g}<br />
When k is <strong>of</strong> characteristic 2, k{1 y , g} is indecomposable.<br />
Whence kC only has two indecomposable projective <strong>and</strong><br />
simple modules.<br />
The category algebra kC, chark = 2<br />
Indecomposable projective modules Simple modules<br />
P x,1 = k{1 x , α} S x,1 = k{1 x }<br />
P y,1 = k{1 y , g} S y,1 = P y,1 /k{1 y + g}<br />
4. A useful example to bear in mind is the following category<br />
(a groupoid) D that is equivalent to •<br />
x<br />
α<br />
α −1<br />
One can write RD = RD · 1 x ⊕ RD · 1 y = R{1 x , α} ⊕<br />
R{1 y , α −1 }. Observe that we have an isomorphism (−) ◦<br />
α −1 : R{1 x , α} → R{1 y , α −1 }. When R = k is a field,<br />
every non-zero element <strong>of</strong> a1 x + bα ∈ kD · 1 x generates the<br />
whole module. Hence kD · 1 x has no non-trivial submodule<br />
<strong>and</strong> thus is simple. Note that it corresponds to the functor<br />
S x,1 which takes values S x,1 (x) = k1 x <strong>and</strong> S x,1 (y) = kα.<br />
Similarly kD · 1 y is also a simple module, corresponding to<br />
S y,1 given by S y,1 (x) = kα −1 <strong>and</strong> S y,1 (y) = k1 y , which is<br />
isomorphic to S x,1 . The algebra RD is semi-simple but it<br />
contains two nilpotent elements α <strong>and</strong> α −1 .<br />
5. Finally let us consider the following category E<br />
y.
124 3 Category algebras <strong>and</strong> their representations<br />
x<br />
α<br />
β<br />
such that αβ = e, βe = β, eα = α <strong>and</strong> e 2 = e. Note that α<br />
<strong>and</strong> β are not invertible in E. Hence E is not a groupoid <strong>and</strong><br />
the two objects are not isomorphic in E. However the category<br />
algebra RE is isomorphic to R{1 x , α, β, e} × R{1 y } ∼ =<br />
RD × R•.<br />
This category is a simple example <strong>of</strong> inverse categories, see<br />
[51]. The category algebra <strong>of</strong> an inverse category is canonically<br />
isomorphic to a direct product <strong>of</strong> groupoid algebras.<br />
Sometimes we may want to use injective modules, so we finish<br />
this section with several remarks concerning the injectives.<br />
In general injective modules behave better than the projectives<br />
in the sense that for any ring A <strong>and</strong> any A-module M,<br />
there exists a minimal injective A-module I M such that M<br />
admits an injection into I M . This module is called the injective<br />
hull <strong>of</strong> M. There are module categories with enough<br />
injective but not with enough projectives. However if A is a<br />
finite-dimensional algebra, then for any M ∈ A-mod there<br />
exists a minimal projective module P M which admits a surjection<br />
onto M. It is called the projective cover <strong>of</strong> M. For<br />
instance, in the tables <strong>of</strong> Example 3.4.7 (3), each row consists<br />
<strong>of</strong> a simple module as well as its projective cover. When we do<br />
representation theory we <strong>of</strong>ten prefer working with projective<br />
modules because simple modules come from their quotients.<br />
As we pointed out earlier one has an anti-isomorphism<br />
(−) ∧ = Hom k (−, k) from A-mod to mod-A. Suppose P is a<br />
projective A-module, then P ∧ is an injective right A-module.<br />
The anti-isomorphism provides a bijection between projective<br />
left (resp. right) A-modules <strong>and</strong> injective right (resp. left) A-<br />
1 y<br />
<br />
<br />
y<br />
e
3.4 EI categories, projectives <strong>and</strong> simples 125<br />
modules. Thus knowing all projectives leads to getting all<br />
injectives. An important case is the group algebra <strong>of</strong> a finite<br />
group G, or more generally a finite-dimensional cocommutative<br />
Hopf algebra. They are self-injectives, which means the<br />
regular module is an injective module. In this case, the sets<br />
<strong>of</strong> projective <strong>and</strong> injective modules coincide.<br />
In the end we mention an important concept in algebra.<br />
Definition 3.4.8. Let A <strong>and</strong> B be two rings. Then A is<br />
Morita equivalent to B if A-mod is equivalent to B-mod.<br />
There are many interesting invariants under a Morita equivalence.<br />
For instance, a Morita equivalence preserves the number<br />
<strong>of</strong> isomorphism classes <strong>of</strong> simple modules. Hochschild<br />
(co)homology is invariant under Morita equivalence. Here we<br />
shall focus on category algebras only. In fact we will prove that<br />
a category equivalence induces a Morita equivalence between<br />
category algebras. Thus in Example 3.4.7 (4) RC is Morita<br />
equivalent to R = R•.<br />
3.4.3 Classifications <strong>of</strong> projectives <strong>and</strong> simples<br />
Now we start describing the projective <strong>and</strong> simple modules <strong>of</strong><br />
an EI category algebra. This part <strong>of</strong> the work is due to Lück<br />
[51], as is described by tom Dieck in [15]. The base ring R is<br />
assumed to be a field or a complete discrete valuation ring.<br />
Let C be a small category <strong>and</strong> x ∈ Ob C an object. Suppose<br />
P x is a projective RC x -module (or in other words a projective<br />
REnd C (x)-module). Then its left Kan extension LK ι P x ,<br />
along ι : C x ↩→ C, is a projective kC-module. Especially<br />
LK ι [REnd C (x)] ∼ = RHom C (x, −) by Proposition 3.2.11.<br />
Let us assume furthermore C is a finite EI category. Then<br />
End C (x) = Aut C (x) for every x ∈ Ob C. From Example 3.4.4<br />
we already learned that RC decomposes into a direct sum
126 3 Category algebras <strong>and</strong> their representations<br />
⊕ x∈Ob C RC · 1 x . Now we try to analyze the indecomposable<br />
direct summ<strong>and</strong>s.<br />
Suppose Is C is the full subcategory <strong>of</strong> C, consisting <strong>of</strong> all<br />
objects <strong>and</strong> all isomorphisms. Then R Is C is a subalgebra <strong>of</strong><br />
RC.<br />
Lemma 3.4.9. If 1 RC = ∑ n<br />
i=1 e i is a primitive decomposition<br />
<strong>of</strong> 1 RC in R Is C, then it is also a primitive decomposition<br />
<strong>of</strong> 1 RC in RC.<br />
Pro<strong>of</strong>. Given any primitive decomposition <strong>of</strong> 1 RC in RC, the<br />
number <strong>of</strong> idempotents in this decomposition is equal to the<br />
number <strong>of</strong> indecomposable direct summ<strong>and</strong>s <strong>of</strong> the regular<br />
module <strong>of</strong> RC, which is equal to the number <strong>of</strong> indecomposable<br />
direct summ<strong>and</strong>s <strong>of</strong> RC/Rad(RC) by Proposition 3.4.5.<br />
Let us take the decomposition 1 RC = ∑ n<br />
i=1 e i. We need to<br />
show it is primitive. To this end, we prove n equals the number<br />
<strong>of</strong> indecomposable direct summ<strong>and</strong>s in RC/Rad(RC).<br />
Since all non-isomorphisms generate an ideal I <strong>of</strong> RC, which<br />
is contained in Rad(RC) <strong>and</strong> which induces an algebra isomorphism<br />
RC/I ∼ = R Is C, from the isomorphism RC/Rad(RC) ∼ =<br />
(RC/I)/(Rad(RC)/I), we know the two sides have the same<br />
numbers <strong>of</strong> indecomposable direct summ<strong>and</strong>s. From definition<br />
one can check that Rad(RC)/I is the radical <strong>of</strong> RC/I. Then<br />
by Proposition 3.4.5, applied to both RC <strong>and</strong> RC/I ∼ = R Is C,<br />
we see the primitive decompositions <strong>of</strong> 1 RC in both RC <strong>and</strong><br />
R Is C must have the same number <strong>of</strong> idempotents. Hence we<br />
are done.<br />
⊓⊔<br />
The category Is C is a disjoint union <strong>of</strong> groupoids, each <strong>of</strong><br />
which comes from an isomorphism class <strong>of</strong> some object. Recall<br />
that, for each object x ∈ Ob C, we denote by [x] the set <strong>of</strong> objects<br />
isomorphic to x, <strong>and</strong> C [x] the full subcategory consisting<br />
<strong>of</strong> these objects.
3.4 EI categories, projectives <strong>and</strong> simples 127<br />
Lemma 3.4.10.1. If x ∼ = y are two isomorphic objects, <strong>and</strong><br />
f y ∈ Hom C (x, y) is an isomorphism, then the assignment<br />
α ↦→ α · f y for each α ∈ RC · e defines an isomorphism <strong>of</strong><br />
RC-modules RC · 1 y → RC · 1 x .<br />
2. If 1 x = ∑ n<br />
i=1 e i is a primitive decomposition in RAut C (x),<br />
then 1 y = ∑ n<br />
i=1 f ye i fy<br />
−1 is a primitive decomposition in<br />
RAut C (y). Furthermore if we fix for each y ∼ = x an isomorphism<br />
f y ∈ Hom C (x, y), then<br />
∑ n∑<br />
f y e i fy<br />
−1<br />
y∈Ob C [x]<br />
i=1<br />
is a primitive decomposition <strong>of</strong> the identity 1 RC[x] in the<br />
groupoid algebra RC [x] .<br />
Pro<strong>of</strong>. The isomorphism is straightforward to prove. Now if<br />
1 x = ∑ n<br />
i=1 e i is a primitive decomposition in RAut C (x), certainly<br />
1 y = ∑ n<br />
i=1 f ye i fy<br />
−1 is a decomposition in RAut C (y). It<br />
has to be primitive, because if it were not, then a primitive<br />
decomposition would be a sum <strong>of</strong> more than n idempotents<br />
in RAut C (y). However fy<br />
−1 (−)f y maps such a primitive decomposition<br />
<strong>of</strong> 1 y to a decomposition <strong>of</strong> 1 x , which contradicts<br />
with the assumption that 1 x = ∑ n<br />
i=1 e i is a primitive decomposition.<br />
⊓⊔<br />
The reader can compare the above statements with Example<br />
3.4.7 (4).<br />
Corollary 3.4.11. Let C be a finite EI category. One can<br />
write<br />
1 RC = ∑ ∑n x<br />
e xj ,<br />
x∈Ob C<br />
∑where n x is a positive integer for each x ∈ Ob C <strong>and</strong> 1 x =<br />
nx<br />
j=1 e xj. As a consequence, RC = ⊕ x∈Ob C ⊕ n x<br />
j=1 RC · e xj<br />
j=1
128 3 Category algebras <strong>and</strong> their representations<br />
for some primitive pairwise orthogonal idempotents e xj ∈<br />
RAut C (x), x ∈ Ob C.<br />
Any projective RC-module is isomorphic to a direct sum <strong>of</strong><br />
indecomposable projective modules <strong>of</strong> the form RC · e, where<br />
e ∈ RAut C (x) is a primitive idempotent, for some x ∈ Ob C.<br />
Given a primitive orthogonal decomposition 1 RC = ∑ i e i<br />
such that each e i belongs to some group algebra RAut C (x),<br />
each summ<strong>and</strong> <strong>of</strong> RC ∼ = ⊕ e i<br />
RC · e i is indeed a left Kan<br />
extension<br />
LK ι [RAut C (x)e i ] = {LK ι [RAut C (x)]}e i<br />
∼ = RHomC (x, −)e i = RCe i<br />
because LK ι commutes with direct sums. Here ι : C x ↩→ C is<br />
the inclusion. In each RCe i , its radical contains ∑ z ≁ =x<br />
RHom x,z e i<br />
which are linear combinations <strong>of</strong> non-isomorphisms in Hom C (x, −).<br />
We continue to characterize the simple RC-modules. Directly<br />
from the EI condition we have seen that a simple module<br />
S has to be atomic. It matches with our description <strong>of</strong><br />
indecomposable projective modules. The quotient <strong>of</strong> RCe i ,<br />
for a primitive idempotent e i ∈ RAut C (x), by its radical is an<br />
atomic module supported on C [x] . Moreover for each y ∼ = x,<br />
S(y) must be a simple RAut C (y)-module. In fact all simple<br />
RC-modules are exactly those simple modules <strong>of</strong> R Is C, which<br />
are obtained from simple modules <strong>of</strong> automorphism group algebras<br />
RAut C (x).<br />
Theorem 3.4.12 (Lück). Let C be a finite EI-category.<br />
The isomorphism classes <strong>of</strong> the simple RC-modules biject<br />
with the pairs ([x], V ), where x ∈ Ob C <strong>and</strong> V is a simple<br />
RAut C (x)-module, taken up to isomorphism.<br />
Pro<strong>of</strong>. First <strong>of</strong> all, we already know that all simple RCmodules<br />
are atomic. Thus simple RC-modules are exactly<br />
those simple RC [x] -modules, with x running over Ob C.
3.4 EI categories, projectives <strong>and</strong> simples 129<br />
Secondly for a fixed x, RC [x] -mod is equivalent to RAut C (y),<br />
for any y ∼ = x, through restrictions induced by the equivalences<br />
C y → C [x] <strong>and</strong> C [x] → C y . Hence simple RC [x] -modules<br />
biject with simple RC y -modules, for any y ∼ = x. Since C y is<br />
the group Aut C (y), we have proved the assertion. ⊓⊔<br />
Because <strong>of</strong> the above theorem, it is natural to denote a simple<br />
RC-module by S x,V , if it comes from a simple RAut C (x)-<br />
module V , for some x ∈ Ob C. For consistency, we use P x,V for<br />
the projective cover <strong>of</strong> S x,V , whose structure is determined by<br />
its value at the object x. If RAut C (x)·e is the projective cover<br />
<strong>of</strong> the simple RAut C (x)-module V , then RC · e is the projective<br />
cover <strong>of</strong> S x,V . The reader may revisit Example 3.4.7 to<br />
get better underst<strong>and</strong>ing <strong>of</strong> our results <strong>and</strong> notations in this<br />
section.<br />
Example 3.4.13. Let k be a field <strong>of</strong> characteristic two <strong>and</strong> C<br />
the following category<br />
{1 x } <br />
x<br />
α<br />
β<br />
y<br />
{1 y ,g}<br />
with g 2 = 1 y , gα = α <strong>and</strong> gβ = β. Indeed the algebra kC has<br />
two (one-dimensional) simples S x,k , S y,k <strong>and</strong> their projective<br />
covers are P x,k = k{1 x , α, β}, <strong>and</strong> P y,k = k{1 y , g}, respectively.<br />
The product P x,k ˆ⊗P x,k<br />
∼ = Px,k ⊕ Sy,k 2 is not projective<br />
because S y,k ≠ P y,k .<br />
Remark 3.4.14. Using the tensor product, one can introduce<br />
a “representation ring” <strong>of</strong> RC, namely a(RC), which consists<br />
<strong>of</strong> Z-linear combinations <strong>of</strong> symbols like [M], representing<br />
an isomorphism class <strong>of</strong> a simple RC-module M. For any<br />
two elements [M] <strong>and</strong> [N], the multiplication is defined by<br />
[M] · [N] = [M ˆ⊗N]. However this product does not exist in<br />
K 0 (RC), which is spanned over the set <strong>of</strong> isomorphism classes<br />
<strong>of</strong> indecomposable projectives.
130 3 Category algebras <strong>and</strong> their representations<br />
With the description <strong>of</strong> indecomposable projectives, we can<br />
show when the trivial module R is projective.<br />
Proposition 3.4.15. Let C be a finite EI-category. Then R<br />
is projective if <strong>and</strong> only if each connected component <strong>of</strong> C<br />
has a unique isomorphism class <strong>of</strong> minimal objects [x], with<br />
the properties that for all y in the same connected component<br />
as x, Aut C (x) acts transitively on Hom(x, y), <strong>and</strong><br />
|Aut C (x)| is invertible in R.<br />
Pro<strong>of</strong>. Without loss <strong>of</strong> generality, we may assume C is connected.<br />
If R is projective then R ∼ = ⊕ P y,W for certain indecomposable<br />
projective modules P y,W . Since R is indecomposable<br />
<strong>and</strong> takes constant value at all objects, we must have that<br />
R ∼ = P x,V for some x ∈ Ob C, x is minimal, <strong>and</strong> all minimal<br />
objects are isomorphic. Moreover because P x,V (x) = R,<br />
the projective cover <strong>of</strong> the simple kAut C (x)-module V , we get<br />
V = R <strong>and</strong> R must be projective as an RAut C (x)-module.<br />
Thus R is projective if <strong>and</strong> only if R ∼ = P x,R , all minimal<br />
objects are isomorphic to x <strong>and</strong> |Aut C (x)| −1 ∈ R.<br />
Now, assume all minimal objects are isomorphic to x <strong>and</strong><br />
|Aut C (x)| −1 ∈ R. By Proposition 3.2.11 RHom C (x, −) ∼ =<br />
LK ι [RAut C (x)]. Then R ∼ = P x,R can<br />
∑<br />
be explicitly constructed,<br />
1<br />
using the idempotent e =<br />
|Aut C (x)| g∈Aut C (x)<br />
g, as<br />
R ∼ = P x,R<br />
∼ = LKι [RAut C (x)e] = {LK ι [RAut C (x)]}e = RHom C (x, −)e<br />
because LK ι commutes with direct sum. It is equivalent to<br />
saying that at each y ∈ Ob C, R ∼ = RHom C (x, y)e. This happens<br />
if <strong>and</strong> only if Aut C (x) to act transitively on Hom C (x, y).<br />
⊓⊔
3.4 EI categories, projectives <strong>and</strong> simples 131<br />
3.4.4 Projective covers, injective hulls <strong>and</strong> their restrictions<br />
Let C be a finite category <strong>and</strong> R = k a field. Then any finitely<br />
generated kC-module M admits a minimal projective resolution<br />
P ∗ → M → 0<br />
in the sense that if P ′ ∗ → M → 0 is another projective resolution<br />
<strong>of</strong> M, then Id M induces a split injection <strong>of</strong> complexes<br />
from the minimal resolution to the latter. Similarly we can<br />
define a minimal injective resolution <strong>of</strong> M, 0 → M → I ∗ .<br />
Proposition 3.4.16. Suppose C is an EI category.<br />
1. If D ⊂ C is a coideal <strong>and</strong> P ∈ kC-mod is an indecomposable<br />
projective module, then Res ι P ∈ kD-mod is either an<br />
indecomposable projective or zero.<br />
2. If D ⊂ C is an ideal <strong>and</strong> I ∈ kC-mod is an indecomposable<br />
injective module, then Res ι I ∈ kD-mod is either an<br />
indecomposable injective or zero.<br />
Pro<strong>of</strong>. Let P = P x,V be an indecomposable projective kCmodule.<br />
If x ∈ Ob C, then Res i P x,V is brutally truncated from<br />
P x,V <strong>and</strong> is indecomposable projective as an kD-module. On<br />
the other h<strong>and</strong>, if x ∉ Ob D, then Res i P x,V = 0.<br />
For the case <strong>of</strong> injective modules we recall (Res i P ) ∧ ∼ =<br />
Res i P ∧ for any right projective module, by Lemma 3.2.12.<br />
Note that Statement 1 stays true for right projective modules<br />
if we replace the term “coideal” by “ideal”. Now we combine<br />
this with the duality between (indecomposable) right projectives<br />
<strong>and</strong> left injectives.<br />
⊓⊔<br />
For example if x is an object in C, then C ≤x is a coideal. Take<br />
any indecomposable projective module P x,V , then Res i P x,V<br />
∼ =<br />
P V , the projective cover <strong>of</strong> the simple kAut C (x)-module V .<br />
This module P V is an indecomposable projective kC ≤x -module.
132 3 Category algebras <strong>and</strong> their representations<br />
Definition 3.4.17. Let C be an EI category. Suppose M is a<br />
kC-module. Then we define C M ⊂ C to be the smallest ideal<br />
satisfying the condition that if x ∉ Ob C M then M(x) = 0.<br />
Suppose M is a kC-module. Then we define C M ⊂ C to be<br />
the smallest coideal satisfying the condition that if x ∉ Ob C M<br />
then M(x) = 0.<br />
Suppose (M, N) is an ordered pair <strong>of</strong> kC-modules. We define<br />
a full subcategory C N M to be C M ∩ C N .<br />
Obviously if N ⊂ M then C N ⊂ C M <strong>and</strong> C N ⊂ C M .<br />
Lemma 3.4.18. Let C be a finite EI category <strong>and</strong> M a kCmodule.<br />
Then the projective cover P M <strong>of</strong> M satisfies the<br />
condition that C PM ⊂ C M , <strong>and</strong> the injective hull I M <strong>of</strong> M<br />
satisfies the condition that C I M<br />
⊂ C M .<br />
This result allows us to give a characterization <strong>of</strong> the minimal<br />
projective <strong>and</strong> injective resolutions <strong>of</strong> a module.<br />
Corollary 3.4.19. Let C be a finite EI category <strong>and</strong> M ∈<br />
kC-mod. Suppose P ∗ <strong>and</strong> I ∗ are minimal projective <strong>and</strong> injective<br />
resolutions <strong>of</strong> M. Then for every n ≥ 0, C Pn ⊂ C M<br />
<strong>and</strong> C I n<br />
⊂ C M .<br />
Pro<strong>of</strong>. The kernel K 0 <strong>of</strong> P 0 ↠ M satisfies C K0 ⊂ C P0 ⊂ C M .<br />
We use the preceding lemma repeatedly. Similar argument<br />
can be made on I ∗ .<br />
⊓⊔<br />
The last corollary will be useful when we compute cohomology<br />
<strong>of</strong> modules.
Chapter 4<br />
<strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
Abstract We begin through investigation <strong>of</strong> (co)homology<br />
theories in this chapter. Extensions <strong>of</strong> modules over a category<br />
algebra is a concept <strong>of</strong> paramount importance here. We<br />
shall discuss various ways to examine Ext groups, multiplicative<br />
structure <strong>and</strong> their relationship with previously defined<br />
simplicial <strong>and</strong> singular (co)homology. A particular important<br />
situation is when the first module is trivial. In this case, on<br />
top <strong>of</strong> the module theoretic tools, simplicial methods are applicable.<br />
We shall provide a discussion <strong>of</strong> the bar resolution<br />
<strong>and</strong> its Kan extensions. Examples are used to illustrate various<br />
computational methods. In the end, the Grothendieck<br />
spectral sequences are introduced <strong>and</strong> we will study them in<br />
a couple <strong>of</strong> special cases.<br />
4.1 General theory<br />
4.1.1 <strong>Cohomology</strong> <strong>of</strong> modules<br />
Since RC-mod is an abelian category with enough projectives<br />
<strong>and</strong> injectives, for any two modules M, N ∈ RC-mod<br />
we can consider the Ext groups Ext i RC(M, N), with i ≥ 0.<br />
It is the i-th right derived functor <strong>of</strong> the left exact functor<br />
Hom RC (M, −) (or Hom RC (−, N)). In general for any<br />
M ∈ RC-mod Ext ∗ RC(M, M) has a ring structure with product<br />
given by the Yoneda splice. Usually it is not graded com-
134 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
mutative. However it is in the case <strong>of</strong> M = R. In this chapter<br />
we shall compare the Yoneda splice with the cup product<br />
introduced earlier.<br />
Similarly for a right RC-module M ′ <strong>and</strong> a left RC-module<br />
N ′ , we can study Tor RC<br />
i (M ′ , N ′ ) as the i-th right derived functors<br />
<strong>of</strong> M ′ ⊗ RC − (or − ⊗ RC N ′ ). In these notes we shall focus<br />
on cohomology. However we shall remark on homology whenever<br />
it is appropriate.<br />
We recall some basics about extensions <strong>of</strong> modules <strong>and</strong> their<br />
relationship with cohomology classes.<br />
Definition 4.1.1. Let M, N be A-modules. An n-fold extension<br />
, n ≥ 1, <strong>of</strong> M by N is an exact sequence <strong>of</strong> A-modules<br />
0 → N → L n−1 → · · · → L 0 → M → 0.<br />
Two n-fold extensions <strong>of</strong> M by N are equivalent if there is<br />
a commutative diagram<br />
0<br />
<br />
N L<br />
n−1 · · ·<br />
<br />
L<br />
0 M 0<br />
0<br />
<br />
N L ′ n−1<br />
<br />
· · ·<br />
<br />
L ′ 0<br />
<br />
M 0<br />
Then we can extend this by symmetry <strong>and</strong> transitivity to<br />
an equivalence relation among n-fold extensions <strong>of</strong> M by N.<br />
Proposition 4.1.2. There is an one-to-one correspondence<br />
between elements <strong>of</strong> Ext n A(M, N) <strong>and</strong> equivalent classes <strong>of</strong><br />
n-fold extensions <strong>of</strong> M by N.<br />
Pro<strong>of</strong>. Let P ∗ → M → 0 be a projective resolution. Then<br />
an extension determines an element in Ext n RC(M, N) by the<br />
following lifting
4.1 General theory 135<br />
· · ·<br />
<br />
P n+1<br />
<br />
P n<br />
∂ n <br />
P n−1<br />
· · ·<br />
<br />
P<br />
0 M 0<br />
f<br />
<br />
0<br />
<br />
N L n−1<br />
<br />
· · ·<br />
<br />
L 0<br />
<br />
M 0<br />
We see from here that two equivalent extensions give rise to<br />
the same element in Ext n A(M, N).<br />
Conversely if two n-fold extensions determine the same elements<br />
in the group Ext n A(M, N), then we can construct a<br />
commutative diagram (by enlarging P ∗ we may assume f is<br />
surjective)<br />
0 N<br />
<br />
L n−1<br />
<br />
L n−2<br />
<br />
· · ·<br />
<br />
L 0<br />
<br />
M 0<br />
0 N P n−1 /∂ n (Kerf)<br />
<br />
0 N<br />
<br />
L ′ n−1<br />
P n−2<br />
<br />
L ′ n−2<br />
· · ·<br />
<br />
P<br />
0 M 0<br />
<br />
· · ·<br />
<br />
L ′ 0<br />
<br />
M 0<br />
Definition 4.1.3. Let M ∈ A-mod. Then we can define the<br />
Yoneda splice on Ext ∗ A(M ′ , M) <strong>and</strong> Ext ∗ A(M, M ′′ ) so that<br />
for any η ∈ Ext i A(M ′ , M) <strong>and</strong> η ′ ∈ Ext j A (M, M ′′ ), η ′ ∗ η ∈<br />
Ext i+j<br />
A<br />
(M ′ , M ′′ ) is given by<br />
0 → M ′′ → N j−1 · · · → N 0 → N ′ i−1 → · · · → N ′ 0 → M ′ → 0,<br />
if η is represented by 0 → M → N ′ i−1 · · · → N ′ 0 → M ′ → 0<br />
<strong>and</strong> η ′ is represented by 0 → M ′′ → N j−1 · · · → N 0 → M →<br />
0.<br />
The Yoneda splice gives Ext ∗ A(M, M) a ring structure which<br />
is not graded commutative in general. Moreover Ext ∗ A(M, M)<br />
acts on Ext ∗ A(M, N) <strong>and</strong> Ext ∗ A(N, M) for another N ∈ A-<br />
mod.<br />
Definition 4.1.4. We call Ext ∗ RC(R, R) = ⊕ i≥0 Exti RC(R, R)<br />
the ordinary cohomology ring <strong>of</strong> the category algebra A. The<br />
⊓⊔
136 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
product in this ring is defined by the Yoneda splice <strong>of</strong> Ext<br />
classes.<br />
We shall show later on that this ring is isomorphic in a<br />
natural way to H ∗ (C, R) ∼ = H ∗ (BC, R) so it deserves the name.<br />
Now based on the tensor structure on RC-mod, we provide<br />
a module theoretic description to the ring Ext ∗ RC(R, R).<br />
In the meantime we pave the road to studying the ring action<br />
<strong>of</strong> Ext ∗ RC(R, R) on various Ext groups. We comment that<br />
since (RC-mod, ˆ⊗, R) is a monoidal category with an exact<br />
tensor product, it gives rise to a suspended monoidal category<br />
(D − (RC), ˆ⊗, R) <strong>and</strong> then following a general statement<br />
[71] on the endomorphisms <strong>of</strong> the identity in a suspended<br />
monoidal category, End D − (RC)(R) is a graded commutative<br />
ring. It will be clear in this section that this endomorphism<br />
ring is isomorphic to what we call the ordinary cohomology<br />
ring Ext ∗ RC(R, R).<br />
Let M, M ′ , N, N ′ ∈ RC-mod which are projective as R-<br />
modules. We will define the cup product to be<br />
∪ : Ext i RC(M, N)⊗Ext j RC (M ′ , N ′ ) → Ext i+j<br />
RC (M ˆ⊗M ′ , N ˆ⊗N ′ ).<br />
Since R is the identity with respect to ˆ⊗, this will give us<br />
a ring structure on Ext ∗ RC(R, R), as well as an action <strong>of</strong><br />
Ext ∗ RC(R, R) on Ext ∗ RC(M, N) for arbitrary M, N ∈ RC-mod.<br />
One shall compare our construction with [, Section 3.2] for<br />
cocommutative Hopf algebras.<br />
Now we are ready to give a precise definition to the cup<br />
product. Suppose ζ ∈ Ext m RC(M, N) is represented by an exact<br />
sequence<br />
0 → N → L m−1 → P m−2 → · · · → P 0 → M → 0,<br />
where P i ’s are projective RC-modules, <strong>and</strong> ζ ′ ∈ Ext n RC(M ′ , N ′ )<br />
is represented by an exact sequence
4.1 General theory 137<br />
0 → N ′ → L ′ n−1 → Q n−2 → · · · → Q 0 → M ′ → 0,<br />
where Q j ’s are projective RC-modules. Since projective RCmodules<br />
are projective R-modules, all modules in these two<br />
exact sequences are projective R-modules. Furthermore an<br />
RC-module L being a projective R-module is equivalent to<br />
L(x) being projective for all x ∈ Ob C. Then applying the<br />
Künnethe formula to the following two complexes <strong>of</strong> projective<br />
R-modules<br />
<strong>and</strong><br />
0 → N(x) → L m−1 (x) → P m−2 (x) → · · · → P 0 (x)<br />
0 → N ′ (x) → L ′ n−1(x) → Q n−2 (x) → · · · → Q 0 (x),<br />
for all x ∈ Ob C, we get exact sequences<br />
0 → (N ˆ⊗N ′ )(x) → (L m−1 ˆ⊗N)(x) ⊕ (N ˆ⊗L ′ n−1)(x) → · · ·<br />
→ (P 0 ˆ⊗Q 0 )(x) → (M ˆ⊗M ′ )(x) → 0,<br />
with x running over Ob C. Thus we get an exact sequence <strong>of</strong><br />
RC-modules<br />
0 → N ˆ⊗N ′ → (L m−1 ˆ⊗N ′ ) ⊕ (N ˆ⊗L ′ n−1) → · · · → P 0 ˆ⊗Q 0 → M ˆ⊗M<br />
which is defined to be the cup product <strong>of</strong> ζ <strong>and</strong> ζ ′ ,<br />
ζ ∪ ζ ′ ∈ Ext m+n<br />
RC (M ˆ⊗M ′ , N ˆ⊗N ′ ).<br />
Lemma 4.1.5. Let ζ, ζ ′ be as above. The cup product ζ ∪ ζ ′<br />
is the Yoneda splice <strong>of</strong><br />
with<br />
ζ ˆ⊗N ′ ∈ Ext i RC(M ˆ⊗N ′ , N ˆ⊗N ′ )<br />
M ˆ⊗ζ ′ ∈ Ext j RC (M ˆ⊗M ′ , M ˆ⊗N ′ ).
138 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
There exists a natural map φ M = − ˆ⊗M : Ext ∗ RC(R, R) →<br />
Ext ∗ RC(M, M) lies in the graded center, for any M ∈ RCmod.<br />
Particularly, Ext ∗ RC(R, R) is graded commutative.<br />
Pro<strong>of</strong>. By using exact sequences representing ζ <strong>and</strong> ζ ′ , one<br />
can easily establish a map between (n + m)-fold extensions<br />
(ζ ˆ⊗Id N ′) ∗ (Id M ˆ⊗ζ ′ ) → ζ ∪ ζ ′ . Let<br />
<strong>and</strong><br />
D : 0 → N → P m−1 = L m−1 → P m−2 → · · · → P 0<br />
C : 0 → N ′ → Q n−1 = L ′ n−1 → Q n−2 → · · · → Q 0<br />
come from the given two extensions. Then we have a map <strong>of</strong><br />
(m + n)-fold extensions<br />
0 N ˆ⊗N ′ <br />
(P m−1 ˆ⊗N ′ ) ⊕ (N ˆ⊗Q n−1 )<br />
<br />
· · ·<br />
<br />
P 0 ˆ⊗Q<br />
0 M ˆ⊗M ′ <br />
0<br />
f m+n−1 <br />
0 N ˆ⊗N ′ <br />
P m−1 ˆ⊗N ′ <br />
· · ·<br />
<br />
M ˆ⊗Q 0<br />
<br />
M ˆ⊗M ′ <br />
0<br />
given by<br />
{<br />
(D ˆ⊗C)<br />
f i :<br />
i → P 0 ˆ⊗Q i → M ˆ⊗Q i , 0 ≤ i ≤ n − 1;<br />
(D ˆ⊗C) i → P i−n ˆ⊗N ′ , n ≤ i ≤ n + m − 1.<br />
Thus cup product is a Yondea splice.<br />
Next for an m-fold extension (all modules chosen to be R-<br />
projective)<br />
ζ : 0 → R → L m−1 → P m−2 → · · · → P 0 → R → 0,<br />
the tensor product with M<br />
0 → M → L m−1 ˆ⊗M → P m−2 ˆ⊗M → · · · → P 0 ˆ⊗M → M → 0,<br />
stays exact because<br />
0 → R = R(x) → L m−1 (x) → P m−2 (x) → · · · → P 0 (x) → R(x) = R<br />
f 0
4.1 General theory 139<br />
is split exact <strong>and</strong> thus<br />
0 → M(x) → L m−1 (x)⊗M(x) → P m−2 (x)⊗M(x) → · · · → P 0 (x)⊗M<br />
is exact for all x ∈ Ob C. Thus − ˆ⊗M gives us a map, commuting<br />
with Yoneda splice,<br />
φ M = − ˆ⊗M : Ext ∗ RC(R, R) → Ext ∗ RC(M, M),<br />
<strong>and</strong> hence is a ring homomorphism.<br />
If ζ ′ ∈ Ext n RC(M, M), then φ M (ζ) ∗ ζ ′ = ζ ∪ ζ ′ <strong>and</strong> ζ ′ ∗<br />
φ M (ζ) = ζ ′ ∪ ζ. We want to show ζ ∪ ζ ′ = (−1) mn ζ ′ ∪ ζ.<br />
This comes from the fact that, given the cocommutativity<br />
τ∆ = ∆, we can establish an isomorphism <strong>of</strong> complexes <strong>of</strong><br />
RC-modules<br />
C ˆ⊗D → D ˆ⊗C,<br />
by a ⊗ b ↦→ (−1) deg a deg b b ⊗ a for any two homogeneous elements.<br />
⊓⊔<br />
In terms <strong>of</strong> projective resolutions, we can describe the cup<br />
product as follows. Let M, M ′ , N, N ′ be RC-modules which<br />
are projective as R-modules. Take two projective resolutions<br />
P ∗ → M → 0 <strong>and</strong> Q ∗ → M ′ → 0. Then by our previous observation,<br />
P ∗ ˆ⊗Q ∗ → M ˆ⊗M ′ → 0 is an exact sequence. This<br />
is usually not a projective resolution as the tensor product <strong>of</strong><br />
two projective is not projective, in contrast to the case <strong>of</strong> a cocommutative<br />
Hopf algebra. However we can build a projective<br />
resolution, unique up to chain homotopy, R ∗ → M ˆ⊗M ′ → 0<br />
such that there exists a chain map δ : R ∗ → P ∗ ˆ⊗Q ∗ <strong>and</strong> a<br />
commutative diagram<br />
R ∗<br />
δ <br />
M ˆ⊗M ′ <br />
0<br />
P ∗ ˆ⊗Q ∗<br />
<br />
M ˆ⊗M ′ <br />
0.
140 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
If for two integers m, n, ζ ∈ Ext m RC(M, N) <strong>and</strong> ζ ′ ∈ Ext n RC(M ′ , N ′ )<br />
are represented by two cocycles f : P m → N <strong>and</strong> g : Q n →<br />
N ′ , then the product ζ ∪ ζ ′ is represented by (f ˆ⊗g) ◦ δ :<br />
R m+n → N ˆ⊗N ′ . In Theorem 4.1.13 we shall show how to establish<br />
the algebra isomorphism Ext ∗ RC(R, R) ∼ = H ∗ (BC, R).<br />
4.1.2 <strong>Cohomology</strong> <strong>of</strong> a small category with coefficients in a functor<br />
In this section we introduce a particular important case, the<br />
Baues-Wirsching cohomology theory <strong>of</strong> small categories, <strong>and</strong><br />
go over some basic properties. The cohomology theory <strong>of</strong> small<br />
categories has been discussed in various places in literature,<br />
see Baues-Wirsching [3], Generalov [29] <strong>and</strong> Oliver []. One<br />
can also find in Gabriel-Zisman [28] <strong>and</strong> Hilton-Stammbach<br />
[35] the homology theory <strong>of</strong> small categories.<br />
Definition 4.1.6. Let C be a small category <strong>and</strong> R a commutative<br />
ring. We define H n (C; M), the n-th cohomology <strong>of</strong><br />
C with coefficients in a covariant functor M : C → R-<br />
mod, as the homology <strong>of</strong> the following cochain complex<br />
{C i (C; M), δ i } i≥0 , where<br />
∏<br />
C i (C; M) = {f : NC i →<br />
M(x i ) ∣ f([x 0 → · · · → x i ]) ∈ M<br />
[x 0 →x 1 →···→x i ]<br />
for all i ≥ 0; <strong>and</strong> for f ∈ C i (C; M),<br />
δ n φ<br />
i∑<br />
(f)([x 0 → · · · → x i→xi+1 ]) = (−1) j f([x 0 → · · · → ˆx j → · · · →<br />
+(−1) i+1 M(φ)(f([x 0 → · · · → x i ]))<br />
We define Hn(C; M), the n-th homology <strong>of</strong> C with coefficients<br />
in a covariant functor M : C → R-mod, as the homology <strong>of</strong><br />
the following chain complex {C i (C; M), δ i } i≥0 , where<br />
j=0
4.1 General theory 141<br />
C i (C; M) =<br />
⊕<br />
[x 0 →x 1 →···→x i ]<br />
M(x 0 ),<br />
for all i ≥ 0; <strong>and</strong> for any m [x0 →x 1 →···→x i ] ∈ C i (C; M) which<br />
is an element in some m ∈ M(x 0 ) indexed by x 0 → · · · →<br />
ˆx j → · · · → x i ,<br />
δ i ( ⊕ m [x0<br />
φ<br />
→x1 →···→x i−1 →x i ] ) = [M(φ)(m)] [x 1 →···→x i−1 ]<br />
+<br />
i∑<br />
(−1) j m [x0 →···→ ˆx j →···→x i ],<br />
j=0<br />
in which m [x0 →···→ ˆx j →···→x i ] means m ∈ M(x 0 ) is considered<br />
as an element <strong>of</strong> C i−1 (C; M) indexed by the (i − 1)-sequence<br />
x 0 → · · · → ˆx j → · · · → x i .<br />
We now give different interpretations to the above homology<br />
<strong>and</strong> cohomology theory. They will give us better underst<strong>and</strong>ing<br />
<strong>of</strong> the preceding definitions <strong>and</strong> then lead us to a more general<br />
cohomology theory. Recall that for any functor u : D → C<br />
between two small categories, we can naturally define two<br />
functors u/− : C → Cat, the category <strong>of</strong> small categories,<br />
<strong>and</strong> B(u/−) : C → T op, the category <strong>of</strong> topological spaces.<br />
Thus for any u : D → C, we may write C ∗ (u/x, R), x ∈ Ob C<br />
as the chain complex coming from the simplicial set associated<br />
to the small category u/x, which can be regarded as the<br />
cellular chain complex on the space B(u/x).<br />
Definition 4.1.7. Let C be a small category. We define the<br />
bar resolution <strong>of</strong> R ∈ RC-mod as B∗<br />
C = C ∗ (Id C /−, R), a<br />
complex <strong>of</strong> RC-modules.<br />
The bar resolution is a complex reconstructed from C ∗ (C, R)<br />
<strong>and</strong> its name is justified as follows.
142 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
Proposition 4.1.8. For any small category C, B C ∗ is a complex<br />
<strong>of</strong> projective RC-modules such that<br />
1. B C 0 ∼ = RC;<br />
2. B C ∗ → R → 0 is an exact sequence <strong>of</strong> RC-modules;<br />
3. There is a degree 1 isomorphism <strong>of</strong> complexes <strong>of</strong> R-modules<br />
from<br />
C ∗ (Id C /−, R) → R → 0<br />
to R[NC] ∗ = C ∗ (C, R) → 0.<br />
4. Hom RC (B C ∗, M) ∼ = ∏ x 0 →···→x i<br />
M(x i ).<br />
Remark 4.1.9. We emphasize that there is no conflict between<br />
2 <strong>and</strong> 3. A complex <strong>of</strong> RC-modules is exact if <strong>and</strong> only if it is<br />
pointwise exact.<br />
Pro<strong>of</strong>. For each i ≥ 0, C i (Id C /−, R) : C → R-mod is a well<br />
defined functor <strong>and</strong> thus an RC-module. There is an RCmap<br />
∂ i : C i+1 (Id C /−, R) → C i (Id C /−, R) as follows. For any<br />
x ∈ Ob C,<br />
∂ i ((x 0 , α 0 ) → · · · → (x j , α j ) → · · · → (x i+1 , α i+1 ))<br />
= ∑ i+1<br />
j=0 (−1)j [(x 0 , α 0 ) → · · · → (x ̂ j , α j ) → · · · → (x i+1 , α i+1 )],<br />
where α j ∈ Hom C (x j , x). Thus for each x ∈ Ob C, C ∗ (Id C /x, R) →<br />
R → 0 is the augmented chain complex <strong>of</strong> Id C /x. When we assemble<br />
these augmented chain complexes together, C ∗ (Id C /−, R) →<br />
R → 0 becomes a complex <strong>of</strong> RC-modules. Since every Id C /x<br />
has a terminal object (x, 1 x ) <strong>and</strong> thus is contractible, the complex<br />
C ∗ (Id C /−, R) → R → 0 is acyclic because its evaluation<br />
at any x, i.e. C ∗ (Id C /x, R) → R → 0 computes<br />
the reduced homology ˜H∗ (Id C /x, R) <strong>and</strong> is acyclic. Moreover,<br />
we can show every C i (Id C /−, R) is projective. Indeed<br />
for each i > 0, C i (Id C /x, R) has base elements <strong>of</strong> the form<br />
(x 0 , α 0 ) → · · · → (x i , α i ). The following bijection provides a<br />
different presentation <strong>of</strong> these base elements
4.1 General theory 143<br />
{(x 0 , α 0 ) β 1<br />
→ · · · →(x βi<br />
α i β i···β 1 α i β i α<br />
i , α i )} {x 0 −→ · · ·<br />
i<br />
−→xi →x}.<br />
This bijection in fact induces an isomorphism <strong>of</strong> complexes <strong>of</strong><br />
R-modules (see Definitions 1.2.5 <strong>and</strong> 2.2.4)<br />
C i (Id C /−, R) ∼ =<br />
−→R[NC] i+1 = C i+1 (C, R), ∀i ≥ 0.<br />
We get immediately C 0 (Id C /−, R) ∼ = C 1 (C, R) ∼ = RC. In general<br />
let us take the new R-basis <strong>of</strong> C i (Id C /−, R), {x 0 → · · · →<br />
x i → x}, <strong>and</strong> regroup them by putting two such sequences<br />
into one subset <strong>of</strong> the basis if, after removing the rightmost<br />
object, they become identical. Then each subset determines<br />
an RC-module<br />
R{x 0 → · · · → x i<br />
α<br />
→−<br />
∣ ∣ s(α) = x i },<br />
isomorphic to the projective module RHom C (x i , −). It means<br />
as RC-modules<br />
C i (Id C /−, R) ∼ ⊕<br />
=<br />
x 0 →x 1 →···→x i<br />
RHom C (x i , −).<br />
Hence B∗<br />
C → R → 0 is a projective resolution. In the end<br />
for any M ∈ RC-mod, there is an isomorphism by Yoneda<br />
lemma<br />
Hom RC (C i (Id C /−, R), M) ∼ =<br />
∏<br />
x 0 →···→x i<br />
M(x i ).<br />
We remind the reader that R can be defined as R Ob C ∼ =<br />
R[NC] 0<br />
∼ = C0 (C, R). Thus we have a degree one isomorphism<br />
<strong>of</strong> chain complexes (<strong>of</strong> R-modules) from C ∗ (Id C /−, R) →<br />
R → 0 to R[NC] ∗ = C ∗ (C, R) → 0 if in the first complex<br />
we insert R in degree -1.<br />
⊓⊔<br />
Example 4.1.10. In Example 2.2.6 (1) we considered the following<br />
category with two objects, two identity <strong>and</strong> non-
144 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
identity morphisms<br />
x<br />
α<br />
β<br />
Its nerve gives rise to a non-normalized chain complex C ∗ (C, R)<br />
which is <strong>of</strong> rank 2 in degree zero, rank 4 in degree 1 <strong>and</strong> rank<br />
2n + 2 at degree n ≥ 2. It is certainly not exact.<br />
The bar resolution B C ∗ can be written down by computing<br />
the two overcategories Id C /x <strong>and</strong> Id C /y. In fact Id C /x is a<br />
trivial category with one object (x, 1 x ), <strong>and</strong> Id C /y is a poset<br />
with three objects (x, α), (x, β) <strong>and</strong> (y, 1 y ).<br />
(x, α)<br />
α<br />
(y, 1 y )<br />
y<br />
β<br />
(x, β)<br />
Hence the bar resolution is given by B C ∗(y) <strong>and</strong> B C ∗(x). The<br />
former has rank 3 in degree zero, rank 5 in degree 1, <strong>and</strong><br />
rank 2n + 3 in degree n ≥ 2; while the latter has rank 1 at<br />
every degree. Thus B C ∗ has rank 4 in degree zero, rank 6 in<br />
degree 1 <strong>and</strong> rank 2n + 4 in degree n ≥ 2. One can see that<br />
Rank(B C i ) =Rank(C i+1 (C, R)) for all i ≥ 0. This equality is<br />
actually induced by the R-isomorphism we explained in the<br />
pro<strong>of</strong> <strong>of</strong> Proposition 3.1.3. Note that both B C ∗(y) <strong>and</strong> B C ∗(x)<br />
are exact because they come from the nerves <strong>of</strong> the two overcategories<br />
Id C /y <strong>and</strong> Id C /x which are contractible. Furthermore<br />
we can explicitly verify that B C 0 ∼ = RC, B C 1 ∼ = P x,1 ⊕P 3 y,1,<br />
etc.<br />
If one examines the normalized complexes <strong>of</strong> C ∗ (C, R),<br />
C ∗ (Id C /y, R) <strong>and</strong> C ∗ (Id C /x, R) then the above equality between<br />
ranks are not true any more.<br />
Remark 4.1.11. Proposition 4.1.8 summarizes to a characterization<br />
<strong>of</strong> the bar resolution in terms <strong>of</strong> the nerve, as well as
4.1 General theory 145<br />
an isomorphism<br />
Hom RC (B C i , M) ∼ =<br />
∏<br />
x 0 →···→x i<br />
M(x i ).<br />
By contrast, when we consider R as a right RC-module. The<br />
bar resolution <strong>of</strong> it is C ∗ (−\Id C , R). We shall also denote it<br />
by B∗ C when it does not cause any confusion. For each i ≥ 0,<br />
C i (−\Id C , R) ∼ ⊕<br />
=<br />
x 0 →x 1 →···→x i<br />
RHom C (−, x 0 ).<br />
Since RHom C (−, x 0 ) = 1 x0 · RC, we can verify that<br />
Bi C ⊗ RC M ∼ ⊕<br />
=<br />
x 0 →x 1 →···→x i<br />
M(x 0 ).<br />
When we examine the special case for M = R, we get isomorphisms<br />
<strong>of</strong> complexes<br />
Hom RC (Bi C , R) ∼ =<br />
∏<br />
<strong>and</strong><br />
B C i ⊗ RC R ∼ =<br />
x 0 →···→x i<br />
R ∼ = Hom R (R[NC] i , R).<br />
⊕<br />
x 0 →···→x i<br />
R ∼ = R[NC] i .<br />
Proposition 4.1.12. Let C be a small category <strong>and</strong> R a<br />
commutative ring. If M : C → R-mod is a functor, then<br />
lim<br />
−→ n M ∼ = Tor RC<br />
C n (R, M) ∼ = Hn(C; M),<br />
<strong>and</strong><br />
lim<br />
←− n M ∼ = Ext n C RC(R, M) ∼ = H n (C; M).<br />
Pro<strong>of</strong>. Let U be an arbitrary R-module. Then we have<br />
Hom R (U, Hom RC (R, M)) ∼ = Hom RC (R⊗ R U, M) ∼ = Hom RC (U, M),
146 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
where U ∈ RC-mod is a constant functor. Hence by Proposition<br />
1.1.23, ←−C<br />
lim M ∼ = Hom RC (R, M).<br />
Similarly<br />
Hom R (R⊗ RC M, U) ∼ = Hom RC (M, Hom R (R, U)) ∼ = Hom RC (M, U),<br />
<strong>and</strong> consequently R ⊗ RC M ∼ = lim −→C<br />
M.<br />
We will only prove the isomorphisms for cohomology because<br />
the statements for homology can be obtained similarly.<br />
Since the kernel <strong>of</strong> the differential δ 0 : C 0 (C; M) → C 1 (C; M)<br />
can be identified with Hom RC (R, M), we get H 0 (C; M) ∼ =<br />
Hom RC (R, M).<br />
n<br />
Since lim ←−C<br />
is left exact <strong>and</strong> ←−<br />
lim is its n-th right derived<br />
C<br />
n<br />
functor, we have ←−<br />
lim M ∼ = Ext n C RC(R, M). Hence we only have<br />
to show Ext n RC(R, M) ∼ = H n (C; M). Because Ext n RC(R, M) can<br />
be computed by using any projective resolution <strong>of</strong> R in RCmod,<br />
we can use the bar resolution to do it. But we have an<br />
isomorphism for each i ≥ 0<br />
Hom RC (Bi C , M) ∼ =<br />
∏<br />
x 0 →···→x i<br />
M(x i ) ∼ = C i (C; M).<br />
Suppose C = G is a group <strong>and</strong> M, N are two left RGmodules,<br />
since a left RG-module can be naturally regarded as<br />
a right RG-module (<strong>and</strong> vice versa), it makes sense to consider<br />
both Tor RG<br />
∗ (M, N) <strong>and</strong> Tor RG<br />
∗ (N, M). Indeed we always have<br />
Tor RG<br />
∗ (M, N) ∼ = Tor RG<br />
∗ (N, M), see [12, Chapter 3]. It is not<br />
the case in general for category homology.<br />
We deduce now that the preceding homology <strong>and</strong> cohomology<br />
theories are the same as the simplicial <strong>and</strong> singular homology<br />
<strong>and</strong> cohomology theories we introduced in Chapter<br />
2.<br />
⊓⊔
4.1 General theory 147<br />
Theorem 4.1.13. We have isomorphisms<br />
<strong>and</strong><br />
Tor RC<br />
∗ (R, R) ∼ = H∗(C; R) ∼ = H∗(C, R) ∼ = H∗(BC, R)<br />
Ext n RC(R, R) ∼ = H ∗ (C; R) ∼ = H ∗ (C, R) ∼ = H ∗ (BC, R).<br />
The latter are algebra isomorphisms.<br />
Pro<strong>of</strong>. In order to prove the isomorphisms between graded R-<br />
modules, we only need to show that H∗(C; R) ∼ = H∗(C, R) <strong>and</strong><br />
H ∗ (C; R) ∼ = H ∗ (C, R). But these follow from Remark 4.1.11<br />
which compare Definitions 2.2.4 with 4.1.6.<br />
In order to prove the isomorphism between cohomology rings<br />
is an algebra isomorphism we just have to compare the cup<br />
products. Here we only need to explain the first algebra isomorphism<br />
because the second is clear.<br />
Using the bar resolution B∗<br />
C → R → 0, we can describe<br />
the cup product on Ext ∗ RC(R, R). In fact, we can explicitly<br />
write out a diagonal approximation map (unique up to chain<br />
homotopy)<br />
R 0<br />
B C ∗<br />
D <br />
B∗ C ˆ⊗B∗<br />
C <br />
R 0<br />
as a natural transformation, given by<br />
α<br />
D x (x<br />
1<br />
n∑<br />
0→x1 → · · · →x αn α α<br />
n→x) = (x<br />
1<br />
α···α i+1<br />
0→ · · · → xi<br />
i=0<br />
→ x)⊗(xi<br />
α i+1<br />
for any x ∈ Ob C <strong>and</strong> integer n. Since for each n ≥ 0,<br />
(B C ∗ ˆ⊗B C ∗) n = ⊕ i+j=n BC i ˆ⊗B C j <strong>and</strong> there is a natural map<br />
Hom RC (B C i , R) ⊗ Hom RC (B C j , R) → Hom RC (B C i ˆ⊗B C j , R),<br />
→ · · ·<br />
α<br />
→
148 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
one can easily see, under the isomorphism Hom RC (B C ∗, R) ∼ =<br />
Hom R (R[NC] ∗ , R), that the diagonal approximation map D<br />
induces a map<br />
Hom RC (B C ∗, R) ⊗ Hom RC (B C ∗, R) → Hom RC (B C ∗, R)<br />
identical to the following<br />
Hom R (R[NC] ∗ , R)⊗Hom R (R[NC] ∗ , R) → Hom R (R[NC] ∗ , R)<br />
induced by the Alex<strong>and</strong>er-Whitney map. Since these two<br />
maps are used to calculate cup products in the two cohomology<br />
rings, our observations imply that Ext ∗ RC(R, R) ∼ =<br />
H ∗ (BC, R) as algebras.<br />
⊓⊔<br />
Recall that the bar resolution can also be constructed via the<br />
nerve <strong>of</strong> overcategories associated with the identity functor<br />
Id C : C → C, as B∗ C ∼ = C ∗ (Id C /−, R) := RN ∗ (Id C /−). In this<br />
form, Bn(x) C ∼ = C n (Id C /x), for each x ∈ Ob C <strong>and</strong> integer<br />
n ≥ 0, consists <strong>of</strong> the following chains as base elements<br />
(x 0 , β 0 ) γ 1<br />
→(x 1 , β 1 ) → · · · → (x n−1 , β n−1 ) γ n<br />
→(x n , β n ),<br />
in which β i is a morphism in Hom C (x i , x), <strong>and</strong> γ i ∈ Hom C (x i−1 , x i )<br />
such that β i−1 = β i γ i−1 . The previously defined diagonal approximation<br />
map D is given by<br />
D x ((x 0 , β 0 ) γ 1<br />
→(x 1 , β 1 ) → · · · → (x n−1 , β n−1 ) γ n<br />
→(x n , β n ))<br />
= ∑ n<br />
i=0 [(x 0, β 0 ) γ 1<br />
→ · · · →(x γi<br />
i , β i )] ⊗ [(x i , β i ) γ i+1<br />
→ · · · →(x γn<br />
n , β n )]<br />
This is exactly the Alex<strong>and</strong>er-Whitney map for the chain<br />
complex coming from the nerve <strong>of</strong> Id C /x.<br />
All homology <strong>and</strong> cohomology theories <strong>of</strong> small categories<br />
that we have introduced so far coincide whenever they are<br />
comparable. Hence we shall only deal with the most general<br />
form Ext ∗ RC(M, N) <strong>and</strong> Tor RC<br />
∗ (M ′ , N) from now on, where<br />
M, N ∈ RC-mod <strong>and</strong> M ′ ∈ mod-RC.
4.1 General theory 149<br />
Proposition 4.1.14. Suppose u : D → C is a functor. Then<br />
we have a restriction Res u : RC-mod → RD-mod. Given two<br />
RC-modules M, N, Res u induces a natural map<br />
res u : Ext ∗ RC(M, N) → Ext ∗ RD(Res u M, Res u N),<br />
called the restriction.<br />
Pro<strong>of</strong>. Take a projective resolution P ∗ <strong>of</strong> M. Then Res u P ∗ →<br />
Res u M → 0 is an exact sequence <strong>of</strong> RD-modules. Hence<br />
for any projective resolution Q ∗ → Res u M → 0, the usual<br />
lifting, a chain map, Q ∗ → Res u P ∗ induces a cochain map<br />
Hom RC (P ∗ , N) → Hom RD (Q ∗ , Res u N) because we have a<br />
cochain map Hom RC (P ∗ , N) → Hom RD (Res u P ∗ , Res u N). This<br />
is a well defined map since it does not depend on the choice<br />
<strong>of</strong> P ∗ <strong>and</strong> Q ∗ .<br />
⊓⊔<br />
When M = N = R, by using bar resolutions, one can easily<br />
see that the above restriction is the same as the one we defined<br />
for simplicial cohomology in Chapter 2.<br />
Remark 4.1.15. Suppose u : D → C is a functor. Let us consider<br />
res u : Ext ∗ RC(R, N) → Ext ∗ RD(R, Res u N).<br />
One way to obtain some information about the restriction is to<br />
examine its action in degree zero. In fact, the restriction is the<br />
same as Hom RC (R, N) → Hom RC (LK u R, N) ∼= →Hom RD (R, Res u N)<br />
induced by the counit LK u R → R. When LK u R ∼ = R, res u<br />
becomes an isomorphism in degree zero. Giving LK u R ∼ = R<br />
is equivalent to saying that every overcategory <strong>of</strong> u is R-<br />
acyclic, that is, ˜H∗ (u/x, R) vanishes for every x ∈ Ob C. In<br />
other words, every overcategory u/x has to be connected, see<br />
Proposition 3.2.7.
150 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
4.1.3 Extensions <strong>of</strong> categories <strong>and</strong> low dimension cohomology<br />
We describe low dimension cohomology groups Ext n RC(M, N)<br />
for n = 1, 2. The results <strong>and</strong> their pro<strong>of</strong>s are taken from<br />
Webb’s notes [80]. Let us recall that there is an augmentation<br />
map ɛ : RC → R by sending α to t(α), its target. The kernel<br />
<strong>of</strong> ɛ is a left ideal <strong>of</strong> RC, which we call the (left) augmentation<br />
ideal, <strong>and</strong> write as IC := Kerɛ.<br />
Lemma 4.1.16. The augmentation ideal IC is a free R-<br />
module with basis the elements α − 1 t(α) , where α runs over<br />
all the non-identity morphisms in C.<br />
Definition 4.1.17. Let M ∈ RC-mod. We define a derivation<br />
d : C → M to be a mapping from Mor C to M so that<br />
d(α) ∈ M(t(α)) <strong>and</strong> so that d(αβ) = M(α)d(β) + d(α). The<br />
set <strong>of</strong> derivations forms an R-module Der(C, M).<br />
Given any set <strong>of</strong> elements {u x ∈ M(x) ∣ ∣x ∈ Ob C} we obtain<br />
a derivation specified by d(α) = M(α)(u s(α) ) − u t(α) . Any<br />
derivation obtained in this way is called an inner derivation.<br />
The inner derivations form an R-module IDer(C, M).<br />
Proposition 4.1.18.1. Der(C, M) ∼ = Hom RC (IC, M) as R-<br />
modules.<br />
2. H 1 (C; M) ∼ = Der(C, M)/IDer(C, M).<br />
Pro<strong>of</strong>. Given a homomorphism δ : IC → M, we can define<br />
a derivation d : C → M by d(α) = δ(α − 1 t(α) ). It makes<br />
sense because M(α)d(β) + d(α) = δ[(α(β − 1 t(β) )] + d(α) =<br />
δ(αβ − α) + d(α) = δ(αβ − 1 t(αβ) )) = d(αβ) since t(αβ) =<br />
t(α). Conversely given a derivation d we can define a RChomomorphism<br />
δ : IC → M by δ(α − 1 t(α) ) = d(α), <strong>and</strong> we<br />
can verify that δ is an RC-map. Hence we have proved (1).<br />
As for (2), consider the short exact sequence 0 → IC →<br />
RC → R → 0 <strong>and</strong> apply Ext ∗ RC(−, M). We obtain an exact
4.1 General theory 151<br />
sequence<br />
0 → Hom RC (R, M) → Hom RC (RC, M) → Hom RC (IC, M) → Ext 1 RC(R<br />
The image <strong>of</strong> Hom RC (RC, M) ∼ = M consists <strong>of</strong> f m | IC , where<br />
f m is the RC-homomorphism determined by some m ∈ M via<br />
the canonical map m ↦→ f m such that f m (1 RC ) = m. Because<br />
f m (α − 1 t(α) ) = [M(α)](m) − [M(1 t(α) )](m) = [M(α)](1 s(α) ·<br />
m) − [M(1 t(α) )](1 t(α) · m), f m | IC is an inner derivation determined<br />
by the set <strong>of</strong> elements {1 x · m ∈ M(x) ∣ ∣ x ∈ Ob C}.<br />
Moreover all inner derivations are <strong>of</strong> this form. We are done.<br />
⊓⊔<br />
In order to characterize H 2 (C; K), we have to introduce category<br />
extensions in the sense <strong>of</strong> G. H<strong>of</strong>f.<br />
Definition 4.1.19 (H<strong>of</strong>f). An extension E <strong>of</strong> a category C<br />
via a category K is a sequence <strong>of</strong> functors<br />
K ι<br />
−→E π<br />
−→C,<br />
which has the following properties:<br />
1. Ob K = Ob E = Ob C, ι is injective <strong>and</strong> π is surjective on<br />
morphisms; <strong>and</strong><br />
2. if π(α) = π(β), for two morphisms α, β ∈ Mor(E), if <strong>and</strong><br />
only if there is a unique g ∈ Mor(K) such that β = ι(g)α.<br />
We may readily deduce some useful information from the<br />
above definition.<br />
Proposition 4.1.20.3. If αι(h) exists for α ∈ Mor(E) <strong>and</strong><br />
h ∈ Mor(K), then there exists a unique h α ∈ Mor(K) such<br />
that ι(h α )α = αι(h). Moreover any α ∈ Hom E (x, y) induces<br />
a group homomorphism K(x) → K(y); <strong>and</strong><br />
4. for any α ∈ Hom C (x, y), K(y) acts regularly (i.e. freely<br />
<strong>and</strong> transitively) on π −1 (α).
152 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
Pro<strong>of</strong>. Suppose αι(h) exists for a given α ∈ Mor E. Then<br />
since π(αι(h)) = π(α), by Definition 4.1.19 (2), there exists<br />
a unique h α ∈ Mor K such that αι(h) = ι(h α )α. If h 1 , h 2 ∈<br />
K(x), then we have αι(h 1 h 2 ) = ι[(h 1 h 2 ) α ]α. On the other<br />
h<strong>and</strong>, we also get αι(h 1 h 2 ) = αι(h 1 )ι(h 2 ) = ι(h 1 ) α [αι(h 2 )] =<br />
ι(h α 1)ι(h α 2)α. Hence h α 1h α 2 = (h 1 h 2 ) α <strong>and</strong> α induces a group<br />
homomorphism.<br />
Now assume β ∈ Hom E (x, y) <strong>and</strong> π(β) = α. Let h ∈ K(y).<br />
If ι(h)β = β, then we have π(ι(h)β) = π(β) = α. Definition<br />
4.1.19 (2) forces h = 1 y because <strong>of</strong> the uniqueness property.<br />
⊓⊔<br />
One can see that K is indeed a disjoint union <strong>of</strong> the groups<br />
π −1 (1 x ) for all 1 x ∈ Mor(C) (regarded as categories), <strong>and</strong> can<br />
be identified with a functor K : E → Groups. Usually from<br />
the context, one knows when we take K to be a category <strong>and</strong><br />
when it is regarded as a functor.<br />
Since in practice one <strong>of</strong>ten crosses a concept dual to the<br />
category extension, for future reference, we first state its definition.<br />
Definition 4.1.21. An opposite extension E <strong>of</strong> C via K is<br />
a sequence <strong>of</strong> functors K ι →E π →C such that the following sequence<br />
is an extension <strong>of</strong> C op<br />
K op ι op<br />
−→E op π op<br />
−→C op ,<br />
Sometimes we just say K → E → C is an opposite extension.<br />
Dually there are characterizations as follows.<br />
1 op . Ob K = Ob E = Ob C, ι is injective <strong>and</strong> π is surjective on<br />
morphisms; <strong>and</strong><br />
2 op . if π(α) = π(β), for two morphisms α, β ∈ Mor(E), if <strong>and</strong><br />
only if there is a unique g ∈ Mor(K) such that β = αι(g).
4.1 General theory 153<br />
3 op . If ι(h)α exists for α ∈ Mor(E) <strong>and</strong> h ∈ Mor(K), then there<br />
exists a unique h ′ ∈ Mor(K) such that αι(h ′ ) = ι(h)α; <strong>and</strong><br />
4 op . for any α ∈ Hom C (x, y), K(x) acts regularly on π −1 (α).<br />
One <strong>of</strong> the main difference between extensions <strong>and</strong> opposite<br />
extensions is that, for an extension, K can be considered as<br />
a covariant functor from C to Groups while, for an opposite<br />
extension, K gives a contra-variant functor. When we study<br />
H 1 <strong>and</strong> H 2 , extensions are used because we deal with left modules.<br />
If dually we want to obtain corresponding statements for<br />
right modules, then we have to replace extensions by opposite<br />
extensions in wherever it applies.<br />
Example 4.1.22.1. Any category C is an extension <strong>and</strong> opposite<br />
extension <strong>of</strong> itself by ⊎ Ob C<br />
•, a disjoint union <strong>of</strong> some<br />
trivial groups.<br />
2. Any extension <strong>of</strong> a group is both an extension <strong>and</strong> an opposite<br />
extension if we consider the group <strong>and</strong> its extension as<br />
categories.<br />
3. The following sequence is an extension <strong>of</strong> the rightmost category<br />
by a group which is the disjoint union <strong>of</strong> a trivial group<br />
<strong>and</strong> a cyclic group <strong>of</strong> order 2:<br />
x<br />
C 2<br />
<br />
{1 y }<br />
y<br />
C 2<br />
<br />
y<br />
ι π<br />
α<br />
<br />
x<br />
C 2<br />
<br />
x<br />
{1 x }<br />
α<br />
C 2<br />
<br />
y<br />
.<br />
4. The following sequence is an opposite extension <strong>of</strong> the rightmost<br />
category by a group which is the disjoint union <strong>of</strong> a<br />
trivial group <strong>and</strong> a cyclic group <strong>of</strong> order 2:
154 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
C 2<br />
<br />
y<br />
C 2<br />
<br />
y<br />
{1 y }<br />
y<br />
x<br />
{1 x }<br />
ι π<br />
α<br />
<br />
x<br />
C 2<br />
<br />
α<br />
x .<br />
C 2<br />
One shall easily see the difference <strong>and</strong> relationship between<br />
the first two examples. We note that in both examples, the<br />
categories at the two ends have finitely generated cohomology<br />
rings, while the ones in the middle do not, see Example<br />
2.2.4 (4). <strong>and</strong> Proposition 2.2.38.<br />
5. Let G be a finite group <strong>and</strong> p a prime dividing the order <strong>of</strong> G.<br />
A collection P <strong>of</strong> p-subgroups <strong>of</strong> G is a set <strong>of</strong> p-subgroups<br />
which is closed under conjugations in G. The transporter<br />
category G ∝ P = Gr G P has various quotient categories,<br />
e.g. the orbit category O P (G), many <strong>of</strong> which form extension<br />
sequences, e.g.<br />
K s → G ∝ P → O P (G),<br />
where K s (H) = H for any H ∈ Ob P = Ob(G ∝ P). We<br />
shall come back to this in Chapter 6.<br />
Definition 4.1.23. Two extensions are equivalent if we have<br />
the following commutative diagram<br />
K E<br />
<br />
C<br />
K E ′ <br />
C<br />
Definition 4.1.24. An extension is split if it admits a functor<br />
s : C → E such that π ◦ s = 1 C .<br />
Proposition 4.1.25. An extension is split if <strong>and</strong> only if it<br />
is equivalent to a Grothendieck construction.
4.1 General theory 155<br />
Pro<strong>of</strong>. Assume an extension K → E → C is split. Then<br />
K can be restricted to a functor C → E → Groups. The<br />
Grothendieck construction Gr C K has the same objects <strong>of</strong> the<br />
form (x, • x ) where x ∈ Ob C <strong>and</strong> • x is the only object <strong>of</strong><br />
K(x). A morphism from (x, • x ) to (y, • y ) is a pair (α, h) with<br />
α ∈ Hom C (x, y) <strong>and</strong> h ∈ K(y). Hence we can define a funtor<br />
Gr C K → E by (x, • x ) ↦→ x <strong>and</strong> (α, h) ↦→ hα.<br />
On the other h<strong>and</strong> for an extension K → Gr C K → C, we can<br />
define a functor C → Gr C K by x ↦→ (x, • x ) <strong>and</strong> α ↦→ (α, 1 y )<br />
if α ∈ Hom C (x, y). Thus this extension is split.<br />
⊓⊔<br />
Proposition 4.1.26. The equivalence classes <strong>of</strong> extensions<br />
<strong>of</strong> C by K are in bijection with H 2 (C; K) in such a way that<br />
the zero element corresponds to the Grothendieck construction<br />
Gr C K.<br />
Pro<strong>of</strong>. We will provide correspondence between equivalence<br />
classes <strong>of</strong> extensions <strong>and</strong> elements <strong>of</strong> H 2 (C; K).<br />
Let K → E → C be an extension. We can choose a section for<br />
E → C, that is, for each morphism α : x → y in C a morphism<br />
q(α) : x → y in E with pq(α) = α. If α : x → y <strong>and</strong> β : y → z<br />
are two morphisms in C, then q(βα) = ι[τ(β, α)]q(β)q(α) for<br />
a unique τ(β, α) ∈ K(z). Thus we obtain a mapping τ :<br />
C × C → C. The associativity <strong>of</strong> morphisms implies the 2-<br />
cocycle condition by direct calculation<br />
τ(γβ, α) + τ(γ, β) = τ(γ, βα) + K(γ)τ(β, α).<br />
If f ∈ Hom RC (RC, M), then mappings <strong>of</strong> the form<br />
˜f(β, α) := f(βα) − f(β) − K(β)f(α)<br />
automatically satisfies the 2-cocycle condition. If we examine<br />
Hom RC (B C ∗, M), used to define H ∗ (C; M) in Definition 3.1.1,<br />
we can verify that all these mappings τ <strong>and</strong> ˜f biject with<br />
the 2-cocycles <strong>and</strong> 2-coboudaries in the the cochain complex
156 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
Hom RC (B∗, C M). In fact we already learned that<br />
Hom RC (B2, C M) ∼ =<br />
∏<br />
K(z)<br />
<strong>and</strong><br />
x α →y β →z<br />
Hom RC (B C 1, M) ∼ = ∏ v γ →w<br />
K(w).<br />
With explicitly given differential we can prove the above bijection.<br />
In the end, we may verify that 2-cocycles are homologous<br />
if <strong>and</strong> only if the extensions which produce them are equivalent,<br />
<strong>and</strong> also a change in the choice <strong>of</strong> a section gives rise to<br />
a cohomologous 2-cocycle. Hence we are done.<br />
⊓⊔<br />
To finish this section, we comment on the connection between<br />
H i (C; M) <strong>and</strong> Ext i RC(R, M), for i = 1, 2. In fact from<br />
the short exact sequence 0 → IC → RC → R → 0, we get<br />
Ext i RC(R, M) ∼ = Ext i−1<br />
RC (IC, M).<br />
Let 0 → M → E → IC → 0 be an RC-module extension<br />
representing an element in Ext 1 RC(IC, M). We can construct<br />
an extension <strong>of</strong> categories<br />
M → Gr C E → Gr C IC.<br />
There is a splitting functor C → Gr C IC, which is identity on<br />
objects <strong>and</strong> sends a morphism α to (α − 1 t(α) , α). Now we<br />
form a pullback diagram<br />
M<br />
<br />
E<br />
<br />
C<br />
M Gr C E Gr C IC
4.2 Classical methods for computation 157<br />
which gives rise to an extension M → E → C, an element in<br />
H 2 (C; M). Note that if 0 → M → E → IC → 0 splits, then<br />
Gr C E ∼ = Gr GrC ICM, in H 1 (C; M).<br />
Conversely if M → E → C is an extension <strong>and</strong> IM is the<br />
kernel <strong>of</strong> the algebra homomorphism RE → RC. Then we can<br />
define an extension <strong>of</strong> RC-modules<br />
0 → M → IE/(IM · IE) → IC → 0,<br />
representing an element in Ext 2 RC(R, M) ∼ = Ext 1 RC(IC, M).<br />
4.2 Classical methods for computation<br />
4.2.1 Minimal resolutions <strong>and</strong> reduction<br />
Recall in Section 3.3.3 we described the minimal projective<br />
<strong>and</strong> injective resolutions <strong>of</strong> a module. With Definition 3.3.13<br />
we establish the following isomorphism.<br />
Proposition 4.2.1. Let C be an EI category, M, N two kCmodules<br />
<strong>and</strong> M ′ a right kC-module. Then<br />
<strong>and</strong><br />
Ext ∗ kC(M, N) ∼ = Ext ∗ kC N M(Res i M, Res i N)<br />
Tor kC<br />
∗ (M ′ , N) ∼ = Tor ∗ (Res<br />
kC M′ i M ′ , Res i N).<br />
Pro<strong>of</strong>. Suppose P ∗ is a minimal projective resolution <strong>of</strong> M.<br />
Then each direct summ<strong>and</strong> P x,V <strong>of</strong> any P n must satisfy<br />
the condition that x ∈ Ob C M by Corollary 3.3.15. Hence<br />
Hom kC (M, N) ∼ = Hom kCM (Res i M, Res i N) <strong>and</strong> Ext ∗ kC(M, N) ∼ =<br />
Ext ∗ kC M<br />
(Res i M, Res i N). Now we apply Corollary 3.3.15 again<br />
to a minimal injective resolution <strong>of</strong> N when computing<br />
<strong>and</strong> the isomorphism follows.<br />
Ext ∗ kC M<br />
(Res i M, Res i N)<br />
N
158 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
To prove the isomorphism for Tor the method is the same.<br />
One just need to take the right projective resolution <strong>of</strong> M ′ .<br />
⊓⊔<br />
This result is useful because one may replace the original<br />
category by a full subcategory before carrying on any computations.<br />
It is particular helpful if both M <strong>and</strong> N are atomic.<br />
Since any module <strong>of</strong> an EI category algebra has a filtration by<br />
atomic modules. Repeatedly using the above corollary may<br />
significantly simplify things. In following examples we will<br />
shall how one combines resolutions, module filtrations <strong>and</strong><br />
reductions through Proposition 4.2.1 to compute various Ext<br />
groups.<br />
Example 4.2.2.1. Suppose C = n is a poset. Then k ∼ =<br />
kHom n (0, −) is a projective module. Thus Ext ∗ kn(k, M) ∼ =<br />
M(0) for any M ∈ kn-mod.<br />
2. Suppose C is the following category<br />
x<br />
α<br />
β<br />
y.<br />
We will only write down non-vanishing Ext groups in this<br />
example.<br />
The minimal projective resolution <strong>of</strong> k is 0 → P y,k →<br />
P x,k → k → 0. Thus for any M ∈ kC-mod Ext ∗ kC(k, M)<br />
is given by the homology <strong>of</strong> 0 → Hom kC (P x,k , M) →<br />
Hom kC (P y,k , M) → 0. If M = k, then Ext 0 kC(k, k) ∼ =<br />
Ext 1 kC(k, k) ∼ = k. If M = S x,k , then Ext 0 kC(k, S x,k ) ∼ = k.<br />
If M = S y,k , then Ext 1 kC(k, S y,k ) ∼ = k.<br />
The minimal projective resolution <strong>of</strong> S x,k is 0 → Py,k 2 →<br />
P x,k → S x,k → 0. We can use it for direct calculation, but<br />
we want to make use <strong>of</strong> Proposition 4.2.1. Ext ∗ kC(S x,k , S x,k ) ∼ =<br />
Ext ∗ k{x}(Res i S x,k , Res i S x,k ) ∼ = Ext ∗ k(k, k). Hence Ext 0 kC(S x,k , S x,k ) ∼ =
4.2 Classical methods for computation 159<br />
k. Proposition 4.2.1 is not useful for computing Ext ∗ kC(S x,k , S y,k )<br />
so we have to compute the homology <strong>of</strong> 0 → Hom kC (P x,k , S y,k ) →<br />
Hom kC (P 2 y,k , S y,k) → 0, which gives Ext 1 kC(S x,k , S y,k ) ∼ = k 2 .<br />
For Ext ∗ kC(S x,k , k) we can use a resolution but we can also<br />
use the long exact sequence<br />
0 → Ext 0 kC(S x,k , k) → Ext 0 kC(S x,k , S x,k ) → Ext 1 kC(S x,k , S y,k ) → Ext 1 kC<br />
from<br />
0 → S y,k → k → S x,k → 0<br />
after applying Ext ∗ kC(S x,k , −). Since Ext 0 kC(S x,k , k) ∼ = Hom kC (S x,k , k)<br />
k, we know immediately Ext 1 kC(S x,k , k) ∼ = k 2 .<br />
3. Suppose k is a field <strong>of</strong> characteristic 2. Consider the following<br />
category D<br />
α <br />
y {1 y ,g}<br />
<br />
{1 x } x<br />
with gα = α. Then here are two one-dimensional simple<br />
modules S x,k <strong>and</strong> S y,k , together with their projective<br />
covers P x,k = k{1 x , α} <strong>and</strong> P y,k = {1 y , g}. Particularly<br />
k ∼ = kHom D (x, −) = P x,k<br />
M ∈ kD-mod, Ext ∗ kD(k, M) ∼ = M(x).<br />
The minimal projective resolution <strong>of</strong> S x,k is<br />
is projective. Thus for any<br />
· · · → P y,k → P y,k → P x,k → S x,k → 0,<br />
which is infinite <strong>and</strong> consists <strong>of</strong> a copy <strong>of</strong> P y,k in each degree<br />
n > 0. Thus Ext 0 kD(S x,k , S x,k ) ∼ = k <strong>and</strong> Ext n kD(S x,k , S y,k ) ∼ = k<br />
for all n > 0.<br />
The minimal projective resolution <strong>of</strong> S y,k is<br />
· · · → P y,k → P y,k → P y,k → S y,k → 0,<br />
which is infinite <strong>and</strong> consists <strong>of</strong> a copy <strong>of</strong> P y,k in each degree.<br />
We get Ext n kD(S y,k , S y,k ) ∼ = k for every n ≥ 0. Moreover<br />
Ext ∗ kD(S y,k , S y,k ) ∼ = Ext ∗ kC 2<br />
(k, k) by Proposition 4.2.1. We
160 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
leave the computation <strong>of</strong> Ext ∗ kD(−, k) <strong>and</strong> <strong>of</strong> all above Ext<br />
groups when k = C to the reader.<br />
One can determine the ring structure <strong>of</strong> Ext ∗ kC 2<br />
(k, k). In<br />
fact C 2 has a very simple bar resolution so one may quickly<br />
find, using simplicial method, that Ext ∗ kC 2<br />
(k, k) ∼ = k[X] is a<br />
polynomial ring with deg X = 1.<br />
Since Ext ∗ kD(k, k) ∼ = k, both Ext ∗ kD(S x,k , S y,k ) <strong>and</strong> Ext ∗ kD(S y,k , S y,k )<br />
are not finitely generated modules.<br />
If we look at the opposite category D op , again we have<br />
Ext ∗ kD op(k, k) ∼ = Ext ∗ kD(k, k) ∼ = k, <strong>and</strong> Ext ∗ kD op(k, S y,k) ∼ =<br />
Ext ∗ kD op(S y,k, S y,k ) is <strong>of</strong> infinite dimensional. It means we do<br />
not have the finite generation <strong>of</strong> Ext ∗ kD(k, M) over Ext ∗ kD(k, k)<br />
even if they can be calculated by using the same projective<br />
resolution.<br />
4.2.2 Examples using classifying spaces<br />
Underst<strong>and</strong>ing the homotopy type <strong>of</strong> a classifying space certainly<br />
will be a great help for compute cohomology rings. Here<br />
we list several cases where direct calculation is possible.<br />
Example 4.2.3. In Example 4.2.2, part (1) is a contractible<br />
category n so we know Ext ∗ kn(k, k) ∼ = k. Part (2) is a category<br />
whose classifying space is the circle S 1 . Part (3) is a<br />
contractible category because it has an initial object.<br />
However, there are more things we can take from homotopy<br />
theory. If we know all Ext ∗ kC(k, −), then we may readily write<br />
down Ext ∗ kCop(k, −).<br />
Now we examine some more sophisticated examples. Although<br />
they do not tell much about cohomology with local<br />
coefficients, they do provide substantial information about the<br />
ordinary cohomology ring <strong>of</strong> a category.<br />
Example 4.2.4.1. Let C be the following category
4.2 Classical methods for computation 161<br />
G<br />
x<br />
α <br />
y H .<br />
Then this is the join <strong>of</strong> G <strong>and</strong> H. Thus its classifying space<br />
is a join <strong>of</strong> spaces BC ≃ BG ∗ BH. Thus its cohomology is<br />
completely determined by G <strong>and</strong> H. Also the cup product<br />
is trivial.<br />
2. Let D be the following category<br />
h<br />
x<br />
1 x<br />
α<br />
β<br />
y {1 y }<br />
,<br />
where h 2 = 1 x <strong>and</strong> βh = α. Then the inclusion i :<br />
Aut D (x) → D induces a homotopy equivalence between<br />
their classifying spaces. In particular, we have a ring isomorphism<br />
Ext ∗ RD(R, R) = Ext ∗ RAut D (x)(R, R) for any ring R.<br />
When R = C, the cohomology ring is just C, while when R<br />
is a field <strong>of</strong> characteristic 2 the ordinary cohomology ring is<br />
a polynomial algebra with indeterminant in degree one.<br />
3. Let E be the following category<br />
h<br />
g<br />
x<br />
1 x<br />
α<br />
β<br />
gh<br />
y {1 y }<br />
,<br />
where g 2 = h 2 = 1 x , gh = hg, αh = α, αg = β, βh =<br />
β, βg = β. Its classifying space is the homotopy pushout <strong>of</strong><br />
BC 2<br />
BAut E (y) ∼ = •<br />
BAut E (x) = B(C 2 × C 2 )<br />
where the cyclic group <strong>of</strong> order 2 is {1 x , h}, the stabilizer <strong>of</strong><br />
Hom E (x, y) = {α, β}. Thus BE ≃ BAut E (x)/BC 2 . In fact,<br />
by Definition 2.4.3 <strong>and</strong> Proposition 2.4.4, we can consider<br />
the subdivision <strong>of</strong> E, which is a Grothendieck construction
162 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
Gr sd[E] M. Then by Theorem 2.4.10, BGr sd[E] M, <strong>and</strong> thus<br />
BE, is homotopy equivalent to hocolim sd[E] BM. Since sd[E]<br />
is a poset • ← • → •, this homotopy colimit is just the<br />
homotopy pushout <strong>of</strong> the above diagram, if one follows Definition<br />
2.4.3 to compute BM.<br />
We can use the exact sequence for computing relative cohomology<br />
(see for instance [34])<br />
H ∗ (BAut E (x), BC 2 ; R)<br />
to find the positive degree cohomology <strong>of</strong> BE, while in degree<br />
zero we just have R because E is connected. In fact<br />
H ∗ (BE, k), for k a field <strong>of</strong> characteristic 2, is a subring <strong>of</strong><br />
the polynomial ring k[X, Y], with deg X = deg Y = 1 <strong>and</strong><br />
with all linear combinations <strong>of</strong> {X i } i>0 removed. This ring<br />
has no nilpotent elements <strong>and</strong> is not finitely generated. In<br />
fact, this is the smallest example in commutative algebra<br />
that a subring <strong>of</strong> a Noetherian ring is not finitely generated.<br />
However when R = C, C is projective (by direct calculation<br />
or by Proposition 3.3.11). That means H 0 (BE, C) = C is<br />
the only non-vanishing cohomology. Note that C as a RE op -<br />
module is not projective, although H ∗ (BE op , C) = C as well.<br />
4.3 Computation via adjoint functors<br />
4.3.1 Adjoint restrictions<br />
It is well-known that if a functor u : D → C has a left adjoint<br />
v : C → D, then Res v is also the left adjoint <strong>of</strong> Res u . These<br />
are very special Kan extensions <strong>and</strong> hence can be regarded<br />
as the first applications <strong>of</strong> Kan extension before we move to<br />
more subtle situations. Because both Res u <strong>and</strong> Res v are exact,<br />
Res u preserves injectives while Res v preserves projectives.
4.3 Computation via adjoint functors 163<br />
Lemma 4.3.1. If u : D → C has a left adjoint v : C → D,<br />
then<br />
Ext ∗ RC(Res v M, N) ∼ = Ext ∗ RD(M, Res u N)<br />
for any M ∈ RD-mod <strong>and</strong> N ∈ RC-mod. If M ′ is a right<br />
RD-module, then<br />
Tor RC<br />
∗ (Res v M ′ , N) ∼ = Tor RD<br />
∗ (M ′ , Res u N)<br />
If furthermore u is an equivalence then we have<br />
Ext ∗ RC(M, N) ∼ = Ext ∗ RD(Res u M, Res u N),<br />
for M, N ∈ RC-mod. In particular res u is a ring isomorphism<br />
Ext ∗ RC(R, R) ∼ = Ext ∗ RD(R, R). If M ′′ is a right RCmodule,<br />
then under the circumstance we get<br />
Tor RC<br />
∗ (M ′′ , N) ∼ = Tor RD<br />
∗ (Res u M ′′ , Res u N)<br />
Pro<strong>of</strong>. These follow directly from the definitions <strong>of</strong> Ext <strong>and</strong><br />
Tor, as well as the adjunction.<br />
⊓⊔<br />
Let us briefly go over an important approach <strong>of</strong> computing<br />
∗<br />
higher limits ←−<br />
lim M. The atomic modules are useful to us<br />
C<br />
because for many interesting category C every module in RCmod<br />
admits a filtration by submodules such that every quotient<br />
is an atomic module. Since the higher limits <strong>of</strong> a module<br />
can be computed via higher limits <strong>of</strong> its atomic subquotients,<br />
∗<br />
one can seek ways <strong>of</strong> computing ←−<br />
lim M for an atomic module<br />
C<br />
M. This method was introduced <strong>and</strong> studied by Oliver [].<br />
∗<br />
We note that usually ←−<br />
lim M, for an atomic module M concentrated<br />
on x ∈ Ob C, is quite different from the group<br />
C<br />
cohomology H ∗ (Aut C (x), M(x)). Given a projective resolution<br />
P <strong>of</strong> R, P(x) must be an exact sequence <strong>of</strong> RAut C (x)-<br />
modules, beginning with R, for any object x ∈ Ob C. But<br />
the modules in P(x) do not have to be projective, <strong>and</strong> even<br />
if P(x) → R → 0 is a projective resolution the cohomology
164 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
groups H ∗ (C, M) <strong>and</strong> H ∗ (Aut C (x), M(x)) are not necessarily<br />
isomorphic. We comment here that the modules in P(x) will<br />
be projective if RAut C (x) happens to be semi-simple. Another<br />
possible way to obtain a projective resolution for the<br />
RAut C (x)-module R is formulated as follows. If Aut C (x) acts<br />
freely on Hom C (y, x) for each y ∈ Ob C with Hom(y, x) ≠ ∅,<br />
then every RHom C (y, x) is a free RAut C (x)-module. As an example<br />
the st<strong>and</strong>ard resolution evaluated at x, P C (x), becomes<br />
a projective resolution for the RAut C (x)-module R.<br />
4.3.2 Kan extensions <strong>of</strong> resolutions<br />
For an arbitrary functor u : D → C we can still construct<br />
the left <strong>and</strong> right adjoints <strong>of</strong> Res u , i.e. the Kan extensions. In<br />
the situation that u admits a left (or right) adjoint v, then<br />
the left (or right) Kan extension enjoys a particularly simple<br />
form Res v , which is exact. In general, the Kan extensions cannot<br />
have simplified forms, <strong>and</strong> consequently computing Kan<br />
extensions <strong>of</strong> an arbitrary module is difficult <strong>and</strong> the new<br />
modules do not come with explicit descriptions. However if a<br />
module is simplicially constructed, in the sense that it comes<br />
from the nerve <strong>of</strong> a category, then we do have some satisfactory<br />
results on their Kan extensions. This is a fundamental<br />
step to study cohomology rings <strong>and</strong> higher limits.<br />
Proposition 4.3.2. Suppose u : D → C is a functor between<br />
small categories. Then<br />
LK u B D ∗<br />
= LK u C ∗ (Id D /−, R) ∼ = C ∗ (u/−, R),<br />
a complex <strong>of</strong> projective RC-modules.<br />
Particularly for pt : D → •, we have<br />
LK pt B D ∗<br />
= LK pt C ∗ (Id D /•, R) ∼ = C ∗ (pt/•, R) ∼ = C ∗ (D, R),
4.3 Computation via adjoint functors 165<br />
Pro<strong>of</strong>. For the sake <strong>of</strong> convenience, we suppress the base ring<br />
R in our notations. Suppose x ∈ Ob C. Fix an integer n ≥ 0.<br />
Then<br />
LK u C n (Id D /−) ∼ = lim −→u/x<br />
C n (Id D /−).<br />
If (y, α) ∈ Ob u/x, that is, there is an α : u(y) → x, then by<br />
definition [C n (Id D /−)](y, α) = C n (Id D /y), <strong>and</strong> we can define<br />
a morphism<br />
θ (y,α) : C n (Id D /y) → C n (u/x)<br />
by [(y 0 , α 0 ) β 1<br />
→ · · · →(y βn<br />
n , α n )] ↦→ [(y 0 , αα 0 ) β 1<br />
→ · · · →(y βn<br />
n , αα n )].<br />
One can easily verify that we have a commutative diagram<br />
for any γ : (y, α) → (z, β)<br />
C n (Id D /y) = [C n (Id D /−)](y, α)<br />
θ (y,α)<br />
γ ∗<br />
C n (u/x)<br />
[C n (Id D /−)](z, β) = C n (Id<br />
θ (z,β)<br />
so that C n (u/x) fits into the limit defining diagram. If L is<br />
another R-module that fits into the limit defining diagram,<br />
equipped with maps θ L (y,α) : [C n(Id D /−)](y, α) → L for all<br />
(y, α) ∈ Ob(u/x). Then we can introduce a morphism Θ :<br />
C n (u/x) → L such that<br />
Θ[(y 0 , αα 0 ) β 1<br />
→ · · · →(y βn<br />
n , αα n )]<br />
:= θ(y L n ,αα n ) [(y 0, u(β n · · · β 1 )) β 1<br />
→(y 0 , u(β n · · · β 2 )) β 2<br />
→ · · · →(y βn<br />
n , 1 u(yn ))].<br />
This is a well defined morphism. In the end, since α n :<br />
(y n , αα n ) → (y, α) in a morphism in u/x, it induces a functor<br />
C n (Id D /y n ) → C n (Id D /y). In particular this functor gives<br />
[(y 0 , u(β n · · · β 1 )) β 1<br />
→(y 0 , u(β n · · · β 2 )) β 2<br />
→ · · · →(y βn<br />
n , 1 u(yn ))]<br />
↦→ [(y 0 , α 0 ) β 1<br />
→ · · · →(y βn<br />
n , α n )],
166 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
by composing with α n . By assumptions on L we get<br />
Θ(y L n ,αα n ) [(y 0, u(β n · · · β 1 )) β 1<br />
→(y 0 , u(β n · · · β 2 )) β 2<br />
→ · · · →(y βn<br />
n , 1 u(yn ))]<br />
= Θ(y,α) L [(y 0, α 0 ) β 1<br />
→ · · · →(y βn<br />
n , α n )].<br />
This equality implies that we can insert Θ into the following<br />
diagram <strong>and</strong> make it commutative<br />
C n (Id D /y) = [C n (Id D /−)](y, α)<br />
θ (y,α)<br />
θ L (y,α)<br />
γ ∗<br />
C n (u/x)<br />
Θ <br />
L<br />
[C n (Id D /−)](z, β) = C n (Id<br />
θ (z,β)<br />
θ L (z,β)<br />
Consequently C n (u/x) ∼ = lim −→u/x<br />
C n (Id D /−) for all x ∈ Ob C.<br />
Since all these isomorphisms assemble to an isomorphism<br />
C ∗ (u/−) ∼ = LK u C ∗ (Id D /−), by naturality, we are done. ⊓⊔<br />
Note the second statement generalizes the obvious fact<br />
LK pt B∗<br />
D = LK pt RD ∼ = lim −→D<br />
RD ∼ = R⊗ RD RD ∼ = R = R Ob D = C 0 (D<br />
Remark 4.3.3. To underst<strong>and</strong> the complex <strong>of</strong> projective RCmodules<br />
C ∗ (u/−), we describe the structure <strong>of</strong> each C n (u/−).<br />
As we observed in the pro<strong>of</strong>, each base element (y 0 , α 0 ) →<br />
· · · → (y n , α n ) ∈ C n (u/−) can be written as α n·[(y 0 , u(β n · · · β 1 )) →<br />
· · · → (y n , 1 u(yn ))]. Thus similar to the special case <strong>of</strong> bar resolution<br />
where u = Id C , we know<br />
LK u Bn D ∼ = C n (u/−, R) ∼ ⊕<br />
=<br />
RHom C (u(y n ), −).<br />
(y 0 ,u(β n···β 1 )) β 1<br />
→···βn<br />
→(y n ,1 u(y n))<br />
For future reference, the dual version for the right bar resolution<br />
is
4.3 Computation via adjoint functors 167<br />
LK u B D n ∼ = C n (−\u, R) ∼ =<br />
⊕<br />
(1 u(y0 ),y 0 ) β 1<br />
→···βn<br />
→(u(β n···β 1 ),y n )<br />
RHom C (−, u(y 0 )).<br />
Recall that a category is R-acyclic if its reduced (simplicial)<br />
homology with coefficients in R vanishes.<br />
Corollary 4.3.4. Suppose u : D → C is a functor, M ∈<br />
RC-mod <strong>and</strong> N ∈ mod-RC. Then<br />
1. if u/x is R-acyclic for every x ∈ Ob C,<br />
Ext ∗ RC(R, M) ∼ = Ext ∗ RD(R, Res u M) <strong>and</strong> Tor RC<br />
∗<br />
2. if x\u is R-acyclic for every x ∈ Ob C,<br />
Ext ∗ RC(R, N) ∼ = Ext ∗ RD(R, Res u N) <strong>and</strong> Tor RC<br />
∗<br />
(N, R) ∼ = Tor RD<br />
∗ (Res<br />
(R, M) ∼ = Tor RD<br />
∗ (R, R<br />
Pro<strong>of</strong>. We prove part (1). Under the assumption, LK u R ∼ = R<br />
<strong>and</strong> LK u B∗<br />
D ∼ = C ∗ (u/−, R) → R → 0 becomes a projective<br />
resolution. Hence the isomorphism follows from<br />
Hom RC (C ∗ (u/−, R), M) ∼ = Hom RD (B D ∗ , Res u M).<br />
For the second isomorphism, we also have<br />
N ⊗ RC C ∗ (u/−, R) ∼ = ⊕ N ⊗ (y 0 ,u(β n···β 1 )) β 1<br />
RC RHom C (u(y n )<br />
→···βn<br />
→(y n ,1 u(y n))<br />
∼<br />
⊕ = (Res (y 0 ,u(β n···β 1 )) β 1<br />
uN)(y n )<br />
→···βn<br />
→(y n ,1 u(y n))<br />
∼<br />
⊕ = β<br />
y 1<br />
(Res u N)(y n )<br />
0 →···βn<br />
→y n<br />
∼ = Resu N ⊗ RD Bn D .<br />
⊓⊔<br />
The above isomorphisms are Eckmann-Shapiro type results.<br />
Example 4.3.5. Let D be the category from Example 4.2.4 (2)<br />
h<br />
x<br />
1 x<br />
α<br />
β<br />
y {1 y }<br />
,
168 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
where h 2 = 1 x <strong>and</strong> βh = α. The inclusion i : Aut D (x) → D<br />
has two overcategories i/x <strong>and</strong> i/y, both contractible. Thus<br />
we have<br />
Ext ∗ RAut D (x)(R, M) ∼ = Ext ∗ RD(R, Res i M).<br />
This calculation can be generalized to all categories that possess<br />
a unique minimal object x whose automorphism group<br />
acts regularly on every non-empty morphism set Hom D (x, y).<br />
Recall from Section 2.3.1 that we introduced the subdivision<br />
S(C) <strong>of</strong> an EI category C <strong>and</strong> showed that their classifying<br />
spaces are homotopy equivalent <strong>and</strong> hence they have the same<br />
simplicial (co)homology.<br />
Corollary 4.3.6. Let C be an EI category <strong>and</strong> S(C) its subdivision.<br />
Suppose M ∈ RC-mod <strong>and</strong> M ′ ∈ RC op -mod. Then<br />
Tor RC<br />
∗ (R, M) ∼ = Tor RS(C)<br />
∗ (R, Res t M)<br />
<strong>and</strong><br />
Ext ∗ RC op(R, M ′ ) ∼ = Tor RS(C)op<br />
∗ (R, Res s M ′ )<br />
where s : S(C) → C is the natural functor.<br />
Pro<strong>of</strong>. In the pro<strong>of</strong> <strong>of</strong> Proposition 2.3., we actually proved<br />
that x\s is contractible for all x ∈ Ob C. This is equivalent<br />
to having s op /x contractible, by Lemma 1.2.8. Hence we can<br />
apply Corollary 4.3.4.<br />
⊓⊔<br />
Remark 4.3.7. Let u : D → C be a functor. We can use LK u<br />
to see how one obtains the maps introduced in Chapter 2<br />
u ∗ : H∗(D, R) → H∗(C, R) <strong>and</strong> u ∗ : H ∗ (C, R) → H ∗ (D, R). In<br />
fact we have the following diagram<br />
D<br />
u <br />
pt<br />
C<br />
pt<br />
•
4.3 Computation via adjoint functors 169<br />
Thus by LK pt = LK pt LK u we get LK pt B∗ D = LK pt LK u B∗ D .<br />
Since LK u B∗<br />
D ∼ = C ∗ (u/−, R) is a complex <strong>of</strong> projective modules<br />
<strong>and</strong> meanwhile there is a canonical map LK u R → R<br />
between RC-modules, there exists a lifting, which is a chain<br />
map unique up to chain homotopy (Comparison Theorem),<br />
· · ·<br />
<br />
LK u B∗<br />
D<br />
LK u R<br />
0<br />
· · ·<br />
<br />
B C ∗<br />
<br />
R<br />
<br />
0<br />
Conceptually we see the left Kan extension induces chain<br />
maps<br />
R ⊗ RD B∗ D ∼ = R ⊗ RC LK u B∗ D → R ⊗ RC B∗<br />
C<br />
because R ⊗ RD B∗ D = LK pt B∗ D = LK pt LK u B∗<br />
D ∼ = R ⊗RC<br />
LK u B∗<br />
D <strong>and</strong><br />
Hom RC (B C ∗, R) → Hom RC (LK u B D ∗ , R) ∼ = Hom RD (B D ∗ , R).<br />
Hence these give exactly the same map we introduced in<br />
Chapter 2. Alternatively by applying LK pt to the previous<br />
commutative diagram we obtain a chain map<br />
<br />
· · · LK pt LK u B∗<br />
D<br />
LK pt LK u R<br />
0<br />
· · ·<br />
<br />
LK pt B C ∗<br />
<br />
LK pt R<br />
<br />
0<br />
The positive degree part LK pt B D ∗ = LK pt LK u B D ∗ → LK pt B C ∗<br />
is exactly the chain map induced by u in Definition 2.2.4, i.e.<br />
C ∗ (D, R) → C ∗ (C, R), in light <strong>of</strong> Proposition 4.3.2.<br />
Remark 4.3.8. The following descriptions balance the underst<strong>and</strong>ing<br />
<strong>of</strong> left <strong>and</strong> right Kan extensions. Especially we know<br />
the right Kan extension <strong>of</strong> a certain injective resolution <strong>of</strong> k.<br />
Since (−) ∧ establishes a one-to-one correspondence between
170 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
right projective kC-modules <strong>and</strong> left injective kC-modules, we<br />
can extend the above lemmas to the bar resolution <strong>of</strong> k ∈modkD<br />
<strong>and</strong> obtain a complex <strong>of</strong> left injective kC-modules. Suppose<br />
B∗<br />
D ∼ → k → 0 is the bar resolution <strong>of</strong> k ∈mod-kD, where<br />
= C ∗ (−\Id D ). Then<br />
B D ∗<br />
0 → k → (B D ∗ ) ∧ ∼ = C∗ (−\Id D ) ∧<br />
is an injective resolution <strong>of</strong> the left kD-module k. Applying<br />
RK u we get a complex <strong>of</strong> injective kC-modules, excluding<br />
RK u k, as follows<br />
0 → RK u k → RK u (B D ∗ ) ∧ .<br />
By Lemma 3.2.13 the complex RK u (B D ∗ ) ∧ ∼ = (LK u B D ∗ ) ∧ ∼ =<br />
C ∗ (−\u) ∧ . Thus we have a simplicially constructed injective<br />
resolution <strong>of</strong> k, <strong>and</strong> we have control on the complex after<br />
applying the right Kan extension<br />
0 → RK u k → (B D ∗ ) ∧ ∼ = C∗ (−\u) ∧ .<br />
Remark 4.3.9. At this stage, the reader shall be well prepared<br />
to go over Chapter 5 where the main results in Section 5.2<br />
are applications <strong>of</strong> Proposition 4.3.2.<br />
4.4 Grothendieck spectral sequences<br />
Let u : D → C be a functor. We can construct Grothendieck<br />
spectral sequences. They generalize the Lydon-Hochschil-<br />
Serre spectral sequences for group extensions as well as the<br />
Leray-Serre spectral sequences for fibrations.<br />
4.4.1 Grothendieck spectral sequences for a functor<br />
Let u : D → C be a functor between two small categories.<br />
Consider the composites <strong>of</strong> following functors
4.4 Grothendieck spectral sequences 171<br />
as well as<br />
RD-mod RK u <br />
RC-mod lim ←−C <br />
R-mod,<br />
RD-mod LK u <br />
RC-mod lim −→C<br />
<br />
R-mod.<br />
The second isomorphism <strong>of</strong> the following is used in Remark<br />
4.3.7.<br />
Lemma 4.4.1. lim ←−C<br />
RK u<br />
∼ = lim ←−D<br />
<strong>and</strong> lim −→C<br />
LK u<br />
∼ = lim −→D<br />
.<br />
Pro<strong>of</strong>. This is a special case <strong>of</strong> Corollary 1.2.12 for u : D → C<br />
<strong>and</strong> pt : C → •.<br />
⊓⊔<br />
Since RK u preserves injectives, there exists a Grothendieck<br />
cohomology spectral sequence for any functor u : D → C<br />
<strong>and</strong> M ∈ RD-mod, which comes from a double complex<br />
E ∗,∗<br />
0<br />
E ∗,∗<br />
2<br />
(M) that we will describe shortly. The spectral sequence<br />
(M) ⇒ E∗,∗ ∞ (M) is<br />
H ∗ (C; H ∗ (−\u; M)) ⇒ H ∗ (D; M).<br />
i<br />
Remember that for any small category E, one has ←−<br />
lim ∼<br />
E =<br />
H i (E; −) ∼ = Ext i RE(R, −). It means H ∗ (−\u; M) is some sort<br />
<strong>of</strong> higher right Kan extension <strong>of</strong> M. Since LK u preserves projectives,<br />
we get a Grothendieck homology spectral sequence<br />
as well. However we will only construct the cohomology spectral<br />
sequence as the construction for the homology spectral<br />
sequence is similar. For future reference, we record the<br />
Grothendieck homology spectral sequence for u : D → C <strong>and</strong><br />
M ∈ RD-mod<br />
H∗(C; H∗(u/−; M)) ⇒ H∗(D; M),<br />
in which H∗(u/−; M) should be considered as higher left Kan<br />
extensions <strong>of</strong> M.
172 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
We shall construct the cohomology spectral sequence <strong>and</strong><br />
then assure the reader that there exists a natural pairing <strong>of</strong><br />
such double complexes E ∗,∗<br />
0 (M) ⊗ E∗,∗ 0 (N) → E∗,∗ 0 (M ˆ⊗N).<br />
With the pairing, we have a product on each page <strong>of</strong> the<br />
Grothendieck cohomology spectral sequences<br />
En i,j (M) ⊗ En s,t (N) → En<br />
i+s,j+t (M ˆ⊗N),<br />
<strong>and</strong><br />
E∞ i,j (M) ⊗ E∞ s,t (N) → E∞<br />
i+s,j+t (M ˆ⊗N),<br />
Thus we have a ring structure on<br />
H ∗ (C; H ∗ (−\u; R)) ⇒ H ∗ (D; R),<br />
over which the following is a module<br />
H ∗ (C; H ∗ (−\u; M)) ⇒ H ∗ (D; M).<br />
Fix an RD-module M we start with a double complex<br />
(M). First take an injective resolution <strong>of</strong> M<br />
E ∗,∗<br />
0<br />
0 → M → I 0 → I 1 → I 2 → · · · .<br />
Then we apply RK u to get a complex <strong>of</strong> injective kC-modules<br />
RK u I 0 → RK u I 1 → RK u I 2 → · · · ,<br />
<strong>and</strong> consequently a commutative diagram<br />
RK u I 0<br />
RK u I 1<br />
RK u I 2<br />
· · ·<br />
J 0,0<br />
J 1,0<br />
J 2,0<br />
· · ·<br />
J 0,1<br />
J 1,1<br />
J 2,1<br />
· · ·<br />
. . . ,
4.4 Grothendieck spectral sequences 173<br />
in which every column is an injective resolution <strong>of</strong> the top<br />
module. Now apply ←−C<br />
lim <strong>and</strong> we obtain a double cochain complex,<br />
denoted by E ∗,∗<br />
0 (M),<br />
lim<br />
←−C J 0,0<br />
lim<br />
←−C J 1,0<br />
lim<br />
←−C J 2,0<br />
· · ·<br />
lim<br />
←−C J 0,1<br />
lim<br />
←−C J 1,1<br />
lim<br />
←−C J 2,1<br />
· · ·<br />
lim<br />
←−C J 0,2<br />
lim<br />
←−C J 1,2<br />
lim<br />
←−C J 2,2<br />
· · ·<br />
. . . ,<br />
<strong>and</strong> it gives rise to the Grothendieck spectral sequence recorded<br />
above. We omit the details as the construction is st<strong>and</strong>ard <strong>and</strong><br />
we are more interested in finding a pairing. Suppose we also<br />
have a double complex E ∗,∗<br />
0 (N) for another RD-module N<br />
lim<br />
←−C J ′ 0,0<br />
lim<br />
←−C J ′ 1,0<br />
lim<br />
←−C J ′ 2,0<br />
· · ·<br />
lim<br />
←−C J ′ 0,1<br />
lim<br />
←−C J ′ 1,1<br />
lim<br />
←−C J ′ 2,1<br />
· · ·<br />
lim<br />
←−C J ′ 0,2<br />
lim<br />
←−C J ′ 1,2<br />
lim<br />
←−C J ′ 2,2<br />
· · ·<br />
. . . ,<br />
<strong>and</strong> furthermore a double complex E ∗,∗<br />
0 (M ˆ⊗N) for the RDmodule<br />
M ˆ⊗N
174 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
lim<br />
←−C J ′′<br />
0,0<br />
lim<br />
←−C J ′′<br />
1,0<br />
lim<br />
←−C J ′′<br />
2,0<br />
· · ·<br />
lim<br />
←−C J ′′<br />
0,1<br />
lim<br />
←−C J ′′<br />
1,1<br />
lim<br />
←−C J ′′<br />
2,1<br />
· · ·<br />
lim<br />
←−C J ′′<br />
0,2<br />
lim<br />
←−C J ′′<br />
1,2<br />
lim<br />
←−C J ′′<br />
2,2<br />
. . . .<br />
We want to establish a natural map<br />
lim J ←−C<br />
i,j ⊗ lim ←−C<br />
J s,t ′ → lim ←−C<br />
J i+s,j+t,<br />
′′<br />
which is compatible with the differentials. In fact, since there<br />
is a unique map, given by the universal property <strong>of</strong> ←−<br />
lim,<br />
lim J ←−C<br />
i,j ⊗ lim ←−C<br />
J s,t ′ → lim ←−C<br />
J i,j ˆ⊗J s,t,<br />
′<br />
we only need to construct a map ←−C<br />
lim J i,j ˆ⊗J s,t ′ → lim ←−C<br />
J i+s,j+t.<br />
′′<br />
Our definition is again based on the universal property <strong>of</strong><br />
lim<br />
←−<br />
, along with the tensor product <strong>of</strong> complexes <strong>of</strong> functors<br />
in Section 3.4.1. We emphasize that ←−C<br />
lim J i,j ⊗ lim ←−C<br />
J s,t<br />
′ →<br />
lim J ←−C<br />
i,j ˆ⊗J s,t ′ respects the differentials in E ∗,∗<br />
0 due to its construction<br />
via the universal property. This is the case when we<br />
define lim ←−C<br />
J i,j ˆ⊗J s,t ′ → lim ←−C<br />
J i+s,j+t ′′ <strong>and</strong> thus we will not verify<br />
the map we are about to construct does respect differentials.<br />
From the two injective resolutions0 → M → I ∗ <strong>and</strong> 0 →<br />
N → I ∗, ′ we can build a commutative diagram<br />
0<br />
<br />
M ˆ⊗N I ∗ ˆ⊗I ′ ∗<br />
<br />
0<br />
<br />
M ˆ⊗N<br />
<br />
I ′′<br />
∗ ,<br />
· · ·
4.4 Grothendieck spectral sequences 175<br />
in which the upper row is an exact sequence <strong>and</strong> the lower<br />
one is the injective resolution used to define E ∗,∗<br />
0 (M ˆ⊗N). Applying<br />
RK u we obtain a chain map RK u (I ∗ ˆ⊗I ∗) ′ → RK u I ∗ ′′ .<br />
Especially we have for any non-negative integers i <strong>and</strong> s a<br />
map RK u (I i ˆ⊗I s) ′ → RK u I i+s. ′′ The universal property <strong>of</strong> ←−<br />
lim<br />
provides a morphism RK u I i ˆ⊗RK u I s ′ → RK u (I i ˆ⊗I s). ′ Thus<br />
we have a natural map<br />
RK u I i ˆ⊗RK u I s ′ → RK u I i+s.<br />
′′<br />
Next we repeat the above tensor construction for the two injective<br />
resolutions 0 → RK u I i → J i,∗ <strong>and</strong> 0 → RK u I s ′ → J s,∗.<br />
′<br />
It follows from our discussions that there is a commutative diagram<br />
0<br />
<br />
RK u I i ˆ⊗RK u I s<br />
′ J i,∗ ˆ⊗J s,∗<br />
′<br />
0<br />
<br />
RK u I ′′<br />
i+s<br />
<br />
J i+s,∗+∗ ′′ .<br />
In particular there exists J i,j ˆ⊗J s,t ′ → J i+s,j+t, ′′ <strong>and</strong> consequently<br />
the desired map lim ←−C<br />
J i,j ˆ⊗J s,t ′ → lim ←−C<br />
J i+s,j+t. ′′ Hence<br />
we do obtain a pairing E ∗,∗<br />
0 (M) ⊗ E∗,∗ 0 (N) → E∗,∗ 0 (M ˆ⊗N).<br />
4.4.2 Spectral sequences <strong>of</strong> category extensions<br />
We introduced category extensions <strong>and</strong> opposite extensions<br />
in Section 4.1.3. Here we want to show that the Grothendieck<br />
spectral sequences for a functor have simpler forms when the<br />
functor fits into an extension sequence.<br />
Lemma 4.4.2. Let K → E → C be an extension. Then there<br />
exists a natural functor ι : K(y) → π/y such that every<br />
undercategory associated with it is contractible. Hence
176 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
1. H∗(π/−; M) ∼ = H∗(K(−); M(−)) in RC-mod for any M ∈<br />
RE-mod; <strong>and</strong><br />
2. H ∗ (−\π op ; N) ∼ = H ∗ (K op (−); N(−)) in RC op -mod for any<br />
N ∈ RE op -mod.<br />
Pro<strong>of</strong>. The category π/y has objects <strong>of</strong> the form (x, α), where<br />
x ∈ Ob E = Ob C <strong>and</strong> α ∈ Hom C (x, y). From the definition<br />
<strong>of</strong> π/y, one can see that the maximal objects are (y, g),<br />
g ∈ Aut C (y), which are isomorphic to each other <strong>and</strong> have<br />
automorphism groups isomorphic to K(y). Let us take the<br />
full subcategory [(y, 1 y )] <strong>of</strong> π/y consisting <strong>of</strong> all maximal objects.<br />
Its skeleton is isomorphic to the group K(y). Using<br />
Quillen’s Theorem A, we show the undercategories associated<br />
with ι : [(y, 1 y )] ↩→ π/y are contractible.<br />
Fix an object (x, α) ∈ Ob(π/y). The undercategory (x, α)\(π/y)<br />
has objects <strong>of</strong> the form (β, (y, g)), where β : (x, α) → (y, g)<br />
is an morphism in π/y satisfying gπ(β) = α. Since π(β) =<br />
g −1 α, by the definition <strong>of</strong> a category extension, β = g −1 αh<br />
for a unique h ∈ K(x). From here we can deduce that<br />
(β, (y, g)) ∼ = (β ′ , (y, g ′ )) for any (y, g ′ ) <strong>and</strong> β ′ : (x, α) →<br />
(y, g ′ ), <strong>and</strong> that (β, (y, g)) ∈ (x, α)\(π/y) has a trivial automorphism<br />
group. These imply (x, α)\(π/y) is equivalent to a<br />
point, <strong>and</strong> hence is contractible.<br />
The first isomorphism follows from Corollary 4.3.4 (2)<br />
Tor R(π/y)<br />
∗<br />
(R, M) ∼ = Tor RK(y) (R, Res π M),<br />
<strong>and</strong> the naturality in y.<br />
The pro<strong>of</strong> <strong>of</strong> the second is the same. By Lemma 1.2.8 the<br />
overcategories associated with ι op : K(y) op → (π/y) op are<br />
contractible. Then we apply Corollary 4.3.4 (1). ⊓⊔<br />
We can obtain similar statements for opposite extensions.<br />
Keep in mind that since K is a group, K ∼ = K op .<br />
∗
4.4 Grothendieck spectral sequences 177<br />
Lemma 4.4.3. Let K → E → C be an opposite extension.<br />
Then there exists a natural functor ι : K(y) → y\π such<br />
that every undercategory associated with it is contractible.<br />
Hence<br />
1. H∗(π op /−; M) ∼ = H∗(K op (−); M(−)) in RC op -mod for any<br />
M ∈ RE op -mod; <strong>and</strong><br />
2. H ∗ (−\π; N) ∼ = H ∗ (K(−); N(−)) in RC-mod for any N ∈<br />
RE-mod.<br />
Pro<strong>of</strong>. Since K op → E op → C op is an extension, we know by<br />
Lemma 4.4.2 that there is a functor ι : K op (y) → π op /y such<br />
that all undercategories are contractible. From Lemma 1.2.8,<br />
this is equivalent to saying that all overcategories associated<br />
to ι op : K(y) → y\π are contractible.<br />
⊓⊔<br />
For brevity we write H ∗ (K; M) etc, instead <strong>of</strong> H ∗ (K(−); M(−))<br />
etc, for the functors in above lemmas.<br />
Proposition 4.4.4. Given a functor M ∈ RE-mod, there<br />
are two spectral sequences associated with an extension K →<br />
E → C as follows:<br />
1. a homology spectral sequence<br />
E 2 ij = Hi(C; Hj(K; M)) ⇒ Hi+j(E; M);<br />
<strong>and</strong><br />
2. a cohomology spectral sequence<br />
E ij<br />
2 = H i (C op ; H j (K op ; M op )) ⇒ H i+j (E op ; M op ).<br />
Dually we have results for opposite extensions. Recall that<br />
RE op -mod = mod-RE. If M ∈ RE-mod, it gives M op ∈ RE op -<br />
mod.<br />
Proposition 4.4.5. Given a functor M ∈ RE-mod, there<br />
are two spectral sequences associated with an opposite extension<br />
K → E → C as follows:
178 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
1. a homology spectral sequence<br />
E 2 ij = Hi(C op ; Hj(K op ; M op )) ⇒ Hi+j(E op ; M op );<br />
<strong>and</strong><br />
2. a cohomology spectral sequence<br />
E ij<br />
2 = H i (C; H j (K; M)) ⇒ H i+j (E; M).<br />
Spectral sequences naturally give rise to some long exact<br />
sequences, <strong>and</strong> we record them below.<br />
Remark 4.4.6. If K → E → C is an extension <strong>and</strong> M ∈ REmod,<br />
one can obtain two five term exact sequences<br />
H2(E; M) → H2(C; M) → H0(C; H1(K; M)) → H1(E; M) → H1(C; M<br />
<strong>and</strong><br />
0 → H 1 (C op ; M op ) → H 1 (E op ; M op ) → H 0 (C op ; H 1 (K op ; M op )) → H 2 (C o<br />
When K → E → C is a group extension then these two exact<br />
sequences are the usual five term sequence in group homology<br />
<strong>and</strong> cohomology.<br />
Similarly if K → E → C is an opposite extension, we have<br />
H2(E op ; M op ) → H2(C op ; M op ) → H0(C; H1(K op ; M op )) → H1(E op ; M op<br />
<strong>and</strong><br />
0 → H 1 (C; M) → H 1 (E; M) → H 0 (C; H 1 (K; M)) → H 2 (C; M) → H 2 (E<br />
In general the finite generation <strong>of</strong> cohomology rings <strong>of</strong> both<br />
K <strong>and</strong> C does not guarantee the cohomology ring <strong>of</strong> E has the<br />
same property. One <strong>of</strong> the examples, C 2 , we used to demonstrate<br />
that the cohomology rings <strong>of</strong> EI-categories are not<br />
finitely generated, is an extension <strong>of</strong> a contractible category:
4.4 Grothendieck spectral sequences 179<br />
1 y<br />
<br />
y<br />
h 2<br />
<br />
y<br />
h 2<br />
<br />
y<br />
ι π<br />
α<br />
<br />
α .<br />
x<br />
h 2<br />
<br />
x<br />
h 2<br />
<br />
However when K is cohomologically trivial, the cohomology<br />
rings <strong>of</strong> E <strong>and</strong> C are isomorphic.<br />
Corollary 4.4.7. Suppose K → E → C is an extension, <strong>and</strong><br />
|K(x)| is invertible in R for every object x. Then for any<br />
M ∈ RE-mod<br />
H∗(E; M) ∼ = H∗(C; H0(K; M)) ∼ = H∗(C; −→K<br />
lim M).<br />
H ∗ (E op ; M) ∼ = H ∗ (C op ; H 0 (K op ; M)) ∼ = H ∗ (C op ; ←−K<br />
lim<br />
opM).<br />
Since lim ←−K<br />
opR ∼ = R in RC op -mod, we have H ∗ (C; R) ∼ =<br />
H ∗ (C op ; R) ∼ = H ∗ (E op ; R) ∼ = H ∗ (E; R) as algebras.<br />
Pro<strong>of</strong>. Under the assumption, the E 2 (resp. E 2 ) page <strong>of</strong> the<br />
cohomology (resp. homology) spectral sequence collapses to<br />
the vertical (resp. horizontal) axis.<br />
⊓⊔<br />
Let K → E → C be an extension. Since there is a natural<br />
correspondence between the subcategories <strong>of</strong> C <strong>and</strong> those <strong>of</strong><br />
E, one would like to exploit further connections between the<br />
homological properties <strong>of</strong> C <strong>and</strong> E. Let D be a subcategory<br />
<strong>of</strong> C <strong>and</strong> E D its “preimage” in E. We show the undercategories<br />
(or overcategories) associated with the inclusions are<br />
equivalent, when K → E → C is an extension (or an opposite<br />
extension) <strong>of</strong> C.<br />
Definition 4.4.8. Let K → E → C be an extension <strong>and</strong><br />
D ⊂ C a subcategory. The subextension <strong>of</strong> D in E via K| D ,<br />
x<br />
1 x
180 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
named E D , is a subcategory <strong>of</strong> E whose object set is the same<br />
as D <strong>and</strong> whose morphism set consists <strong>of</strong> morphisms in E<br />
which are preimages <strong>of</strong> morphisms in D.<br />
If D is a full subcategory <strong>of</strong> C then E D is a full subcategory <strong>of</strong><br />
E. Given an extension K → E → C, Aut K (x) → Aut E (x) →<br />
Aut C (x) is a subextension for any x ∈ Ob C.<br />
Proposition 4.4.9. Let K → E → C a sequence <strong>of</strong> functors<br />
<strong>and</strong> D a full subcategory <strong>of</strong> C with the inclusion ι D : D → C.<br />
Then<br />
1. if E is an extension <strong>of</strong> C, K| D → E D → D is the subextension<br />
<strong>and</strong> ι ED : E D → E is the inclusion, then for any<br />
y ∈ Ob C = Ob E, the undercategory y\ι D is isomorphic to<br />
a subcategory <strong>of</strong> the undercategory y\ι ED , which is equivalent<br />
to y\ι ED ;<br />
2. if E is an opposite extension <strong>of</strong> C, K| D → E D → D is<br />
the opposite subextension <strong>and</strong> ι ED : E D → E is the inclusion,<br />
then for any y ∈ Ob C = Ob E, the overcategory ι D /y<br />
is isomorphic to a subcategory <strong>of</strong> the overcategory ι ED /y,<br />
which is equivalent to ι ED /y.<br />
Pro<strong>of</strong>. We will only prove (2). In ι ED /y, any two objects (x, α)<br />
<strong>and</strong> (x, β) are isomorphic if <strong>and</strong> only if π(α) = π(β). Let<br />
ι ED /y ⊂ ι ED /y be the full subcategory consisting <strong>of</strong> one object<br />
from each isomorphism class <strong>of</strong> objects described above.<br />
Then ι ED /y <strong>and</strong> ι ED /y are equivalent. We prove the former is<br />
isomorphic to ι D /y.<br />
There is a natural bijection between objects sets <strong>of</strong> these<br />
two categories (x, α) → (x, π(α)) (π is surjective on morphisms).<br />
We show there is a bijection between the morphism<br />
sets <strong>and</strong> the bijections extend to a functor which gives an<br />
isomorphism between two categories. Any (x, α) →(z, γ β) in<br />
Mor(ι ED /y) gives rise to a morphism (x, π(α)) π(γ)<br />
→ (z, π(β)) in
4.4 Grothendieck spectral sequences 181<br />
ι D /y. On the other h<strong>and</strong>, a morphism (x, π(α)) π(γ)<br />
→ (z, π(β))<br />
in ι D /y implies π(β)π(γ) = π(α), which means there exists a<br />
unique g ∈ K(x) such that βγ = αg. Thus we have a uniquely<br />
defined morphism (x, α) g−1<br />
→(x, αg) →(z, γ β) = (x, α) γg−1<br />
→ (z, β)<br />
in Mor(ι ED /y). Note that a different γ ′ such that π(γ ′ ) = π(γ)<br />
gives the same morphism (x, α) γg−1<br />
→ (z, β), so the map from<br />
Mor(ι D /y) to Mor(ι ED /y) is well-defined. It’s straightforward<br />
to check these two assignments on morphisms are mutually<br />
inverse to each other.<br />
In order to show the bijections on objects <strong>and</strong> morphisms<br />
defining an isomorphism between categories, we need to verify<br />
they preserve composition <strong>and</strong> identity. We will just prove<br />
the former <strong>and</strong> leave the pro<strong>of</strong> <strong>of</strong> preserving identity to the<br />
reader. Suppose (x, α) → (z, β) → (w, γ) is a composite<br />
<strong>of</strong> two morphisms in ι ED /y. Then our map naturally sends<br />
it to a composite <strong>of</strong> morphisms (x, π(α)) → (z, π(β)) →<br />
(w, π(γ)). Conversely, if (x, π(α)) π(u)<br />
→ (z, π(β)) π(v)<br />
→ (w, π(γ)) =<br />
(x, π(α)) π(vu)<br />
→ (w, π(γ)) is the composite <strong>of</strong> two morphisms in<br />
ι D /y, then we need to show the two morphisms (x, α) ug−1<br />
→ (z, β) vh−1<br />
→ (w, γ<br />
= (x, α) vh−1 ug<br />
→<br />
−1<br />
(w, γ) <strong>and</strong> (x, α) vut−1<br />
→ (w, γ) are equal, where<br />
g, h, t are isomorphisms, described in the preceding paragraph.<br />
Since π(vh −1 ug −1 ) = π(vut −1 ), there is a unique isomorphism<br />
s satisfying vh −1 ug −1 = vut −1 s. But then we have<br />
α = γvut −1 = γvh −1 ug −1 = γvut −1 s, <strong>and</strong> this forces s = 1<br />
because Definition 4.1.21 (4 op ). Hence we get vh −1 ug −1 =<br />
vut −1 .<br />
⊓⊔<br />
The following corollary is a useful outcome <strong>of</strong> the proposition.
182 4 <strong>Cohomology</strong> <strong>of</strong> categories <strong>and</strong> modules<br />
Corollary 4.4.10. Let K → E → C be a sequence <strong>of</strong><br />
functors <strong>and</strong> D ⊂ C a full subcategory with the inclusion<br />
ι D : D → C. Then<br />
1. if E is an extension <strong>of</strong> C, then y\ι D is contractible (or<br />
R-acyclic or connected) if <strong>and</strong> only if y\ι ED is;<br />
2. if E is an opposite extension <strong>of</strong> C, then ι D /y is contractible<br />
(or R-acyclic or connected) if <strong>and</strong> only if ι ED /y is.<br />
Example 4.4.11. Let K → E → C be an extension with a<br />
unique maximal object x such that Aut C (x) acts freely <strong>and</strong><br />
transitively on Hom C (y, x) for any y ∈ Ob C. Then it’s easy<br />
to check that ι : Aut C (x) ↩→ C induces a homotopy equivalence<br />
since all undercategories associated to it are contractible.<br />
Hence we know Aut E (x) ↩→ E is a homotopy equivalence as<br />
well.<br />
Since any category can be regarded as a trivial extension<br />
<strong>of</strong> itself, the following result is some sort <strong>of</strong> generalization <strong>of</strong><br />
Corollary 4.3.4.<br />
Corollary 4.4.12. Suppose there is an extension K → E →<br />
C. If ι D : D ↩→ C is an inclusion such that y\ι D is contractible<br />
for every y ∈ Ob C, then H ∗ (E; M) ∼ = H ∗ (E D ; M)<br />
for any M ∈ RE op -mod, <strong>and</strong> H∗(E; N) ∼ = H∗(E D ; N) for any<br />
N ∈ RE-mod. Here E D is the subextension corresponding to<br />
D.<br />
Suppose there is an opposite extension K → E → C. If<br />
ι D : D ↩→ C is an inclusion such that ι D /y is contractible for<br />
every y ∈ Ob C, then H ∗ (E; M) ∼ = H ∗ (E D ; M) for any M ∈<br />
RE-mod, <strong>and</strong> H∗(E; N) ∼ = H∗(E D ; N) for any N ∈ RE op -<br />
mod. Here E D is the opposite subextension corresponding to<br />
D.<br />
Pro<strong>of</strong>. We prove the statements for cohomology. Since y\ι D<br />
is contractible for every y ∈ Ob C, y\ι ED is contractible for
4.4 Grothendieck spectral sequences 183<br />
every y ∈ Ob E as well by Corollary 4.4.10. Then we apply<br />
Corollary 4.3.4.<br />
When we have an opposite extension, using Corollary 4.4.10<br />
again we get that ι ED /y is contractible for every y ∈ Ob E.<br />
Hence Corollary 4.3.4 can be used to obtain the isomorphism.<br />
⊓⊔<br />
As an example when K → E → C is an extension (resp.<br />
an opposite extension) <strong>and</strong> C has a unique maximal (resp.<br />
minimal) object x <strong>and</strong> Aut C (x) acts regularly on Hom C (y, x)<br />
(or Hom C (x, y)) for any y ∈ Ob C, we have H ∗ (C; M) ∼ =<br />
H ∗ (Aut C (x), M(x)) hence H ∗ (E; M) ∼ = H ∗ (Aut E (x), M(x))<br />
for any contra-variant (resp. covariant) functor M.
Chapter 5<br />
Hochschild cohomology<br />
Abstract We consider Hochschild cohomology <strong>of</strong> category<br />
algebras here. The upshot is that we can translate Hochschild<br />
cohomology into functor cohomology, which provide a context<br />
for comparing Hochschild <strong>and</strong> ordinary cohoomology <strong>of</strong><br />
a small category. A very general theorem on the relationship<br />
<strong>of</strong> these two cohomology theories will be stated <strong>and</strong> proved,<br />
combining module theoretic <strong>and</strong> simplicial methods. Some<br />
examples are computed to help the reader to underst<strong>and</strong>.<br />
5.1 Hochschild homology <strong>and</strong> cohomology<br />
Let A be an associative R-algebra with identity. Then we<br />
consider A as an A e -module. We shall assume A is a finitely<br />
presented flat R-module. This implies that A is a projective<br />
R-module.<br />
5.1.1 Definition <strong>and</strong> general properties<br />
Definition 5.1.1. The acyclic Hochschild complex <strong>of</strong> A is<br />
{B A n } n≥−1 such that<br />
B A n = A ⊗ R · · · ⊗ R A<br />
} {{ }<br />
(n+2)−copies<br />
<strong>and</strong> such that the differential ∂ n : A ⊗n → A ⊗(n−1) is given by
186 5 Hochschild cohomology<br />
∂ n (a 0 ⊗ a 1 ⊗ · · · ⊗ a n ) ↦→<br />
n∑<br />
(−1) i a 0 ⊗ · · · ⊗ a i a i+1 ⊗ · · · ⊗ a n .<br />
i=0<br />
By direct calculations, one can see ∂ 2 = 0 <strong>and</strong> so we do have<br />
a complex <strong>of</strong> A e -modules. Moreover since we can establish a<br />
chain contraction s n : B A n−1 → B A n by s n (a 0 ⊗ · · · ⊗ a n ) =<br />
a 0 ⊗ · · · ⊗ a n ⊗ 1, this complex is exact.<br />
Suppose M ∈ A e -mod. Then we can regard it as a right A e -<br />
module by m·(a, a ′ ) := a ′·m·a. Thus we shall not distinguish<br />
left <strong>and</strong> right A e -modules in this chapter.<br />
Definition 5.1.2. For any M ∈ A e -mod, we define<br />
1. the Hochschild homology HH ∗ (A, M) to be the homology <strong>of</strong><br />
· · · → M ⊗ A<br />
e B A n → · · · → M ⊗ A<br />
e B A 1 → M ⊗ A<br />
e B A 0 → 0;<br />
<strong>and</strong><br />
2. the Hochschild cohomology HH ∗ (A, M) to be the homology<br />
<strong>of</strong><br />
0 → Hom A e(B A 0 , M) → Hom A e(B A 1 , M) → · · · → Hom A e(B A n , M) →<br />
Remark 5.1.3. Since for any n ≥ 0 there is an isomorphism<br />
<strong>of</strong> A e -modules<br />
B A n → A e ⊗ R ˜BA n ,<br />
in which ˜B A 0 = R <strong>and</strong> ˜B A n for n ≥ 1 is<br />
}<br />
A ⊗ R ·<br />
{{<br />
· · ⊗ R A<br />
}<br />
,<br />
n−copies<br />
<strong>and</strong> furthermore because ˜B A n is a projective R-module B A n is<br />
a projective A e -module, we actually obtain a complex <strong>of</strong> A e -<br />
modules with A in degree −1<br />
· · · → B A n → · · · → B A 1 → B A 0 → A → 0.
5.1 Hochschild homology <strong>and</strong> cohomology 187<br />
We call B A ∗ the bar resolution <strong>of</strong> the A e -module A. Note that<br />
M ⊗ A eBn A ∼ = M ⊗ R ˜BA n <strong>and</strong> Hom A e(Bn A , M) ∼ = Hom R (˜B n A , M).<br />
Because <strong>of</strong> our assumption on A we can also define Hochschild<br />
homology <strong>and</strong> cohomology as Tor Ae<br />
n (M, A) <strong>and</strong> Ext n Ae(A, M),<br />
respectively. When M = A, we usually write HH n (A) <strong>and</strong><br />
HH n (A) for the Hochschild homology <strong>and</strong> cohomology.<br />
When we regard the bar resolution B∗<br />
A → A → 0 as a<br />
complex <strong>of</strong> right A-modules, it splits. Hence for any left A-<br />
module M, Bn A ⊗ A M → A⊗ A M → 0 is a projective resolution<br />
<strong>of</strong> the left A-module M. Based on this observation, we prove<br />
the following well-known results [14].<br />
Proposition 5.1.4. Let M, N ∈ A-mod <strong>and</strong> M ′ ∈ mod-A.<br />
Suppose R is a field. Then Hom R (M, N) <strong>and</strong> N ⊗ R M ′ are<br />
A e -modules such that<br />
Ext ∗ A(M, N) ∼ = H ∗ (A, Hom R (M, N)) ∼ = Ext ∗ A e(A, Hom R(M, N)).<br />
<strong>and</strong><br />
Tor A ∗ (M ′ , N) ∼ = H∗(A, N ⊗ R M ′ ) ∼ = Tor Ae<br />
∗ (N ⊗ R M ′ , A).<br />
Pro<strong>of</strong>. We only prove the isomorphism for Ext. The A e -<br />
module structure on Hom R (M, N) is given by (a 1 , a 2 )f(m) =<br />
a 1 f(a 2 m) for any (a 1 , a 2 ) ∈ A e <strong>and</strong> m ∈ M. Let Bn<br />
A → A<br />
be the previously introduced projective resolution <strong>of</strong> the A e -<br />
module A. Then<br />
Hom A e(B A n , Hom R (M, N)) = Hom R (˜B A n , Hom R (M, N))<br />
∼ = HomR (˜B A n ⊗ R M, N)<br />
∼ = HomA (B A n ⊗ A M, N).
188 5 Hochschild cohomology<br />
Since R is a field, Bn A ⊗ A M is projective <strong>and</strong> thus B∗ A ⊗ A M<br />
becomes a projective resolution <strong>of</strong> M. Consequently the last<br />
term above computes Ext ∗ A(M, N).<br />
⊓⊔<br />
Let B∗<br />
A → A → 0 be the bar resolution <strong>of</strong> the A e -module<br />
A. There is a chain map D : B∗ A → B∗ A ⊗ A B∗ A given by<br />
n∑<br />
D(a 0 ⊗a 1 ⊗· · ·⊗a n+1 ) = (a 0 ⊗· · · a i ⊗1)⊗(1⊗a i+1 ⊗· · ·⊗a n+1 ).<br />
i=0<br />
Let ζ ∈ H n (A) <strong>and</strong> ζ ′ ∈ H m (A). Then the cup product ζ ∪ ζ ′<br />
is given by<br />
B∗<br />
A D<br />
→B∗ A ⊗ A B∗<br />
A ζ⊗ζ<br />
−→A ′<br />
⊗ A A = A.<br />
More explicitly ζ ∪ ζ ′ : Bn+m A → A is given by<br />
(ζ∪ζ ′ )(a 0 ⊗a 1 ⊗· · ·⊗a n+m+1 ) = ζ(a 0 ⊗· · · a n ⊗1)ζ ′ (1⊗a n+1 ⊗· · ·⊗a n+m+<br />
Based on previous construction, HH ∗ (A) forms a ring, called<br />
the Hochschild cohomology ring <strong>of</strong> A. The multiplicative<br />
structure was known to Yoneda when R is a field.<br />
By [67], one can use an arbitrary projective resolution P ∗ →<br />
A → 0 to construct the cup product, <strong>and</strong> there exists a unique<br />
chain map D : P ∗ → P ∗ ⊗ A P ∗ such that (∂ 0 ⊗ ∂ 0 )D = ∂ 0 . In<br />
fact one just have to show that P ∗ ⊗ A P ∗ → A → 0 is also a<br />
projective resolution.<br />
Theorem 5.1.5 (Gerstenhaber). With the above cup product<br />
the Hochschild cohomology ring<br />
HH ∗ (A) = ⊕ i≥0<br />
HH i (A)<br />
is graded commutative. Moreover the cup product coincides<br />
with the Yondea splice.
5.1 Hochschild homology <strong>and</strong> cohomology 189<br />
Pro<strong>of</strong>. Let 0 → A → L n−1 → · · · → L 0 → A → 0 <strong>and</strong><br />
0 → A → M m−1 → · · · → M 0 → A → 0 be two n-fold <strong>and</strong><br />
m-fold extensions. They uniquely determine two cohomology<br />
classes by liftings<br />
· · ·<br />
<br />
B A n<br />
B A n−1<br />
· · ·<br />
<br />
B A 0<br />
A 0<br />
<strong>and</strong><br />
ζ=ζ n <br />
ζ n−1<br />
<br />
0<br />
<br />
A = L n<br />
<br />
L n−1<br />
<br />
· · ·<br />
<br />
L 0<br />
<br />
A 0<br />
· · ·<br />
<br />
B A m<br />
B A m−1<br />
ζ 0 <br />
· · ·<br />
<br />
B A 0<br />
A<br />
<br />
0<br />
ξ=ξ m <br />
ξ m−1<br />
<br />
0<br />
<br />
A = M m<br />
<br />
M m−1<br />
<br />
· · ·<br />
<br />
M 0<br />
<br />
A 0.<br />
The Yoneda splice ζ ∗ ξ is given by any lifting <strong>of</strong> ξ = ξ m<br />
· · ·<br />
<br />
Bn+m A Bn+m−1<br />
A · · ·<br />
<br />
Bm<br />
A<br />
θ n <br />
θ n−1 <br />
θ 0 ξ<br />
0<br />
<br />
A<br />
<br />
L<br />
n−1 · · ·<br />
<br />
L 0<br />
<br />
A<br />
ξ 0 <br />
B A m−1<br />
· · ·<br />
<br />
A 0<br />
<br />
M m−1<br />
<br />
· · ·<br />
<br />
A 0<br />
0 0 .<br />
To finish our pro<strong>of</strong>, we will define two explicit liftings. On one<br />
h<strong>and</strong>, the composition <strong>of</strong> the following chain maps<br />
θ ∗ ′ : B∗<br />
A D<br />
−→B∗ A ⊗ A B∗<br />
A Id⊗ξ<br />
−→B∗ A ⊗ A A[m] → B∗ A [m].<br />
provides a lifting <strong>of</strong> ξ by<br />
θ i ′ : Bi+m<br />
A D<br />
−→(B ∗ A ⊗ A B∗ A Id⊗ξ<br />
) i+m −→Bi<br />
A ⊗ A A[m] → Bi A [m] ξ i<br />
→L i ,<br />
for each 0 ≤ i ≤ n. Here [m] denotes a shift <strong>of</strong> the chain<br />
complex. From its definition we can check that ζ ∗ ξ = ζ ◦<br />
(Id ⊗ θ n) ′ ◦ D = ζ ∪ ξ.
190 5 Hochschild cohomology<br />
On the other h<strong>and</strong> we can define ξ ′′<br />
i : B A n+i → BA i<br />
ξ ′′<br />
i (a 0 ⊗· · ·⊗a n+i+1 ) = (−1) ni ξ(a 0 ⊗· · ·⊗a n ⊗1)⊗a n+1 ⊗· · ·⊗a n+i+1 .<br />
They give rise to another lifting <strong>of</strong> ξ. Then we have<br />
ζ ∗ ξ(a 0 ⊗ · · · ⊗ a n+m+1 ) = ζθ ′′ m(a 0 ⊗ · · · ⊗ a n+m+1 )<br />
= (−1) nm ζ(ξ(a 0 ⊗ · · · ⊗ a n ⊗ 1) ⊗ a n+1 ⊗ · ·<br />
= (−1) nm ξ(a 0 ⊗ · · · ⊗ a n ⊗ 1)ζ(a n+1 ⊗ · · · ⊗<br />
= (−1) nm (ξ ∪ ζ)(a 0 ⊗ · · · ⊗ a n+m+1 ).<br />
The pro<strong>of</strong> is taken from [70].<br />
Theorem 5.1.6 (Dennis,). If A <strong>and</strong> B are Morita equivalent,<br />
then HH ∗ (A) ∼ = HH ∗ (B) <strong>and</strong> there is a ring isomorphism<br />
HH ∗ (A) ∼ = HH ∗ (B).<br />
Pro<strong>of</strong>. The pro<strong>of</strong> to the general situation is not difficult, but<br />
one needs to know the functors realizing the given equivalence<br />
between two module categories. See []<br />
⊓⊔<br />
In the situation <strong>of</strong> category algebras, if D ≃ C are two categories,<br />
then the Hochschild (co)homology <strong>of</strong> RD <strong>and</strong> RC are<br />
isomorphic. These isomorphisms can be seen after we express<br />
Hochschild (co)homology as certain ordinary (co)homology.<br />
Hochschild homology <strong>and</strong> cohomology are important invariants<br />
<strong>of</strong> rings. However they are very difficult to compute. The<br />
main result in this chapter is that when A = RC is a category<br />
algebra, we can express HH ∗ (RC) by ordinary cohomology,<br />
<strong>and</strong> this allows effective calculation.<br />
by<br />
⊓⊔<br />
5.1.2 Ring homomorphisms from the Hochschild cohomology ring<br />
In general, if A <strong>and</strong> B are two associative k-algebras <strong>and</strong> M is<br />
a A ⊗ k B op -module, or equivalently a A-B-bimodule, we can
5.1 Hochschild homology <strong>and</strong> cohomology 191<br />
define a ring homomorphism induced by the tensor product<br />
− ⊗ A M<br />
φ M : Ext ∗ A e(Λ, Λ) → Ext∗ A⊗ R Bop(M, M).<br />
Let 0 → A → L n−1 → P n−2 → · · · → P 0 → A → 0 represent<br />
an element in Ext n A e(A, A). We may assume P i are projective<br />
A e -modules. Then considered as a complex <strong>of</strong> right<br />
A-modules, it is split exact. Thus tensoring with M gives an<br />
exact sequence <strong>of</strong> A-B-modules<br />
0 → M → L n−1 ⊗ A M → P n−2 ⊗−AM → · · · → P 0 ⊗ A M → M → 0.<br />
This induces a ring homomorphism φ M : Ext ∗ Ae(A, A) →<br />
Ext ∗ A⊗ R B op(M, M). If N is another A ⊗ R B op -module, we<br />
see Ext ∗ A⊗ R B op(M, N) has an Ext∗ Ae(A, A)-module structure<br />
via the ring homomorphisms φ M <strong>and</strong> φ N together with the<br />
Yoneda splice. We quote the following theorem <strong>of</strong> Snashall<br />
<strong>and</strong> Solberg [], which generalizes Gerstenhaber’s theorem.<br />
Theorem 5.1.7. Let A <strong>and</strong> B be two associative R-algebras.<br />
Let η be an element in Ext n Ae(A, A) <strong>and</strong> θ an element in<br />
Ext m A⊗ R Bop(M, N) for two A-B-bimodules M <strong>and</strong> N. Then<br />
φ N (η)θ = (−1) mn θφ M (η).<br />
Pro<strong>of</strong>. Let us fix an element η for n ≥ 1<br />
0 → A → L n−1 → P n−2 → · · · → P 0 → A → 0<br />
with P i projective.<br />
First we consider the case for θ ∈ Hom A⊗B op(M, N) (m = 0).<br />
When n = 0, Ext 0 Ae(A, A) = Z(A). It means each element in<br />
Ext 0 Ae(A, A) is a multiplication by some a ∈ Z(A). Such a<br />
map induces an element in Hom A⊗B op(M, M) by m ↦→ am.<br />
One can easily verify that (η ⊗ A N)θ = θ(η ⊗ A M).<br />
When n = 1, we can construct the following commutative<br />
diagram
192 5 Hochschild cohomology<br />
0<br />
<br />
A ⊗ A M L 0 ⊗ A M<br />
A⊗θ <br />
L 0 ⊗θ<br />
A ⊗ A M 0<br />
<br />
0 A ⊗ A N X A ⊗ A M<br />
A⊗θ<br />
0<br />
<br />
A ⊗ A N<br />
<br />
L 0 ⊗ A N<br />
<br />
A ⊗ A N 0<br />
where the middle row is (η ⊗ A N)θ = θ(η ⊗ A M).<br />
Next let θ ∈ Ext 1 A⊗ R Bop(M, N) represented by 0 → N →<br />
Y → M → 0. Since all syzygies <strong>of</strong> A are projective as right<br />
A-modules, we have the following commutative diagram<br />
0<br />
<br />
Ω i (A) ⊗ A N<br />
<br />
P i−1 ⊗ A N<br />
0<br />
Ω i−1 (A) ⊗ A N<br />
0<br />
0<br />
<br />
Ω i (A) ⊗ A Y<br />
<br />
P i−1 ⊗ A Y<br />
Ω i−1 (A) ⊗ A Y<br />
0<br />
0<br />
<br />
Ω i (A) ⊗ A M P i−1 ⊗ A M Ω i−1 (A) ⊗ A M 0<br />
for all i with Ω 0 (A) = A. Denote the upper row by σ i , the<br />
rightmost column by θ i , the leftmost column by θ i+1 <strong>and</strong> the<br />
lower row by σ ′ i. Then by the 3 × 3-splice <strong>of</strong> [52, Lemma 3.3],<br />
we get σ i θ i = −θ i+1 σ ′ i for all i ≥ 1. Since<br />
η⊗ A N = (0 → A⊗ A N → L 0 ⊗ A N → Ω n−1 (A)⊗ A N → 0)σ n−1 · · · σ 1<br />
<strong>and</strong><br />
η⊗ A M = (0 → A⊗ A M → L 0 ⊗ A M → Ω n−1 (A)⊗ A M → 0)σ ′ n−1 · · · σ<br />
we obtain an equality (η ⊗ A N)θ = (−1) n θ(η ⊗ A M) by combining<br />
all above.<br />
When m > 1, since every θ ∈ Ext m Ae(M, N) is the Yoneda<br />
splice <strong>of</strong> m extensions, it follows directly that (η ⊗ A N)θ =<br />
(−1) mn θ(η ⊗ A M). We are done.<br />
⊓⊔
5.2 Hochschild (co)homology <strong>of</strong> category algebras 193<br />
5.2 Hochschild (co)homology <strong>of</strong> category algebras<br />
5.2.1 Basic ideas <strong>and</strong> examples<br />
Let C be a small category. Recall from Section 3.1.1 that we<br />
call C e = C × C op the enveloping category <strong>of</strong> a small category<br />
C. We also showed in Example 3.1.5 that there is a natural isomorphism<br />
kC e ∼ = (kC) e . As a functor, kC(x, y) = kHom C (y, x)<br />
if Hom C (y, x) ≠ ∅ <strong>and</strong> kC(x, y) = 0 otherwise. Here (x, y) ∈<br />
Ob C e . This result is just a simple observation. It implies the<br />
enveloping algebra <strong>of</strong> a category algebra <strong>of</strong> C is the category<br />
algebra <strong>of</strong> its enveloping category, so later on we will just use<br />
the terminology kC e when dealing with Hochschild cohomology.<br />
This identification enables us to apply functor cohomology<br />
theory to the investigation <strong>of</strong> the Hochschild cohomology<br />
theory <strong>of</strong> category algebras.<br />
A key ingredient is F (C), the category <strong>of</strong> factorizations in<br />
C. Recall that the category F (C) has the morphisms in C as<br />
its objects. In order to avoid confusion, we write an object in<br />
F (C) as [α], whenever α ∈ Mor C. A morphism from [α] ∈<br />
Ob F (C) to [α ′ ] ∈ Ob F (C) is given by a pair <strong>of</strong> u, v ∈ Mor C,<br />
making the following diagram commutative<br />
x<br />
u <br />
α<br />
y<br />
x ′ y ′ .<br />
α ′<br />
In other words, there is an morphism from [α] to [α ′ ] if <strong>and</strong><br />
only if α ′ = uαv for some u, v ∈ Mor C, or equivalently α is a<br />
factor <strong>of</strong> α ′ in Mor C. The category F (C) admits two natural<br />
covariant functors to C <strong>and</strong> C op<br />
v op<br />
C<br />
t s<br />
F (C) <br />
C op ,
194 5 Hochschild cohomology<br />
where t <strong>and</strong> s send an object [α] to its target <strong>and</strong> source,<br />
respectively. Using his Theorem A <strong>and</strong> its corollary, Quillen<br />
showed these two functors induce homotopy equivalences <strong>of</strong><br />
the classifying spaces. We will be interested in the functor<br />
∇ = (t, s) : F (C) → C e = C × C op ,<br />
sending an [α] ∈ Ob F (C) to (x, y) ∈ Ob C e if α ∈ Hom C (y, x)<br />
<strong>and</strong> a morphism (u, v op ) ∈ Mor F (C) to (u, v op ) ∈ Mor(C e ).<br />
The importance <strong>of</strong> the functor ∇ : F (C) → C e lies in the<br />
fact that its target category gives rise to the Hochschild cohomology<br />
ring <strong>of</strong> C, while its source category determines the<br />
ordinary cohomology ring <strong>of</strong> C ≃ F (C). In the situation <strong>of</strong><br />
(finite) posets <strong>and</strong> groups, the functor is well-understood <strong>and</strong><br />
in the group case it has been implicitly used to establish the<br />
homomorphism from the Hochschild cohomology ring to the<br />
ordinary cohomology ring.<br />
Example 5.2.1.1. When C is a poset, ∇ : F (C) → C e sends<br />
F (C) isomorphically onto a full category C e ∆ ⊂ Ce , where<br />
Ob C e ∆ = {(x, y) ∈ Ob C e ∣ ∣ Hom C (y, x) ≠ ∅}<br />
(the full subcategory C∆ e is well-defined whenever C is EI).<br />
One can easily see that RC as a functor only takes nonzero<br />
values at objects in Ob C∆ e . Furthermore as a RCe ∆ -<br />
module, RC ∼ = R is the trivial module. Since C∆ e ∼ = F (C)<br />
is a co-ideal in the poset C e , we obtain Ext ∗ RC e(RC, RC) ∼ =<br />
Ext ∗ RC (RC, RC) ∼ = Ext ∗<br />
∆ e RF (C)(k, k) ∼ = Ext ∗ kC(k, k), where the<br />
last isomorphism comes from the fact that BF (C) ≃ BC.<br />
This isomorphism between the two cohomology rings was<br />
first established in [30];<br />
2. When C is a group, the category F (C) is a groupoid <strong>and</strong> is<br />
equivalent to a subcategory <strong>of</strong> the one object category C e<br />
with morphism set
5.2 Hochschild (co)homology <strong>of</strong> category algebras 195<br />
{(g, g −1op ) ∣ ∣ g ∈ Mor C} ⊂ Mor C e .<br />
Based on this description, one can prove the existence <strong>of</strong> the<br />
surjective homomorphism from the Hochschild cohomology<br />
ring to the ordinary cohomology ring <strong>of</strong> a group, which is<br />
basically the same as the classical approach. See for example<br />
[].<br />
5.2.2 Hochschild (co)homology as ordinary (co)homology<br />
The two examples <strong>of</strong> last section hint that we may express<br />
Hochschild (co)homology in terms <strong>of</strong> ordinary (co)homology.<br />
Moreover they show us the way to establish such an expression.<br />
Let us examine the following commutative diagram <strong>of</strong><br />
small categories<br />
F (C)<br />
t<br />
∇<br />
<br />
C e = C × C op<br />
p<br />
C ,<br />
where p is the projection onto the first component. Recall from<br />
Section 3.2.2 that the preceding diagram leads to another<br />
R<br />
RF (C)-mod<br />
LK ∇<br />
RC e -mod<br />
RC<br />
LK t<br />
LK p<br />
∼ =−⊗RC R<br />
RC-mod .<br />
R<br />
In the rest <strong>of</strong> this section, we will establish <strong>and</strong> describe<br />
the following maps, induced by the three left Kan extensions<br />
LK t , LK p <strong>and</strong> LK ∇ respectively,
196 5 Hochschild cohomology<br />
t ∗ : Ext ∗ RF (C)(R, R) → Ext ∗ RC(R, R),<br />
p ∗ : Ext ∗ RC e(RC, RC) → Ext∗ RC(R, R)<br />
∇ ∗ : Ext ∗ RF (C)(R, R) → Ext ∗ RCe(RC, RC).<br />
Theorem 5.2.2. Let C be a small category. For any functor<br />
M ∈ RC e -mod, we have<br />
Ext ∗ RC e(RC, M) ∼ = Ext ∗ RF (C)(R, Res ∇ M).<br />
The maps t ∗ is an ring isomorphism, p ∗ ∼ = φ C is a split<br />
surjective ring homomorphism <strong>and</strong> ∇ ∗ is a split injective<br />
ring homomorphism.<br />
The theorem is proved in a series <strong>of</strong> lemmas.<br />
Lemma 5.2.3. The following complex <strong>of</strong> RC e -module<br />
LK ∇ B F (C)<br />
∗ → LK ∇ R → 0<br />
is a projective resolution <strong>of</strong> the RC e -module RC.<br />
Pro<strong>of</strong>. Let B∗<br />
F (C) → R → 0 be the bar resolution. By Proposition<br />
4.3.2, LK ∇ B∗<br />
F (C) ∼ = C∗ (∇/−, R). In Proposition 2.3.13,<br />
due to Quillen, we asserted that the category F (C) is a c<strong>of</strong>ibred<br />
category over C e . More explicitly for any (x, y) ∈ Ob C e<br />
the overcategory ∇/(x, y) is homotopy equivalent to the fibre<br />
∇ −1 (x, y), which is the discrete category Hom C (y, x). In other<br />
words, for any (x, y) ∈ Ob C e ,<br />
is exact. Thus<br />
C ∗ (∇/(x, y), R) → RHom C (y, x) → 0<br />
C ∗ (∇/−, R) → RC → 0<br />
is a projective resolution <strong>of</strong> the RC e -module RC. Furthermore<br />
we have LK ∇ R ∼ = RC by direct calculation<br />
(LK ∇ R)(x, y) = lim −→∇/(x,y)<br />
R ∼ = lim −→HomC (y,x) R ∼ = RHom C (y, x).
5.2 Hochschild (co)homology <strong>of</strong> category algebras 197<br />
⊓⊔<br />
For any functor M ∈ RC e -mod, the isomorphism<br />
Ext ∗ RC e(RC, M) ∼ = Ext ∗ RF (C)(R, Res ∇ M)<br />
is a direct consequence <strong>of</strong> Lemma 5.2.3.<br />
Before we study the ring homomorphisms, we give a interesting<br />
result on Res ∇ RC. For an arbitrary u : D → C, since<br />
LK u is the left adjoint <strong>of</strong> Res u , there are natural transformations<br />
Id → Res u LK u <strong>and</strong> LK u Res u → Id. We pay attention<br />
to the case <strong>of</strong> ∇ : F (C) → C e . There exists an RF (C)-<br />
homomorphism R → Res ∇ LK ∇ (R) = Res ∇ (RC) as well as<br />
a RC e -homomorphism RC = LK ∇ Res ∇ (R) → R. The latter<br />
gives rise to a RF (C)-homomorphism<br />
Res ∇ (RC) = Res ∇ LK ∇ Res ∇ (R) → R = Res ∇ R.<br />
In case that C is a poset, one has R = Res ∇ (RC). When C is<br />
a group, F (C) is a groupoid, equivalent to the automorphism<br />
group <strong>of</strong> [1 C ] ∈ Ob F (C), that is, {(g, g −1op ) ∣ g ∈ Mor C}.<br />
If we name the full subcategory <strong>of</strong> F (C), consisting <strong>of</strong> one<br />
object [1 C ], by ˜∆C <strong>and</strong> the inclusion (an equivalence) by i :<br />
˜∆C ↩→ F (C). Then Res ∇i (RC) = Res ∇ (RC)([1 C ]) is a R ˜∆Cmodule<br />
with the action (g, g −1op )·a = gag −1 , a ∈ Res ∇i (RC).<br />
Thus Res ∇i (RC) = ⊕ Rc g , where c g is the conjugacy class <strong>of</strong><br />
g ∈ Mor C. In particular R = Rc 1C is a direct summ<strong>and</strong> <strong>of</strong><br />
Res ∇i (RC) <strong>and</strong> it implies R ∣ Res ∇ (RC) as RF (C)-modules<br />
because i is an equivalence <strong>of</strong> categories.<br />
Lemma 5.2.4. Let C be a small category. Then R ∣ ∣ Res ∇ (RC)<br />
as RF (C)-modules.<br />
Pro<strong>of</strong>. One needs to keep in mind that the restriction <strong>of</strong> a<br />
module usually has a larger R-rank than the module itself<br />
since ∇ is not injective on objects. We define a RF (C)-
198 5 Hochschild cohomology<br />
homomorphism (a natural transformation) ι : R → Res ∇ (RC)<br />
by the assignments ι [α] (1 R ) = α ∈ Res ∇ (RC)([α]) for each<br />
[α] ∈ Ob F (C). If [β] is another object in Ob F (C) <strong>and</strong><br />
(u, v op ) ∈ Hom F (C) ([α], [β]) is an arbitrary morphism, then<br />
by the definition <strong>of</strong> an F (C)-morphism, (u, v op ) · α = uαv =<br />
β. Hence ι maps R isomorphically onto a submodule <strong>of</strong><br />
Res ∇ (RC). On the other h<strong>and</strong>, we may define a RF (C)-<br />
homomorphism ɛ : Res ∇ (RC) → R such that, for any<br />
[α] ∈ Ob F (C), ɛ [α] : Res ∇ (RC)([α]) → R([α]) = R sends<br />
each base element in Res ∇ (RC)([α]) = RHom C (y, x) to 1 R .<br />
One can readily check the composite <strong>of</strong> these two maps is the<br />
identity<br />
R ι →Res ∇ (RC) ɛ →R,<br />
<strong>and</strong> this means R ∣ Res ∇ (RC) or Res ∇ (RC) = R ⊕ N C for<br />
some RF (C)-module N C .<br />
⊓⊔<br />
The module N C as a functor can be described by<br />
N C ([α]) = R{β − γ ∣ ∣ β, γ ∈ Hom C (y, x)},<br />
if [α] ∈ Ob F (C) <strong>and</strong> α ∈ Hom C (y, x). It will be useful to our<br />
computation since it determines the “difference” between the<br />
Hochschild <strong>and</strong> ordinary cohomology rings <strong>of</strong> a category. The<br />
next lemma finishes <strong>of</strong>f our pro<strong>of</strong> <strong>of</strong> the main theorem.<br />
Using adjunction between LK ∇ <strong>and</strong> Res ∇ , along with Lemmas<br />
5.2.3 <strong>and</strong> 5.2.4, we get a commutative diagram<br />
Hom RF (C) (B∗ F (C) , R)<br />
splitting<br />
Hom RF (C) (B F (C)<br />
∗<br />
, Res ∇ RC)<br />
LK ∇<br />
∼ =<br />
<br />
Hom RC e(LK ∇ B F (C)<br />
∗ , LK ∇ R)<br />
<br />
Hom RC e(LK ∇ B∗<br />
F (C) , RC)<br />
The top row gives rise to ∇ ∗ . From this diagram one can see<br />
it can also obtained as
5.2 Hochschild (co)homology <strong>of</strong> category algebras 199<br />
Ext ∗ RF (C)(R, R) → Ext ∗ RF (C)(R, Res ∇ RC) ∼ = Ext ∗ RCe(RC, RC).<br />
Now we turn to establish the ring homomorphisms.<br />
Lemma 5.2.5. The map t ∗ : Ext ∗ RF (C)(R, R) → Ext ∗ RC(R, R)<br />
is a ring isomorphism.<br />
Pro<strong>of</strong>. As we explained in Remark 4.3.7, t ∗ is induced by<br />
LK t , <strong>and</strong> is the same as the map between simplicial/singular<br />
cohomology, induced by the functor t in Chapter 2. Since<br />
F (C) ≃ C, it is an isomorphism.<br />
⊓⊔<br />
Lemma 5.2.6. The map p ∗ : Ext ∗ RC e(RC, RC) → Ext∗ RC(R, R)<br />
is equal to φ C , induced by − ⊗ RC R.<br />
Pro<strong>of</strong>. Suppose M ∈ RC e -mod. We show LK p M ∼ = M ⊗ RC R<br />
as RC-modules. Let x ∈ Ob C. Then<br />
1 x · LK p M = (LK p M)(x) = lim −→p/x<br />
M.<br />
Because p/x ∼ = (Id C /x) × C op ≃ {(x, 1 x )} × C op , we have<br />
lim M ∼ = lim<br />
−→p/x −→IdC<br />
M<br />
/x×C<br />
∼ op = lim −→{(x,1x<br />
M<br />
)}×C<br />
∼ op = lim −→C opM(x, −)<br />
∼ = lim −→C<br />
op1 x · M<br />
∼ = R ⊗RC op (1 x · M)<br />
∼ = 1x · M ⊗ RC R.<br />
In particular it implies LK p (RC e ) ∼ = RC e ⊗ RC R ∼ = RC ⊗ R<br />
R. Thus if ˜P∗ → RC → 0 is the projective resolution in<br />
Section 5.1.1, it splits when regarded as a complex <strong>of</strong> right RCmodules.<br />
Consequently LK p<br />
∼ = − ⊗RC R maps any projective<br />
resolution <strong>of</strong> RC to an exact sequence <strong>of</strong> left RC-modules<br />
LK p ˜P∗ → LK p (RC) ∼ = R → 0,
200 5 Hochschild cohomology<br />
which is a projective resolution. Hence LK p induces a chain<br />
map<br />
Hom RC e( ˜P∗ , RC) → Hom RC (LK p ˜P∗ , R),<br />
<strong>and</strong> the ring homomorphism<br />
p ∗ = φ C : Ext ∗ RC e(RC, RC) → Ext∗ RC(R, R).<br />
Lemma 5.2.7. Let C be a small category. Then ∇ ∗ is a ring<br />
homomorphism <strong>and</strong> there is another ring homomorphism ɛ ∗<br />
Ext ∗ RC e(RC, RC) ↠ Ext∗ RF (C)(R, R),<br />
such that ɛ ∗ ∇ ∗ ∼ = 1 <strong>and</strong> p ∗ ∼ = t ∗ ɛ ∗ . It means ∇ ∗ <strong>and</strong> ∇ ∗ (t ∗ ) −1<br />
are injective ring homomorphisms while p ∗ <strong>and</strong> ɛ ∗ = (t ∗ ) −1 p ∗<br />
are surjective ring homomorphisms.<br />
Pro<strong>of</strong>. We prove the map ∇ ∗ is a ring homomorphism. Then<br />
from p ∗ ∇ ∗ = t ∗ we get [(t ∗ ) −1 p ∗ ]∇ ∗ = 1 Ext<br />
∗<br />
RF (C)<br />
(R,R) <strong>and</strong> thus<br />
we can define ɛ ∗ = (t ∗ ) −1 p ∗ which is a surjective ring homomorphism.<br />
Take the bar resolution B∗<br />
F (C) → R → 0. We have seen that<br />
LK ∇ B∗<br />
F (C) → LK ∇ R ∼ = RC → 0 is a projective resolution <strong>of</strong><br />
the RC e -module RC. Let f, g be two cocycles representing two<br />
cohomology classes. Then we construct the following diagram<br />
B F (C)<br />
∗<br />
LK ∇ B F (C)<br />
∗<br />
D F (C)<br />
<br />
B F (C)<br />
∗<br />
LK ∇ D F (C)<br />
<br />
LK ∇ (B F (C)<br />
LK ∇ B∗<br />
F (C) LK<br />
D Ce ∇ B∗<br />
F (C)<br />
∗<br />
Θ ∇<br />
<br />
ˆ⊗B F (C)<br />
∗<br />
ˆ⊗B F (C)<br />
∗<br />
⊗ RC LK ∇ B F (C)<br />
∗<br />
f⊗g<br />
⇓LK ∇<br />
) LK ∇(f⊗g)<br />
LK ∇ (f)⊗LK ∇ (g)<br />
<br />
R ˆ⊗R<br />
⊓⊔<br />
∼ =<br />
<br />
LK ∇ (R ˆ⊗R) ∼ =<br />
<br />
Θ 0<br />
∼ =<br />
LK ∇ (R) ⊗ RC LK ∇ (R)<br />
The first row represents f ∪ g, the cup product <strong>of</strong> f <strong>and</strong> g.<br />
The left Kan extension LK ∇ maps it to the second row <strong>of</strong><br />
<br />
R<br />
<br />
RC<br />
∼ =<br />
<br />
RC
5.2 Hochschild (co)homology <strong>of</strong> category algebras 201<br />
RC e -modules which represents the image <strong>of</strong> the cup product,<br />
<strong>and</strong> we want to show it gives rise to the cup product <strong>of</strong><br />
LK ∇ (f) <strong>and</strong> LK ∇ (g) as Hochschild cohomology classes. Since<br />
we have LK ∇ (D F (C) ) : LK ∇ B∗<br />
F (C) → LK ∇ (B∗<br />
F (C) ˆ⊗B∗ F (C) ),<br />
<strong>and</strong> LK ∇ (B∗<br />
F (C) ) <strong>and</strong> LK ∇ (B∗<br />
F (C) ) ⊗ RC LK ∇ (B∗<br />
F (C) ) are chain<br />
homotopy equivalent as both <strong>of</strong> them are projective resolutions<br />
<strong>of</strong> RC, we can construct chain maps D Ce <strong>and</strong> Θ ∇ , unique<br />
up to chain homotopy, such that the above diagram is commutative.<br />
Because the Hochschild diagonal approximation map<br />
always exists <strong>and</strong> is unique up to chain homotopy, independent<br />
<strong>of</strong> the choice <strong>of</strong> a projective resolution <strong>of</strong> RC [67], D Ce will<br />
serve as the Hochschild diagonal approximation map. Then<br />
since the lower two rows form a commutative diagram, we<br />
know they represent the same cohomology class, i.e. the cup<br />
product LK ∇ (f) ∪ LK ∇ (g), in Ext ∗ RCe(RC, RC). ⊓⊔<br />
The surjective ring homomorphism<br />
ɛ ∗ = (t ∗ ) −1 p ∗ : Ext ∗ RC e(RC, RC) → Ext∗ RF (C)(R, R)<br />
is the composite <strong>of</strong><br />
Ext ∗ RC e(RC, RC) ∼= →Ext ∗ RF (C)(R, Res ∇ RC)↠Ext ∗ RF (C)(R, R).<br />
Remark 5.2.8. Slightly modifying the previous argument, we<br />
can also demonstrate the action <strong>of</strong> Ext ∗ RC e(RC, RC) on Ext∗ RCe(RC, M)<br />
alternatively via ˆ⊗ on RF (C)-mod. For any M ∈ RC e -mod,<br />
one gets<br />
Ext ∗ RC e(RC, M) ∼ = Ext ∗ RF (C)(R, Res ∇ M).<br />
It was also shown that the RF (C)-module Res ∇ RC natural<br />
splits as R ⊕ N C for some N C ∈ RF (C)-mod. This provides<br />
a surjective homomorphism ρ : Res τ RC ˆ⊗Res ∇ M → Res ∇ M,<br />
<strong>and</strong> hence a map
202 5 Hochschild cohomology<br />
ρ ∗ : Ext ∗ RF (C)(R, Res ∇ RC ˆ⊗Res ∇ M) → Ext ∗ RF (C)(R, Res ∇ M).<br />
The latter fits into the following commutative diagram<br />
Ext ∗ RF (C) (R, Res ∇RC) ⊗ Ext ∗ RF (C) (R, Res ∇M)<br />
∪<br />
<br />
Ext ∗ RC e(RC, RC) ⊗ Ext∗ RCe(RC, M)<br />
Ext ∗ RF (C) (R, Res ∇RC ˆ⊗Res ∇ M)<br />
ρ ∗<br />
<br />
Ext ∗ RF (C) (R, Res ∇M)<br />
Ext ∗ RC<br />
∪<br />
e(RC, M),<br />
which reduces to [67, Proposition 3.1] when C = G is a<br />
group. The top row is the so-called cup product with respect<br />
to the pairing ρ. Since Ext ∗ RF (C)(R, R) is a direct summ<strong>and</strong><br />
<strong>of</strong> Ext ∗ RF (C)(R, Res ∇ RC), it also exhibits the action <strong>of</strong><br />
Ext ∗ RC(R, R) on Ext ∗ RCe(RC, M), via its identification with<br />
Ext ∗ RF (C)(R, R).<br />
Note that when C is an abelian group, we obtain an isomorphism<br />
[36, ]<br />
Ext ∗ RC e(RC, RC) ∼ = Ext ∗ RF (C)(R, Res ∇ (RC)) ∼ = RC⊗ R Ext ∗ RC(R, R).<br />
Finally we comment on Hochschild homology. It is a direct<br />
consequence <strong>of</strong> Lemma 5.2.3. Its pro<strong>of</strong> is entirely analogues<br />
to that <strong>of</strong> Corollary 4.3.4 (1). We do not have a counterpart<br />
for Corollary 4.3.4 (2) because we do not know the structure<br />
<strong>of</strong> (y, x)\∇ ( for posets ).<br />
Theorem 5.2.9. Suppose N ∈ RC e -mod. Then we have<br />
Particularly<br />
Tor RCe<br />
∗<br />
Thus Tor RC<br />
∗<br />
Tor RCe<br />
∗<br />
Tor RCe<br />
∗<br />
(RC, RC) ∼ = Tor<br />
(N, RC) ∼ = Tor<br />
RF (C)<br />
∗<br />
RF (C)<br />
∗<br />
(Res ∇ N, R).<br />
(Res ∇ RC, R) ∼ = Tor<br />
RF (C)<br />
∗<br />
(R, R)⊕Tor<br />
(R, R) is isomorphic to a direct summ<strong>and</strong> <strong>of</strong><br />
(RC, RC).<br />
RF (C)<br />
∗
5.2 Hochschild (co)homology <strong>of</strong> category algebras 203<br />
Remark 5.2.10. Suppose u : D → C is an equivalence. By<br />
Proposition 2.3.9 we naturally obtain another equivalence<br />
F (u) : F (D) → F (C). Because we have a commutative diagram<br />
F (D) F (u) <br />
F (C)<br />
∇ <br />
D e u e <br />
C e<br />
we can deduce that the Hochschild (co)homology <strong>of</strong> RD <strong>and</strong><br />
RC are isomorphic. For instance, given any M ∈ RC e -mod,<br />
we have<br />
Tor RCe<br />
∗<br />
(RC, RC) ∼ RF (C)<br />
= Tor∗ (Res ∇ RC, R)<br />
∼ = Tor<br />
RF (D)<br />
∇<br />
∗ (Res F (u) Res ∇ RC, R)<br />
∼ = Tor<br />
RF (D)<br />
∗ (Res ∇ Res u eRC, R)<br />
∼ = Tor<br />
RD e<br />
∗ (Res u eRC, RD).<br />
Finally using Hochschild (co)homology, we can establish the<br />
following useful isomorphisms between ordinary (co)homology.<br />
Part (1) extends Proposition 3.3.10.<br />
Theorem 5.2.11. Suppose k is a field <strong>and</strong> C is a small category.<br />
Let M, N ∈ kC-mod <strong>and</strong> M ′ ∈ mod-kC. Then we<br />
have<br />
1. Ext ∗ kC(M, N) ∼ = Ext ∗ kF (C)(k, Res ∇ Hom k (M, N)); <strong>and</strong><br />
2. Tor kC<br />
∗ (M ′ , N) ∼ kF (C)<br />
= Tor∗<br />
(Res ∇ (N ⊗ k M ′ ), k).<br />
Pro<strong>of</strong>. By Proposition 5.1.4, Ext ∗ kC(M, N) ∼ = Ext ∗ kC e(kC, Hom k(M, N))<br />
From Theorem 5.2.2, Ext ∗ kC e(kC, Hom k(M, N)) ∼ = Ext ∗ kF (C)(k, Res ∇ Ho<br />
Similarly by Proposition 5.1.4 <strong>and</strong> Theorem 5.2.9, we get<br />
the isomorphism for homology. Note that it is natural to give<br />
Res ∇ (N ⊗ M ′ ) a right RF (C)-module structure. ⊓⊔
204 5 Hochschild cohomology<br />
The above isomorphisms generalize the well known isomorphisms<br />
in group cohomology Ext ∗ kG(M, N) ∼ = Ext ∗ kG(k, Hom k (M, N))<br />
<strong>and</strong> Tor kG<br />
∗ (M, N) ∼ = Tor kG<br />
∗ (N ⊗ k M, k).<br />
Proposition 3.3.10 does not generalize to cohomology. Let<br />
B∗ C → k → 0 be the bar resolution. Then the lefting B∗<br />
F (C) →<br />
Res t B∗ C induces<br />
Hom kC (B C ∗, RK t Res ∇ Hom k (M, N)) ∼ = Hom kF (C) (Res t B C ∗, Res ∇ Hom k (M<br />
→ Hom kF (C) (B F (C)<br />
∗ , Res ∇ Hom R (M<br />
<strong>and</strong> thus<br />
Ext ∗ kC(k, RK t Res ∇ Hom k (M, N)) → Ext ∗ kF (C)(k, Res ∇ Hom R (M, N))<br />
∼ = Ext<br />
∗<br />
kC (M, N).<br />
These maps have no reason to be isomorphisms in general.<br />
Similarly we have a map<br />
Tor kC<br />
∗ (M ′ , N) ∼ kF (C)<br />
= Tor∗ (Res ∇ (N ⊗ k M ′ ), k)<br />
→ Tor kC<br />
∗ (LK t Res ∇ (N ⊗ k M ′ ), k),<br />
by Corollary 4.3.4 (1) <strong>and</strong> Lemma 4.4.1.<br />
5.2.3 EI categories<br />
In this section, we assume our categories are EI. The purpose<br />
is to compare Hochschild (co)homology <strong>of</strong> C with that <strong>of</strong> an<br />
automorphism group <strong>of</strong> an object.<br />
Suppose A is the full subcategory <strong>of</strong> C which consists <strong>of</strong> all<br />
objects <strong>and</strong> all isomorphisms in C. The category A is a disjoint<br />
union <strong>of</strong> finitely many finite groups. Its category algebra<br />
RA = ⊕ x∈Ob C RAut C(x) is an RC e -module, a direct sum <strong>of</strong><br />
atomic modules supported on minimal objects in C RC . There<br />
is a surjective RC e -morphism π : RC → RA, with kernel written<br />
as ker π. Considered as a functor ker π ⊂ RC takes non-
5.2 Hochschild (co)homology <strong>of</strong> category algebras 205<br />
zero values only at (x, y) for which there exists a C-morphism<br />
from y to x <strong>and</strong> x ≁ = y. Since there is an inclusion functor<br />
i : A → C, we have maps between ordinary (co)homology i ∗ :<br />
H ∗ (C, R) → H ∗ (A, R) <strong>and</strong> i ∗ : H∗(A, R) → H∗(C, R). Here we<br />
show there are maps between their Hochschild (co)homology<br />
as well.<br />
The short exact sequence <strong>of</strong> RC e -modules<br />
induces a long exact sequence<br />
0 → ker π → RC π →RA → 0<br />
· · · → Ext n RC e(RC, ker π) → Extn RC e(RC, RC) ˜π →Ext n RC e(RC, RA) η → · · ·<br />
By Proposition 4.2.1, one can see Ext ∗ RCe(RC, RA) is naturally<br />
isomorphic to<br />
Ext ∗ RAe(RA, RA),<br />
which is isomorphic to the direct product <strong>of</strong> the Hochschild<br />
cohomology rings <strong>of</strong> the automorphism groups <strong>of</strong> objects in<br />
C: ∏<br />
Ext ∗ RAut C (x) e(RAut C(x), RAut C (x)),<br />
[x]⊂Ob C<br />
where [x] st<strong>and</strong>s for the isomorphism class <strong>of</strong> x ∈ Ob C. The<br />
following map will still be written as ˜π<br />
˜π : Ext ∗ RC e(RC, RC) → Ext∗ RAe(RA, RA).<br />
We show ˜π can be identified with the algebra homomorphism<br />
induced by − ⊗ RC RA, where as a left RC-module RA is a<br />
direct sum <strong>of</strong> atomic modules.<br />
φ A : Ext ∗ RC e(RC, RC) → Ext∗ RC e(RA, RA) ∼ = Ext ∗ RAe(RA, RA).<br />
Hence we do not need to distinguish the maps φ A <strong>and</strong> ˜π.<br />
Lemma 5.2.12. The following diagram is commutative
206 5 Hochschild cohomology<br />
Ext ∗ RC e(RC, RC) ˜π <br />
Ext ∗ RCe(RC, RA)<br />
φ A<br />
∼<br />
<br />
=<br />
Ext ∗ RC e(RA, RA) <br />
∼ =<br />
Ext ∗ RAe(RA, RA).<br />
Pro<strong>of</strong>. This can be seen on the cochain level. Suppose R ∗ →<br />
RC → 0 is the minimal projective resolution <strong>of</strong> the RC e -<br />
module RC. Then Ext ∗ RCe(RC, RC) is the cohomology <strong>of</strong> the<br />
cochain complex Hom RC e(R ∗ , RC). The tensor product −⊗ RC<br />
RA induces a map<br />
Hom RC e(R ∗ , RC) → Hom RC−RA (R ∗ ⊗ RC RA, RA).<br />
Since B∗<br />
RC ⊗ RC RA is a projective resolution <strong>of</strong> the RC-RAmodule<br />
RA, R ∗ ⊗ RC RA is also a projective resolution <strong>of</strong> RA.<br />
Moreover because R ∗ is minimal, it is supported on C RC . It<br />
implies R ∗ ⊗ RC RA is also supported on C RC . But the RC e -<br />
module RA is supported on minimal objects <strong>of</strong> C RC , we have<br />
Hom RC−RA (R ∗ ⊗ RC RA, RA) ∼ = Hom RA e(Res i (R ∗ ⊗ RC RA), RA)<br />
∼ = HomRA e(Res i R ∗ , RA),<br />
which gives rise to φ A . Here i : A e → C e is the inclusion,<br />
induced by another inclusion, also written as i : A → C. On<br />
the other h<strong>and</strong> ˜π is exactly given by the same chain map<br />
Hom RC e(R ∗ , RC) → Hom RC e(R ∗ , RA) ∼ = Hom RA e(Res i R ∗ , RA).<br />
Thus we are done.<br />
We have the following commutative diagram, involving four<br />
cohomology rings.<br />
Theorem 5.2.13. Let C be an EI-category. Then we have<br />
the following commutative diagram<br />
⊓⊔
5.2 Hochschild (co)homology <strong>of</strong> category algebras 207<br />
Ext ∗ RC e(RC, RC) φ A =˜π <br />
φ C <br />
Ext ∗ RAe(RA, RA)<br />
φ A<br />
Ext ∗ RC(R, R)<br />
Res i<br />
<br />
Ext ∗ RA(R, R).<br />
Pro<strong>of</strong>. We prove it on the cochain level. Let R ∗ → RC → 0<br />
be the minimal projective resolution <strong>of</strong> the RC e -module RC.<br />
Then we have the following commutative diagram<br />
Hom RC e(R ∗ , RC) −⊗ RCRA <br />
Hom RC−RA (R ∗ ⊗ RC RA, RA)<br />
−⊗ RC R<br />
<br />
Hom RC (R ∗ ⊗ RC R, R)<br />
−⊗ RA R<br />
Hom RC (R ∗ ⊗ RC R, R)<br />
Hom RC (R ′ ∗, R)<br />
Res i<br />
<br />
Hom RA (Res i R ′ ∗, R)<br />
<br />
Hom RA (R ′′<br />
∗, R),<br />
in which R ′ ∗ → R → 0 <strong>and</strong> R ′′<br />
∗ → R → 0 are the projective<br />
resolutions <strong>of</strong> the trivial RC- <strong>and</strong> RA-modules satisfying<br />
the following commutative diagrams <strong>of</strong> RC-modules <strong>and</strong> RAmodules,<br />
respectively,<br />
R ′ ∗<br />
R 0 R ′′<br />
∗<br />
R ∗ ⊗ RC R R 0 <strong>and</strong> Res i R ′ ∗<br />
R<br />
<br />
0<br />
<br />
R 0.<br />
In the main diagram, upper left cochain complex computes<br />
Ext ∗ RC e(RC, RC), upper right corner computes Ext∗ RAe(RA, RA)<br />
by Lemma 5.2.12, lower left corner computes Ext ∗ RC(R, R) <strong>and</strong><br />
lower right corner computes Ext ∗ RA(R, R). Hence our statement<br />
follows.<br />
⊓⊔
208 5 Hochschild cohomology<br />
We note that in the theorem the category A may be replaced<br />
by any full subcategory <strong>of</strong> it. Especially, we have a<br />
commutative diagram for each Aut C (x) ⊂ A<br />
Ext ∗ RC e(RC, RC)φ RAut C (x)<br />
Ext ∗ RAut C (x) e(RAut C(x), RAut C (x))<br />
φ C <br />
φ AutC (x)<br />
<br />
Ext ∗ RC(R, R)<br />
Res i<br />
<br />
Ext ∗ RAut C (x)(R, R).<br />
5.3 Examples <strong>of</strong> the Hochschild cohomology rings <strong>of</strong> categories<br />
In this section we calculate the Hochschild cohomology rings<br />
for four finite EI-categories, with base field k <strong>of</strong> characteristic<br />
2.<br />
5.3.1 The category E 0<br />
In [87] we presented an example, by Aurélien Djament, Laurent<br />
Piriou <strong>and</strong> the author, <strong>of</strong> the mod-2 ordinary cohomology<br />
ring <strong>of</strong> the following category E 0<br />
h<br />
g<br />
x<br />
1 x<br />
α<br />
β<br />
gh<br />
y {1 y }<br />
,<br />
where g 2 = h 2 = 1 x , gh = hg, αh = βg = α, <strong>and</strong> αg = βh =<br />
β. The ordinary cohomology ring Ext ∗ kE 0<br />
(k, k) is a subring <strong>of</strong><br />
the polynomial ring H ∗ (Z 2 × Z 2 , k) ∼ = k[u, v], removing all<br />
u n , n ≥ 1, <strong>and</strong> their scalar multiples. It has no nilpotents <strong>and</strong><br />
is not finitely generated. By Theorem 2.3.4, it implies that the<br />
Hochschild cohomology ring Ext ∗ kE e 0 (kE 0, kE 0 ) is not finitely<br />
generated either. We compute its Hochschild cohomology ring<br />
using Proposition 2.3.5.
5.3 Examples <strong>of</strong> the Hochschild cohomology rings <strong>of</strong> categories 209<br />
The category <strong>of</strong> factorizations in E 0 , F (E 0 ), has the following<br />
shape<br />
[α]<br />
[β]<br />
[1 x ]<br />
[1 y ]<br />
[h]<br />
[gh]<br />
[g]<br />
in which [1 x ] ∼ = [h] ∼ = [g] ∼ = [gh] <strong>and</strong> [α] ∼ = [β]. For the purpose<br />
<strong>of</strong> computation, we use the skeleton F ′ (E 0 ) <strong>of</strong> F (E 0 ) (which<br />
is equivalent to F (E 0 ) hence the two category algebras <strong>and</strong><br />
their module categories are Morita equivalent)<br />
{(α,1 op<br />
x ),(α,h op ),(β,g op ),(β,(gh) op )}<br />
{(1 y ,1 op<br />
x )}<br />
<br />
[α]<br />
[1 x ]<br />
[1<br />
y ].<br />
{(1 x ,1 op<br />
x ),(h,h op ),(g,g op ),(gh,(gh) op )}<br />
{(1 y ,α op )}<br />
{(1 y ,1 op<br />
y )}<br />
In the above category, next to each arrow is the set <strong>of</strong> homomorphisms<br />
in F ′ (E 0 ) from one object to another. The module<br />
N E0 ∈ kF ′ (E 0 )-mod (see Proposition 2.3.5) takes the following<br />
values<br />
N C ([1 x ]) = k{1 x + h, g + gh, 1 x + g} , N C ([h]) = k{1 x + h, g + gh, 1 x<br />
N C ([g]) = k{1 x + h, g + gh, 1 x + g} , N C ([gh]) = k{1 x + h, g + gh, 1 x<br />
N C ([α]) = k{α + β}<br />
, N C ([β]) = k{α + β},<br />
N C ([1 y ]) = 0.<br />
,
210 5 Hochschild cohomology<br />
Thus N E0 = S [1x ],k(1 x +h)⊕S [1x ],k(g+gh)⊕k ′ 1 x +g, where S [1x ],k(1 x +h)<br />
<strong>and</strong> S [1x ],k(g+gh) are simple kF ′ (E 0 )-modules such that S [1x ],k(1 x +h)([1 x ])<br />
k(1 x + h) <strong>and</strong> S [1x ],k(g+gh)([1 x ]) = k(g + gh), <strong>and</strong> k ′ 1 x +g is a<br />
kF ′ (E 0 )-module such that k ′ 1 x +g([1 x ]) = k(1 x +g), k ′ 1 x +g([α]) =<br />
k(α + β) <strong>and</strong> k ′ 1 x +g([1 y ]) = 0. Note that S [1x ],k(1 x +h)([1 x ]) =<br />
k(1 x + h), S [1x ],k(g+gh)([1 x ]) = k(g + gh) <strong>and</strong> k ′ 1 x +g([1 x ]) =<br />
k(1 x + g) are all isomorphic to the trivial kAut F ′ (E 0 )([1 x ])-<br />
module, <strong>and</strong> have the same trivial ring structure in the sense<br />
that the product <strong>of</strong> any two elements is zero. Hence we have<br />
(along with the result quoted in Section 2.4, paragraph two)<br />
Ext ∗ kF ′ (E 0 )(k, S [1x ],k(1 x +h)) ∼ = k(1 x + h) ⊗ k Ext ∗ kAut F ′ (E0 ) ([1 x])(k, k)<br />
<strong>and</strong><br />
Ext ∗ kF ′ (E 0 )(k, S [1x ],k(g+gh)) ∼ = k(g + gh) ⊗ k Ext ∗ kAut F ′ (E0 ) ([1 x])(k, k)<br />
as rings, in which k(1 x + h) <strong>and</strong> k(g + gh) are concentrated<br />
in degree zero in each ring. From the structure <strong>of</strong> F (E 0 ), one<br />
has Aut F ′ (E 0 )([1 x ]) ∼ = Z 2 × Z 2 .<br />
For computing Ext ∗ kF ′ (E 0 )(k, k ′ 1 x +g), we use the following<br />
short exact sequence <strong>of</strong> kF (E 0 )-modules<br />
0 → k ′ 1 x +g → k → S [1y ],k → 0.<br />
It induces a long exact sequence in which one can find<br />
Ext 0 kF ′ (E 0 )(k, S [1y ],k) = k <strong>and</strong> Ext n kF ′ (E 0 )(k, S [1y ],k) = 0 if n ≥ 1.<br />
Thus Ext 0 kF ′ (E 0 )(k, k ′ 1+g) = 0 while Ext n kF ′ (E 0 )(k, k) ∼ = Ext n kF ′ (E 0 )(k, k ′ 1+g)<br />
for each n ≥ 1. Hence as a ring<br />
Ext ∗ kF ′ (E 0 )(k, k ′ 1 x +g) ∼ = k(1 x +g)⊗ k Ext ∗>0<br />
kF ′ (E 0 ) (k, k) ∼ = k(1 x +g)⊗ k Ext ∗>0<br />
kE 0<br />
(<br />
All in all, we have<br />
Ext 0 kE e 0 (kE 0, kE 0 ) ∼ = Ext 0 kE 0<br />
(k, k) ⊕ k(1 x + h) ⊕ k(g + gh),
5.3 Examples <strong>of</strong> the Hochschild cohomology rings <strong>of</strong> categories 211<br />
<strong>and</strong> if n ≥ 1<br />
Ext n kE e 0 (kE 0, kE 0 )<br />
∼ = Ext<br />
n<br />
kE0 (k, k) ⊕ {k(1 x + g) ⊗ k Ext n kE 0<br />
(k, k)}<br />
⊕{k(1 x + h) ⊗ k Ext ∗ k(Z 2 ×Z 2 )(k, k)} ⊕ {k(g + gh) ⊗ k Ext n k(Z 2 ×Z 2 )(k,<br />
Combining all the information we obtained, the surjective ring<br />
homomorphism<br />
φ E0 : Ext ∗ kE e 0 (kE 0, kE 0 ) ↠ Ext ∗ kE 0<br />
(k, k)<br />
has its kernel consisting <strong>of</strong> all nilpotents. Consequently this<br />
Hochschild cohomology ring modulo nilpotents is not finitely<br />
generated, against the finite generation conjecture in []. We<br />
comment that the category algebra kE 0 is not a self-injective<br />
algebra (hence is not Hopf, by []). Nicole Snashall points out<br />
to the author that this algebra is Koszul since both kE 0 <strong>and</strong><br />
Ext ∗ kE 0<br />
(kE 0 , kE 0 ) as graded algebras are generated in degrees<br />
zero <strong>and</strong> one, where kE 0 = kE 0 /Rad(kE 0 ) ∼ = S x,k ⊕ S y,k .<br />
5.3.2 The category E 1<br />
The following category E 1 has a terminal object <strong>and</strong> hence is<br />
contractible:<br />
h<br />
x<br />
1 x<br />
α <br />
y {1 y } ,<br />
where h 2 = 1 x <strong>and</strong> αh = α. The contractibility implies the<br />
ordinary cohomology ring is simply the base field k. In this<br />
case F (E 1 ) is the following category
212 5 Hochschild cohomology<br />
[1 x ]<br />
(h,1 op<br />
y )<br />
(α,Aut E1 (x) op )<br />
(1 x ,1 op<br />
x ) <br />
(h,h op )<br />
(1 x ,h op )<br />
(h,1 op<br />
x )<br />
[α]<br />
[h]<br />
(1 x ,h op )<br />
(1 x ,1 op<br />
y )<br />
(h,1 op<br />
x )<br />
(1 y ,α op )<br />
(α,Aut E1 (x) op )<br />
[1 y ]<br />
(1 y ,1 op<br />
y )<br />
We calculate its Hochschild cohomology ring. By proposition<br />
2.3.5, we only need to compute Ext ∗ kF (E 1 )(k, N E1 ), where N E1<br />
has the following value at objects <strong>of</strong> F (E 1 )<br />
N E1 ([1 x ]) = k{1 x + h} , N E1 ([h]) = k{1 x + h},<br />
N E1 ([1 y ]) = 0 , N E1 ([α]) = 0.<br />
One can easily see that N E1 = S [1x ],k(1 x +h) is a simple module <strong>of</strong><br />
dimension one with a specified value k(1 x + h) at [1 x ]. Since<br />
[1 x ] ∼ = [h] ∈ Ob F (E1 ) are minimal objects, using quoted<br />
result in Section 2.4 paragraph two, we get<br />
Ext ∗ kF (E 1 )(k, N E1 ) ∼ = Ext ∗ kAut F (E1 )([1 x ])(k, k(1 x +h)) ∼ = k(1 x +h)⊗ k Ext ∗ kZ 2<br />
(<br />
which is isomorphic to k(1 x + h) ⊗ k k[u]. Here k[u] is a polynomial<br />
algebra with an indeterminant u at degree one <strong>and</strong><br />
k(1 x + h) is at degree zero. Thus<br />
Ext ∗ kE e 1 (kE 1, kE 1 ) ∼ = Ext ∗ kE 1<br />
(k, k)⊕Ext ∗ kF (E 1 )(k, N E1 ) ∼ = k⊕{k(1 x +h)⊗ k k<br />
The kernel <strong>of</strong> φ E1<br />
cohomology ring.<br />
consists <strong>of</strong> all nilpotents in the Hochschild<br />
5.3.3 The category E 2<br />
The following category has its classifying space homotopy<br />
equivalent to the join, BZ 2 ∗ BZ 2 = Σ(BZ 2 ∧ BZ 2 ) =
5.3 Examples <strong>of</strong> the Hochschild cohomology rings <strong>of</strong> categories 213<br />
Σ[B(Z 2 × Z 2 )/(BZ 2 ∨ BZ 2 )], <strong>of</strong> the classifying spaces <strong>of</strong> the<br />
two automorphism groups:<br />
h<br />
x<br />
1 x<br />
α<br />
g<br />
1 y<br />
<br />
y ,<br />
<br />
where h 2 = 1 x , αh = α = gα <strong>and</strong> g 2 = 1 y . As direct consequences,<br />
its ordinary cohomology groups are equal to k, 0, 0<br />
at degrees zero, one <strong>and</strong> two, <strong>and</strong> k n−2 at each degree n ≥ 3,<br />
<strong>and</strong> furthermore the cup product in this ring is trivial [87].<br />
We compute its Hochschild cohomology ring. The category<br />
F (E 2 ) is as follows<br />
(1 x ,1 op<br />
x ) <br />
(h,h op )<br />
[1 x ]<br />
(α,Aut E2 (x) op )<br />
(α,Aut E2 (x) op )<br />
(1 x ,h op )<br />
[h]<br />
(Aut E2 (x),Aut E2 (y) op )<br />
<br />
(h,1 op<br />
x )<br />
[α]<br />
(1 y ,1 op<br />
y )<br />
(Aut E2 (y),α op )<br />
(Aut E2 (y),α op )<br />
[1 y ]<br />
(g,g op )<br />
[g]<br />
(1 y ,g op )<br />
(g,1 op<br />
y )<br />
By Proposition 2.3.5, we need to compute Ext ∗ kE 2<br />
(k, N E2 ). In<br />
this case we have<br />
N E2 ([1 x ]) = k{1 x + h} , N E2 ([h]) = k{1 x + h},<br />
N E2 ([1 y ]) = k{1 y + g} , N E2 ([g]) = k{1 y + g},<br />
N E2 ([α]) = 0.<br />
It means N E2 = S [1x ],k(1 x +h) ⊕ S [1y ],k(1 y +g) <strong>and</strong> thus by Proposition<br />
2.2.5<br />
Ext ∗ kE 2<br />
(k, N E2 ) ∼ = Ext ∗ kAut F (E2 )([1 x ])(k, k(1 x + h)) ⊕ Ext ∗ kAut F (E2 )([1 y ])(k, k(1<br />
∼ = {k(1x + h) ⊗ k Ext ∗ kZ 2<br />
(k, k)} ⊕ {k(1 y + g) ⊗ k Ext ∗ kZ 2<br />
Hence
214 5 Hochschild cohomology<br />
Ext ∗ kE e 2 (kE 2, kE 2 ) ∼ = Ext ∗ kE 2<br />
(k, k)⊕{k(1 x +h)⊗ k k[u]}⊕{k(1 y +g)⊗ k k[v]}<br />
where k[u] <strong>and</strong> k[v] are two polynomial algebras with indeterminants<br />
in degree one. Both the Hochschild <strong>and</strong> ordinary<br />
cohomology rings modulo nilpotents are isomorphic to the<br />
base field k.<br />
5.3.4 The category E 3<br />
The following category has a classifying space homotopy<br />
equivalent to that <strong>of</strong> Aut E3 (x) ∼ = Z 2 (by Quillen’s Theorem A<br />
[], or see [86])<br />
h<br />
x<br />
1 x<br />
α<br />
β<br />
y {1 y }<br />
,<br />
where h 2 = 1 x <strong>and</strong> αh = β. We compute its Hochschild<br />
cohomology ring. The category F (E 3 ) is as follows (not all<br />
morphisms are presented since only its skeleton is needed)<br />
(1 y ,1 op<br />
x )<br />
(α,1 op<br />
x ),(β,h op )<br />
(h,1 op<br />
[α]<br />
(1 y ,h op )<br />
[β]<br />
(1 y ,1 op<br />
x )<br />
<br />
x ) [1 x ]<br />
[1 y ]<br />
(1 x ,h op )<br />
(1 x ,h op )<br />
[h]<br />
(h,1 op<br />
x )<br />
(1 y ,β op )<br />
.<br />
The module N E3 takes the following values<br />
(1 y ,1 op<br />
y )<br />
N E2 ([1 x ]) = k{1 x + h} , N E2 ([h]) = k{1 x + h},<br />
N E2 ([1 y ]) = 0 , N E2 ([α]) = k{α + β},<br />
N E2 ([α]) = k{α + β}.
5.3 Examples <strong>of</strong> the Hochschild cohomology rings <strong>of</strong> categories 215<br />
Thus N E3 fits into the following short exact sequence <strong>of</strong><br />
kF (E 3 )-modules<br />
0 → N E3 → k → S [1y ],k → 0.<br />
Just like in our first example, using the long exact sequence<br />
coming from it, we know Ext 0 kE 2<br />
(k, N E3 ) = 0 <strong>and</strong> Ext ∗>0<br />
kE 2<br />
(k, N E3 ) ∼ =<br />
k(1 x +h)⊗ k Ext ∗>0<br />
kF (E 3 ) (k, k) ∼ = k(1 x +h)⊗ k Ext ∗>0<br />
kE 3<br />
(k, k). Hence<br />
Ext ∗ kE e 3 (kE 3, kE 3 ) ∼ = Ext ∗ kE 3<br />
(k, k)⊕{k(1 x +h)⊗ k Ext ∗>0<br />
kE 3<br />
(k, k)}.<br />
The kernel <strong>of</strong> φ E3<br />
cohomology ring.<br />
contains all nilpotents in the Hochschild
Chapter 6<br />
Connections with group representations <strong>and</strong> cohomology<br />
Abstract We study various local categories <strong>of</strong> finite groups.<br />
The purpose is to compare representations <strong>and</strong> cohomology<br />
<strong>of</strong> groups <strong>and</strong> <strong>of</strong> these categories. The transporter categories<br />
play a key role to bridge up these two concepts. In fact a<br />
group <strong>and</strong> all its subgroups are some sort <strong>of</strong> transporter categories.<br />
We pay attention to the transporter categories <strong>and</strong><br />
try to demonstrate their similarities with <strong>and</strong> differences from<br />
groups, in terms <strong>of</strong> homological properties. We shall apply<br />
tools developed in the preceding two chapters to investigate<br />
transporter categories, a special kind <strong>of</strong> finite EI categories<br />
which partially motivate modern research on category representations<br />
<strong>and</strong> cohomology. In this chapter all categories are<br />
finite <strong>and</strong> all modules are finitely generated.<br />
6.1 Local categories<br />
Let G be a finite group <strong>and</strong> p a positive prime that divides<br />
|G|, the order <strong>of</strong> G. We study various collections <strong>of</strong> subgroups<br />
<strong>of</strong> G <strong>and</strong> resulting categories. <strong>Finite</strong> categories naturally appear<br />
in both group representation <strong>and</strong> homotopy theory. For<br />
a finite group G <strong>and</strong> a positive prime p that divides the order<br />
<strong>of</strong> G, various posets <strong>of</strong> p-subgroups have been investigated intensively<br />
to establish connections between representations <strong>of</strong><br />
G <strong>and</strong> those <strong>of</strong> its local subgroups [61, 7, 76, 80, 81, 69]. Over<br />
the past two decades, mathematicians realized that categories
218 6 Connections with group representations <strong>and</strong> cohomology<br />
build upon those previously mentioned posets should play an<br />
important role in modular representation theory [60]. Indeed,<br />
to any such poset P, it naturally comes with a G-action.<br />
There is a Grothendieck construction on P which results in<br />
a category G ∝ P, called a transporter category, containing<br />
P as a subcategory. A transporter category admits some interesting<br />
quotient categories, namely Brauer categories, Puig<br />
categories <strong>and</strong> orbit categories [75]. For many good reasons,<br />
we shall call any quotient category <strong>of</strong> a transporter category a<br />
local category, as it reveals some p-local information about G.<br />
<strong>Representations</strong> <strong>and</strong> cohomology <strong>of</strong> these local categories are<br />
currently in the center <strong>of</strong> group representation theory. In homotopy<br />
theory <strong>of</strong> classifying spaces, the geometric realization<br />
<strong>of</strong> the nerve <strong>of</strong> a transporter category is a Borel construction,<br />
EG × G BP. It was shown by several authors, especially<br />
Dwyer [17], that transporter categories play a key role in homology<br />
decompositions <strong>of</strong> classifying spaces. Recently, Broto,<br />
Levi <strong>and</strong> Oliver [10, 11] developed a theory <strong>of</strong> p-local finite<br />
groups, which is motivated by the use <strong>of</strong> local categories associated<br />
with BG (or its p-completion). In their terminology,<br />
a p-local finite group is a triple (S, F, L) such that S is a<br />
finite p-group, F is a finite category called a fusion system<br />
on S, <strong>and</strong> L is an extension <strong>of</strong> a full subcategory <strong>of</strong> F. The<br />
(p-completion <strong>of</strong>) classifying space <strong>of</strong> L behaves like the classifying<br />
space <strong>of</strong> a finite group. In this theory, cohomology <strong>of</strong><br />
small categories is an essential ingredients. See the new book<br />
[2] for fusion systems in group theory, homotopy theory <strong>and</strong><br />
representation theory.
6.1 Local categories 219<br />
6.1.1 G-categories<br />
Definition 6.1.1. Consider G as a category with one object<br />
•. A G-category is a functor F : G → Cat.<br />
In other words, a G-category is a category C, equipped with<br />
a group homomorphism G = Aut G (•) → Aut Cat (C). The<br />
simplest example is a point with trivial action by G. Recall<br />
that a set is regarded as a poset with trivial relations, <strong>and</strong> a<br />
poset is regarded as a category.<br />
Example 6.1.2. Suppose H ⊂ G is a subgroup. Consider the<br />
discrete set <strong>of</strong> left cosets G/H = {gH ∣ ∣ g ∈ G}. Then G acts<br />
on it by left multiplication, permuting these cosets.<br />
Example 6.1.3. Suppose H ⊂ G is a subgroup. Then the discrete<br />
set <strong>of</strong> conjugacy class G H = { g H ∣ ∣ g ∈ G} is also a<br />
G-category with G acting by conjugations.<br />
The above two examples are the examples <strong>of</strong> G-sets.<br />
Definition 6.1.4. A collection <strong>of</strong> subgroups <strong>of</strong> G is a set <strong>of</strong><br />
subgroups <strong>of</strong> G, closed under conjugations by elements in G.<br />
Example 6.1.5. A collection <strong>of</strong> subgroups <strong>of</strong> G is naturally<br />
a poset with inclusions as relations. Hence any collection <strong>of</strong><br />
subgroups is a G-poset.<br />
There are various interesting collections <strong>of</strong> subgroups <strong>of</strong> G.<br />
Example 6.1.6.1. The collection <strong>of</strong> all p-subgroups <strong>of</strong> G, denoted<br />
by S e p.<br />
2. The collection <strong>of</strong> all non-identity p-subgroups <strong>of</strong> G, denoted<br />
by S p . Below is a concrete example <strong>of</strong> the poset <strong>of</strong> all nontrivial<br />
2-subgroups <strong>of</strong> Σ 4 , S 2 (Σ 4 ):
220 6 Connections with group representations <strong>and</strong> cohomology<br />
· · ·<br />
· · ·<br />
D 8 D 8 D 8<br />
<br />
C 2 × C 2<br />
<br />
C 4<br />
<br />
V<br />
C 2<br />
<br />
C 2<br />
Here D 8 is the dihedral group <strong>of</strong> order 8, V is a Kleine four<br />
group <strong>and</strong> those C are cyclic groups with order specified<br />
in the subscripts. Due to the size, we do not record the full<br />
poset. In deed we only write down the subgroups <strong>of</strong> one <strong>of</strong><br />
the three D 8 . However, in order to obtain the full poset, one<br />
just needs to copy the same thing under the leftmost D 8 <strong>and</strong><br />
put it for each omitted part, where the dots appear. Note<br />
that V is a subgroup <strong>of</strong> all three D 8 .<br />
3. The collection <strong>of</strong> all non-identity elementary p-subgroups <strong>of</strong><br />
G, denoted by A p .<br />
4. The collection <strong>of</strong> all p-radical subgroups <strong>of</strong> G, denoted by<br />
B p .<br />
Moreover, there are various refinements <strong>of</strong> some <strong>of</strong> the above<br />
posets.<br />
Example 6.1.7.1. Let b be a p-block <strong>of</strong> kG. The the b-Brauer<br />
pairs S b form a G-poset. When b is the principal block, it is<br />
isomorphic to S e p.<br />
2. Let A be an interior G-algebra. Then the pointed subgroups<br />
<strong>of</strong> G form a G-poset.<br />
C 2<br />
C 2<br />
C 2<br />
6.1.2 Homology representations <strong>of</strong> kG<br />
Suppose C is a small G-category. Its nerve N ∗ C is a simplicial<br />
set, from which one can construct a chain complex C ∗ (C, k)
6.1 Local categories 221<br />
such that, for each i > 0, C i (C, k) is a k-vector space with a<br />
basis the set <strong>of</strong> i-chains <strong>of</strong> morphisms in C, while C 0 (C, k) =<br />
k Ob C. Then one can see that G acts on each C i (C, k) <strong>and</strong><br />
C i (C, k) by permutating its base elements. Consequently we<br />
obtains G-modules Hi(C, k) <strong>and</strong> H i (C, k).<br />
Definition 6.1.8. Let C be a small category. Assume the normalized<br />
complex C † ∗(C, k) is finite. Then the Euler characteristic<br />
χ(C) = χ(C, k) is defined to be ∑ i≥0 (−1)i dim k C † i (C, k).<br />
Note that C † ∗(C, k) being finite is equivalent to BC being<br />
a finite CW complex. Since ∑ i≥0 (−1)i dim k C † i (C, k) =<br />
∑<br />
i≥0 (−1)i dim k Hi(C, k), χ(C, k) is invariant under homotopy<br />
equivalence.<br />
Example 6.1.9.1. In Example 6.1.2, C ∗ (G/H) is a stalk complex<br />
C ∗ (G/H) = C 0 (G/H) ∼ = k ↑ G H with χ(G/H) = [G :<br />
H].<br />
2. In Example 6.1.3, C ∗ ( G H) is a stalk complex C ∗ ( G H) =<br />
C 0 ( G H) ∼ = k ↑ G N G (H) with χ(G H) = [G : N G (H)].<br />
3. In Example 6.1.4 (1), Sp e is not a stalk complex. However<br />
since Sp e has an initial object <strong>and</strong> thus is contractible,<br />
χ(Sp) e = 1.<br />
4. The inclusions A p ⊂ S p <strong>and</strong> B p ⊂ S p induce homotopy<br />
equivalences, by using Quillen’s Theorem A. Hence<br />
χ(S p , k) = χ(A p , k) = χ(B p , k).<br />
6.1.3 Transporter categories as Grothendieck constructions<br />
Various G-posets <strong>of</strong> subgroups <strong>of</strong> G demonstrate certain local<br />
structural information <strong>of</strong> G. However it is not complete. For<br />
example given H ∈ G <strong>and</strong> the discrete set G H, any two nonidentical<br />
objects are conjugate in G while this relationship is
222 6 Connections with group representations <strong>and</strong> cohomology<br />
not seen in G H. Thus we want a category that presents all intrinsic<br />
connections among any chosen collection <strong>of</strong> subgroups<br />
<strong>of</strong> G.<br />
Definition 6.1.10. Let G be a group <strong>and</strong> P a G-poset. We<br />
define the transporter category on P to be a Grothendieck<br />
construction G ∝ P := Gr G P. More precisely, G ∝ P has<br />
the same objects as P, that is, Ob(G ∝ P) = Ob P. For<br />
x, y ∈ Ob(G ∝ P), a morphism from x to y is a pair (g, gx ≤<br />
y) for some g ∈ G.<br />
In the literature the transporter categories are mostly considered<br />
as auxiliary constructions before passing to various<br />
quotient categories <strong>of</strong> them. Here we want to stress on the perhaps<br />
unique property, among various categories constructed<br />
from a group, that transporter categories admit natural functors<br />
to the group itself. It singles out this particular type <strong>of</strong><br />
categories <strong>and</strong> is the starting point <strong>of</strong> this chapter. Here in<br />
order to emphasize the similarities <strong>and</strong> connections between<br />
transporter categories <strong>and</strong> subgroups, we follow a definition<br />
which is well known to some algebraic topologists. The symbol<br />
G ∝ P is not st<strong>and</strong>ard <strong>and</strong> is used because this particular<br />
Grothendieck construction resembles a semidirect product,<br />
yet is different.<br />
This neat but seemingly abstract definition can be easily<br />
seen to give the usual transporter categories. For example,<br />
when P is the poset <strong>of</strong> non-trivial p-subgroups, we get<br />
G ∝ P = Tr p (G), the p-transporter category <strong>of</strong> G. The advantage<br />
<strong>of</strong> taking our approach is shown by the upcoming<br />
examples, where each subgroup <strong>of</strong> G is identified as a transporter<br />
category, up to a category equivalence.<br />
From Definition 6.1.10 one can easily see that there is a<br />
natural embedding ι P : P ↩→ G ∝ P via (x ≤ y) ↦→ (e, x ≤<br />
y). On the other h<strong>and</strong>, the transporter category admits a
6.1 Local categories 223<br />
natural functor π P : G ∝ P → G, given by x ↦→ • <strong>and</strong><br />
(g, gx ≤ y) ↦→ g. Thus we always have a sequence <strong>of</strong> functors<br />
P ι P<br />
↩→G ∝ P π P<br />
→G<br />
such that π P ◦ ι P (P) is the trivial subgroup or subcategory<br />
<strong>of</strong> G. For convenience, in the rest <strong>of</strong> this chapter we <strong>of</strong>ten<br />
neglect the subscript P <strong>and</strong> write ι = ι P , π = π P .<br />
Example 6.1.11.1. If G acts trivially on P, then G ∝ P =<br />
G × P.<br />
2. Let G be a finite group <strong>and</strong> H a subgroup. We consider<br />
the set <strong>of</strong> left cosets G/H which can be regarded as a G-<br />
poset: G acts via left multiplication. The transporter category<br />
G ∝ (G/H) is a connected groupoid whose skeleton<br />
is isomorphic to H. In this way one can recover all subgroups<br />
<strong>of</strong> G, up to category equivalences. Consequently we<br />
have k(G ∝ G/H) ≃ kH as well as a homotopy equivalence<br />
B(G ∝ G/H) ≃ BH.<br />
3. From Example 6.1.6 (2) we build a concrete transporter category<br />
Σ 4 ∝ S 2 (Σ 4 ) :<br />
C 2<br />
4<br />
8<br />
C 2 × C 2<br />
8<br />
4 <br />
4<br />
8<br />
C 2<br />
4<br />
8<br />
8<br />
· · ·<br />
· · ·<br />
<br />
8<br />
D 8<br />
8<br />
8 D 8<br />
8<br />
8 D 8<br />
8 24 24 24<br />
C 4 8 V 24<br />
8<br />
C 2<br />
8<br />
24<br />
8 <br />
8<br />
24<br />
C 2<br />
Here the numbers are the numbers <strong>of</strong> morphisms from one<br />
object to another. Note that again this is part <strong>of</strong> the whole<br />
category. However, it contains a skeleton so we know what<br />
is missing.<br />
8<br />
24<br />
8 <br />
8<br />
C 2<br />
8
224 6 Connections with group representations <strong>and</strong> cohomology<br />
Remark 6.1.12. One can check directly that if Hom G∝P (x, y) ≠<br />
∅ then both Aut G∝P (x) <strong>and</strong> Aut G∝P (y) act freely on Hom G∝P (x, y).<br />
Another notable structural fact is that G ∝ P is a category<br />
with subobjects, which means each morphism can be<br />
uniquely factorized as an isomorphism followed by a morphism<br />
in P (regarded as a subcategory <strong>of</strong> G ∝ P). In fact we<br />
have (g, gx ≤ y) = (e, gx ≤ y) ◦ (g, gx = gx) as shown in the<br />
diagram<br />
y<br />
(g,gx≤y)<br />
x<br />
(e,gx≤y)<br />
(g,gx=gx) gx.<br />
Definition 6.1.13. We call B(G ∝ P) the Borel construction<br />
on P.<br />
For any G-space X, one can define a Borel construction<br />
EG × G X. In our definition, we actually have B(G ∝ P) ≃<br />
EG × G BP. It explains the concept. In particular B(G ∝<br />
(G/H)) ≃ BH. Forming the transporter category over a G-<br />
poset eliminates the G-action, as an algebraic analogy <strong>of</strong> introducing<br />
a Borel construction over a G-space.<br />
6.1.4 Local categories<br />
Our concept <strong>of</strong> a transporter category is quite general because<br />
each subgroup H <strong>of</strong> G can be recovered as a transporter category<br />
G ∝ (G/H), for the G-poset G/H, up to a category<br />
equivalence. Thus it makes sense if we deem transporter categories<br />
as generalized subgroups for a fixed finite group.<br />
Definition 6.1.14. Any quotient category C <strong>of</strong> G ∝ P is<br />
called a local category <strong>of</strong> G. When P consists <strong>of</strong> p-subgroups<br />
<strong>of</strong> G, for a prime p ∣ ∣ |G|, we also call such a quotient category<br />
a p-local category.
6.1 Local categories 225<br />
A local category is connected with the group by the following<br />
diagram<br />
G ∝ P<br />
π<br />
G<br />
C<br />
Transporter categories were implicitly considered by Mark<br />
Ronan <strong>and</strong> Steve Smith [65] in the 1980s for constructing<br />
group modules, <strong>and</strong> later on played a key role in Bill Dwyer’s<br />
work [17] on homology decomposition <strong>of</strong> classifying spaces.<br />
Dwyer used this diagram to establish connections among various<br />
homotopy colimits (e.g. classifying spaces), while Ronan<br />
<strong>and</strong> Smith constructed kG-modules via representations<br />
<strong>of</strong> G ∝ P (using the language <strong>of</strong> G-presheaves on P).<br />
Example 6.1.15.1. The p-transporter category Tr p (G) = G ∝<br />
S p has all non-identity subgroups as its objects. For any<br />
p-subgroups P, Q, the morphism set is <strong>of</strong>ten written as<br />
Hom G (P, Q) = { g P ⊂ Q ∣ ∣ g ∈ G}. In particular AutG (P ) =<br />
N G (P ), the normalizer.<br />
2. The p-fusion system F p (G) is a quotient category <strong>of</strong> G ∝ S p ,<br />
given by Hom Fp (G)(P, Q) = Hom G (P, Q)/C G (P ).<br />
3. The p-orbit category O p (G) is a quotient category <strong>of</strong> G ∝ P,<br />
given by Hom Op (G)(P, Q) = Q\Hom G (P, Q).<br />
Example 6.1.16. Let b be a p-block <strong>of</strong> kG.<br />
1. The b-transporter category Tr b (G) = G ∝ S b has all nonidentity<br />
subgroups as its objects. For any p-subgroups P, Q,<br />
the morphism set is <strong>of</strong>ten written as Hom G∝Sb ((P, e P ), (Q, e Q )) =<br />
{ g (P, e P ) ⊂ (Q, e Q ) ∣ ∣ g ∈ G}. In particular AutG∝Sb (P ) =<br />
N G (P, e P ).<br />
2. The Brauer category, or p-fusion system, F b (G) is a quotient<br />
category <strong>of</strong> G ∝ S b , given by Hom Fb (G)(P, Q) =<br />
Hom G∝Sb (P, Q)/C G (P ).<br />
ρ
226 6 Connections with group representations <strong>and</strong> cohomology<br />
3. The b-orbit category O b (G) <strong>of</strong> F b (G) is a quotient category<br />
<strong>of</strong> F b (G), given by Hom Ob (G)(P, Q) = Q\Hom Fb (G)(P, Q).<br />
When b is the principal block, then the first two categories<br />
are isomorphic to the first two in Example 6.1.15. The b-orbit<br />
category is quite different from the p-orbit category because it<br />
is commonly believed that in non-principal block situation the<br />
latter construction is better than the former in capturing p-<br />
local information <strong>of</strong> the block b. This explains why transporter<br />
categories are important but still not enough <strong>and</strong> so we have<br />
to examine various quotient categories.<br />
6.2 Properties <strong>of</strong> local categories<br />
In this section, we illustrate the role <strong>of</strong> transporter categories<br />
in group representations. Then we continue to give several results<br />
on representations <strong>of</strong> transporter categories, which further<br />
demonstrate close connections between representations <strong>of</strong><br />
groups <strong>and</strong> transporter categories. This section will end with<br />
two applications, <strong>of</strong> representation theory, to cohomology <strong>of</strong><br />
transporter category algebras.<br />
6.2.1 Two diagrams <strong>of</strong> categories<br />
We have seen that given a G-poset P there exists a diagram<br />
<strong>of</strong> categories <strong>and</strong> functors<br />
π<br />
G ∝ P<br />
G<br />
C<br />
for any given local category C. Diagrams similar to this<br />
have been (implicitly) considered by Ronan-Smith [65, 6] <strong>and</strong><br />
Dwyer [17]. Here we pursue a direction that is closely related<br />
to the work <strong>of</strong> Ronan <strong>and</strong> Smith. The reader is referred to [17]<br />
ρ
6.2 Properties <strong>of</strong> local categories 227<br />
for Dwyer’s beautiful results on homology decompositions <strong>of</strong><br />
classifying spaces <strong>of</strong> groups.<br />
In practice one <strong>of</strong>ten finds that the functor G ∝ P → C<br />
is part <strong>of</strong> an extension (or an opposite extension) sequence<br />
<strong>of</strong> categories. (Among examples are various orbit categories,<br />
Brauer categories <strong>and</strong> Puig categories.) It means that for such<br />
a quotient category C there exists a category K which is a<br />
disjoint union <strong>of</strong> subgroups <strong>of</strong> Aut G∝P (x), x running over<br />
Ob P = Ob(G ∝ P), such that we can add K into the picture<br />
K<br />
π<br />
<br />
G ∝ P<br />
G<br />
C<br />
<strong>and</strong> moreover K ↩→ G ∝ P ↠ C satisfies some natural conditions<br />
provided in Section 4.1.. It will helps us to underst<strong>and</strong><br />
relationship between k(G ∝ P)-mod <strong>and</strong> kC-mod.<br />
For any M ∈ kG-mod, sometimes we denote by κ M the<br />
restriction Res π M <strong>and</strong> call it a constant value module.<br />
ρ<br />
6.2.2 Frobenius Reciprocity<br />
Applying classical tools in homological algebra, particularly<br />
the Kan extensions, we obtain a Frobenius Reciprocity between<br />
kG-mod <strong>and</strong> kC-mod, where kC is the (k-)category<br />
algebra <strong>of</strong> C. It implies, to some extent, comparing kG-mod<br />
with kC-mod provides an extended context for local representation<br />
theory. This observation illustrates a possible way to<br />
underst<strong>and</strong> group representations <strong>and</strong> cohomology via those<br />
<strong>of</strong> suitable categories. From the preceding diagram we obtain<br />
a diagram <strong>of</strong> module categories
228 6 Connections with group representations <strong>and</strong> cohomology<br />
LK π ,RK π<br />
k(G ∝ P)-mod<br />
LK ρ ,RK ρ<br />
<br />
Res π<br />
Res ρ<br />
kG-mod<br />
kC-mod.<br />
The adjunctions between the restrictions <strong>and</strong> Kan extensions<br />
have the following consequences.<br />
Proposition 6.2.1 (Frobenius Reciprocity). Suppose P<br />
is a G-poset <strong>and</strong> C is a quotient category <strong>of</strong> G ∝ P as in<br />
the preceding diagrams. Let M, N ∈ kG-mod <strong>and</strong> m, n ∈ kCmod.<br />
Then<br />
1. Hom kG (M, RK π Res ρ n) ∼ = Hom kC (LK ρ Res π M, n);<br />
2. Hom kG (LK π Res ρ m, N) ∼ = Hom kC (m, RK ρ Res π N).<br />
By direct calculations, these particular Kan extensions in<br />
Proposition 6.2.1 are simplified:<br />
•LK π<br />
∼ = lim −→P<br />
, RK π<br />
∼ = lim ←−P<br />
(used by Ronan-Smith. See<br />
Corollary 6.3. for a pro<strong>of</strong>);<br />
•LK ρ<br />
∼ =↑<br />
kC<br />
k(G∝P)<br />
(the induction), RK ρ<br />
∼ =⇑<br />
kC<br />
k(G∝P)<br />
(the coinduction),<br />
since ρ induces an algebra homomorphism k(G ∝<br />
P) → kC when C is a quotient category.<br />
Remark 6.2.2. For any n ∈ kC-mod, we shall write the k(G ∝<br />
P)-module Res ρ n as n because they share the same underlying<br />
vector space. Recall that κ M = Res π M. Then the Frobenius<br />
Reciprocity can be rewritten as<br />
(i’) Hom kG (M, lim ←−P<br />
n) ∼ = Hom kC (κ M ↑ kC<br />
k(G∝P) , n);<br />
(ii’) Hom kG (lim −→P<br />
m, N) ∼ = Hom kC (m, κ N ⇑ kC<br />
k(G∝P) ).<br />
When P = G/H for some subgroup H, we have natural<br />
isomorphisms ←−G/H<br />
lim ∼ = lim ∼<br />
−→G/H<br />
=↑<br />
G<br />
H . Then the above<br />
isomorphisms certainly become the usual adjunctions between<br />
↑ G H <strong>and</strong> ↓G H (the usual Frobenius Reciprocity) with
6.3 The functor π: group representations via transporter categories 229<br />
C = G ∝ (G/H) <strong>and</strong> ρ = Id, in light <strong>of</strong> the Morita equivalence<br />
between kC <strong>and</strong> kH.<br />
Remark 6.2.3. Our Frobenius reciprocity is different from a<br />
similar result <strong>of</strong> Ronan-Smith, see [6, 7.2.4], where they (implicitly)<br />
had a diagram <strong>of</strong> the same shape. However their<br />
C = G ∝ Q, not necessarily a quotient <strong>of</strong> G ∝ P, is another<br />
transporter category <strong>and</strong> ρ is induced by a G-map P → Q.<br />
This prohibits us from considering various quotients <strong>of</strong> transporter<br />
categories. Moreover since a G-map P → Q usually<br />
does not induce an algebra homomorphism from k(G ∝ P)<br />
to k(G ∝ Q), their Kan extensions cannot be interpreted as<br />
induction <strong>and</strong> coinduction.<br />
The functors ↑ kC<br />
k(G∝P)<br />
<strong>and</strong> ⇑kC<br />
k(G∝P)<br />
admit interesting interpretations<br />
when G ∝ P → C is part <strong>of</strong> an extension (or an<br />
opposite extension) sequence <strong>of</strong> categories. Under the circumstance<br />
↑ kC<br />
k(G∝P)<br />
<strong>and</strong> ⇑kC<br />
k(G∝P)<br />
on certain k(G ∝ P)-modules can<br />
be very well understood. We shall discuss it in Section 5.<br />
6.3 The functor π: group representations via transporter categories<br />
6.3.1 Homology representations via transporter categories<br />
Suppose P is a G-poset <strong>and</strong> π is the natural functor from the<br />
transporter category G ∝ P to G, regarded as a category with<br />
one object •. The naive fibre <strong>of</strong> the functor π, i.e. π −1 (•), is<br />
exactly P. On the other h<strong>and</strong>, the overcategory π/• provides<br />
the (topological) fibre <strong>of</strong> the map Bπ in the sense that the<br />
sequence <strong>of</strong> categories<br />
π/• → G ∝ P → G<br />
corresponds to a fibration after passing to classifying spaces<br />
B(π/•) → B(G ∝ P) → BG.
230 6 Connections with group representations <strong>and</strong> cohomology<br />
It leads to an action <strong>of</strong> G = π 1 (BG) on B(π/•), which is<br />
realized by a G-action on the category π/•, described shortly<br />
after Proposition 6.3.1. We shall see that there exists a functor<br />
π/• → P as well as an inclusion P → π/•, inducing a<br />
category equivalence. Interestingly the former is a G-functor,<br />
while the latter is not.<br />
The objects <strong>of</strong> π/• are <strong>of</strong> the form (x, h), in which x ∈<br />
Ob(G ∝ P) = Ob P <strong>and</strong> h ∈ G. A morphism from (x, h)<br />
to (x ′ , h ′ ) is a morphism (g, gx ≤ x ′ ) ∈ Mor(G ∝ P) such<br />
that h ′ g = h or equivalently g = h ′−1 h. It implies that each<br />
object (x, h) ∼ = (x ′ , h ′ ) if <strong>and</strong> only if x ′ = gx <strong>and</strong> h ′ = hg −1<br />
for some g ∈ G. Indeed the objects isomorphic to (x, h) are<br />
{(gx, hg −1 ) ∣ ∣ g ∈ G}. Particularly (x, h) ∼ = (hx, e) for the<br />
identity e ∈ G.<br />
There is also an undercategory •\π. One can put it in place<br />
<strong>of</strong> the overcategory <strong>and</strong> all our observations stay true. The<br />
objects <strong>of</strong> •\π are <strong>of</strong> the form (h, x), in which x ∈ Ob(G ∝<br />
P) = Ob P <strong>and</strong> h ∈ G. A morphism from (h, x) to (h ′ , x ′ ) is<br />
a morphism (g, gx ≤ x ′ ) ∈ Mor(G ∝ P) such that gh ′ = h.<br />
It implies that each object (h, x) ∼ = (h ′ , x ′ ) if <strong>and</strong> only if<br />
x ′ = gx <strong>and</strong> h ′ = g −1 h for some g ∈ G. Indeed the objects<br />
isomorphic to (h, x) are {(g −1 h, gx) ∣ ∣ g ∈ G}. Particularly<br />
(h, x) ∼ = (e, hx) for the identity e ∈ G.<br />
When P = •, G ∝ • ∼ = G <strong>and</strong> the functor π can be identified<br />
with Id G . Consequently π/• ∼ = Id G /•. The following<br />
result generalizes this special situation <strong>and</strong> gives a characterization<br />
<strong>of</strong> the overcategory <strong>and</strong> undercategory coming from<br />
G ∝ P → G.<br />
Proposition 6.3.1. The category π/• is isomorphic to P ×<br />
(Id G /•), <strong>and</strong> •\π is isomorphic to P×(•\Id G ). Consequently<br />
π/• ∼ = •\π.
6.3 The functor π: group representations via transporter categories 231<br />
Pro<strong>of</strong>. We establish an isomorphism φ : P × (Id G /•) → π/•<br />
as follows. The objects <strong>of</strong> P × (Id G /•) are {(x, (•, g)) ∣ g ∈ G, x ∈ Ob P}. We define φ((x, (•, g))) = (g −1 x, g) ∈<br />
Ob(π/•). For a morphism (x 1 ≤ x 2 , g2 −1 g 1) : (x 1 , (•, g 1 )) →<br />
(x 2 , (•, g 2 )) we put<br />
φ((x 1 ≤ x 2 , g2 −1 g 1)) := (g2 −1 g 1, (g2 −1 g 1)(g1 −1 x 1) ≤ g2 −1 x 2),<br />
a morphism from (g1 −1 x 1, g 1 ) to (g2 −1 x 2, g 2 ). We can write<br />
down its inverse ψ : π/• → P ×(Id G /•), given by ψ((y, h)) :=<br />
(hy, (•, h)). For any morphism in π/•, (h −1<br />
2 h 1, h −1<br />
2 h 1y 1 ≤ y 2 ) :<br />
(y 1 , h 1 ) → (y 2 , h 2 ), we define ψ((h −1<br />
2 h 1, h −1<br />
2 h 1y 1 ≤ y 2 )) :=<br />
(h 1 y 1 ≤ h 2 y 2 , h2 −1 h 1) : (h 1 y 1 , (•, h 1 )) → (h 2 y 2 , (•, h 2 )).<br />
The isomorphism for •\π can be similarly obtained.<br />
In the end, from Example 1.2.7 (1), we have an isomorphism<br />
Id G /• ∼ = •\Id G <strong>and</strong> hence the isomorphism between the over<br />
category <strong>and</strong> undercategory.<br />
⊓⊔<br />
As we have shown in Example 1.2.7 (2) that both •\Id G <strong>and</strong><br />
Id G /• are equivalent to •. For instance, there is the canonical<br />
functor pt : Id G /• → • as well as a functor • → Id G /•, given<br />
by • ↦→ (•, e), where e is the identity <strong>of</strong> G. The reader can<br />
quickly verify that they provide a category equivalence. These<br />
two categories are actually the Cayley graph.<br />
Corollary 6.3.2. There exists a natural embedding P ↩→<br />
π/• (or P ↩→ •\π) making P a skeleton <strong>of</strong> π/• (or •\π).<br />
Consequently LK π m ∼ = lim −→P<br />
m <strong>and</strong> RK π m ∼ = lim ←−P<br />
m for any<br />
m ∈ k(G ∝ P)-mod.<br />
Pro<strong>of</strong>. By the preceding proposition, π/• ∼ = P × Id G /•. Since<br />
we know explicitly the equivalence-inducing functors between<br />
Id G /• <strong>and</strong> •, we can easily translate them for π/•.<br />
The natural functor P → π/•, given by x ↦→ (x, e) <strong>and</strong><br />
x ≤ y ↦→ (e, x ≤ y), is an embedding, sending P to a skeleton
232 6 Connections with group representations <strong>and</strong> cohomology<br />
<strong>of</strong> π/•. It is straightforward to check that there is a natural<br />
surjective functor π/• → P, induced by (x, h) ↦→ hx. For any<br />
morphism in π/•, (h −1<br />
2 h 1, h −1<br />
2 h 1y 1 ≤ y 2 ) : (y 1 , h 1 ) → (y 2 , h 2 ),<br />
we define ψ((h −1<br />
2 h 1, h2 −1 h 1y 1 ≤ y 2 )) := h 1 y 1 ≤ h 2 y 2 .<br />
These two functors provide an equivalence between π/• <strong>and</strong><br />
P. Same equivalence can be established between P <strong>and</strong> •\π.<br />
The existence <strong>of</strong> equivalences between these categories forces<br />
lim m ∼ = lim<br />
−→π/• −→P<br />
m <strong>and</strong> ←−•\π<br />
lim m ∼ = lim ←−P<br />
m.<br />
⊓⊔<br />
Next we shall show that both π/• <strong>and</strong> •\π are G-categories.<br />
Moreover we study whether or not the above functors, inducing<br />
equivalences with P, are compatible with G-action. Before<br />
we do so, we need to specify what we mean by this compatibility.<br />
Definition 6.3.3. Let D, C be two G-categories <strong>and</strong> u : D →<br />
C a functor. We say u is a G-functor if for any g ∈ G <strong>and</strong><br />
x ∈ Ob D we have u(gx) = gu(x), <strong>and</strong> for any α ∈ Mor D,<br />
gu(α) = u(gα).<br />
We first describe the G-actions on Id G /• <strong>and</strong> on •\Id G . They<br />
are given on objects by u · (•, g ′ ) = (•, ug ′ ) <strong>and</strong> u · (g ′′ , •) =<br />
(g ′′ u −1 , •), respectively. On morphisms,<br />
while<br />
u · [(•, g 1 ) g−1 2 g 1<br />
−→(•, g 2 )] = (•, ug 1 ) g−1 2 g 1<br />
−→(•, ug 2 )<br />
u · [(g 1, ′ •) g′ 2 g′ −1<br />
1<br />
−→ (g 2, ′ •)] = (g 1u ′ −1 , •) g′ 2 g′ −1<br />
1<br />
−→ (g 2u ′ −1 , •).<br />
Immediately we see that the surjection Id G /• → • is a G-<br />
functor, <strong>and</strong> by contrast the embedding • ↩→ Id G /• is not a<br />
G-functor, because g• = • ↦→ (•, e) by the embedding while<br />
g(•, e) = (•, g), for any g ∈ G. Hence we do not expect the<br />
embedding P ↩→ π/• to be a G-functor.
6.3 The functor π: group representations via transporter categories 233<br />
Based on the G-actions on Id G /• <strong>and</strong> •\Id G <strong>and</strong> Proposition<br />
6.3.1, we can define G-actions on π/• <strong>and</strong> •\π. For any object<br />
(x, h) ∈ Ob(π/•) <strong>and</strong> u ∈ G, we have u · (x, h) = (x, uh),<br />
<strong>and</strong> for any morphism (g, gx ≤ x ′ ) : (x, h) → (x ′ , h ′ ) we<br />
have u · (g, gx ≤ x ′ ) = (g, gx ≤ x ′ ) : (x, uh) → (x ′ , uh ′ ).<br />
Note that g = h ′−1 h = (uh ′ ) −1 (uh). Similarly For any object<br />
(x, h) ∈ Ob(•\π) <strong>and</strong> u ∈ G, we have u · (x, h) = (x, hu −1 ),<br />
<strong>and</strong> for any morphism (g, gx ≤ x ′ ) : (x, h) → (x ′ , h ′ ) we<br />
have u · (g, gx ≤ x ′ ) = (g, gx ≤ x ′ ) : (x, hu −1 ) → (x ′ , h ′ u −1 ),<br />
with g = h ′ h −1 = (h ′ u −1 )(hu −1 ) −1 . From our constructions <strong>of</strong><br />
G-actions on π/• <strong>and</strong> •\π, we reach the following statements.<br />
Corollary 6.3.4.1. The isomorphisms π/• ∼ = P ×Id G /• <strong>and</strong><br />
•\π ∼ = P × •\Id G are G-isomorphisms.<br />
2. The natural functors π/• ↠ P <strong>and</strong> •\π ↠ P are G-<br />
functors.<br />
As we mentioned earlier, a G-category C gives rise to a complex<br />
<strong>of</strong> kG-modules C ∗ (C, k). Furthermore if u : D → C is<br />
a G-functor between two G-categories, then it induces a G-<br />
simplicial map u ∗ : ND ∗ → NC ∗ , <strong>and</strong> hence a chain map<br />
between complexes <strong>of</strong> kG-modules u : C ∗ (D, k) → C ∗ (C, k).<br />
Example 6.3.5. If H ⊂ K are subgroups <strong>of</strong> G, then there<br />
is a G-functor G/H → G/K. Hence we have a chain map<br />
C ∗ (G/H, k) → C ∗ (G/K, k). Since both complexes concentrate<br />
in degree zero, this chain map consists <strong>of</strong> only one kGmap:<br />
k(G/H) = C 0 (G/H, k) → k(G/K) = C 0 (G/K, k).<br />
Corollary 6.3.6. The G-functors π/• → P <strong>and</strong> •\π → P<br />
induce chain maps between complexes <strong>of</strong> kG-modules. The<br />
complex LK π B∗<br />
G∝P ∼ = C∗ (π/•, k) is a projective resolution<br />
<strong>of</strong> the finite complex <strong>of</strong> kG-modules C ∗ (P).<br />
Pro<strong>of</strong>. Since LK π preserves projectives, LK π B∗<br />
G∝P ∼ = C∗ (π/•, k)<br />
is a complex <strong>of</strong> projective kG-modules. The existing G-
234 6 Connections with group representations <strong>and</strong> cohomology<br />
functor π/• → P gives rise to a chain map <strong>of</strong> complexes<br />
<strong>of</strong> kG-modules LK π B∗<br />
G∝P ∼ = C∗ (π/•, k) → C ∗ (P, k). However<br />
since that G-functor is a category equivalence, it induces<br />
an isomorphism between the homology <strong>of</strong> complexes. ⊓⊔<br />
Remark 6.3.7. Any left kG-module is naturally a right kGmodule.<br />
If we consider C ∗ (P, k) as a complex <strong>of</strong> right kGmodules<br />
through (−) · g := g −1 · (−), then a projective resolution<br />
can be obtained as C ∗ (•\π, k) ∼ = RK π B∗<br />
G in which<br />
we take the right bar resolution <strong>of</strong> k ∈mod-k(G ∝ P). If we<br />
regard the complex <strong>of</strong> right modules C ∗ (•\π, k) as a complex<br />
<strong>of</strong> left modules via g · (−) := (−) · g −1 , then it is isomorphic<br />
to C ∗ (π/•, k). In fact, the isomorphism comes from the<br />
isomorphism <strong>of</strong> categories π/• ∼ = •\π.<br />
It should be useful to underst<strong>and</strong> the complexes C ∗ (π/•, k)<br />
<strong>and</strong> C ∗ (•\π, k).<br />
Example 6.3.8. By Proposition 6.3.1, if P = G/G = •, then<br />
π G/G = Id G <strong>and</strong> π G/G /• = Id G /• is the Cayley graph, giving<br />
rise to the total space EG whose complex is the bar resolution<br />
B G ∗ . More generally for P = G/H, we have an isomorphism<br />
<strong>of</strong> complexes <strong>of</strong> kG-modules C ∗ (π/•) ∼ = B G ∗ ⊗k(G/H), which<br />
provide a projective resolution <strong>of</strong> k(G/H) = C 0 (G/H) =<br />
C ∗ (G/H).<br />
For any small category C, there is an augmentation map<br />
ɛ : C ∗ (C, k) → k, which is zero on positive degrees <strong>and</strong> which<br />
maps every base element in C 0 (C, k) = k Ob C to 1 ∈ k.<br />
Corollary 6.3.9.1. The G-isomorphisms π/• ∼ = P × Id G /•<br />
<strong>and</strong> •\π ∼ = P × •\Id G induce isomorphisms between complexes<br />
<strong>of</strong> kG-modules.<br />
2. We have equivalences <strong>of</strong> complexes <strong>of</strong> kG-modules C ∗ (π/•, k) ≃<br />
C ∗ (P, k) ⊗ C ∗ (Id G /•, k) <strong>and</strong> C ∗ (•\π, k) ≃ C ∗ (P, k) ⊗
6.3 The functor π: group representations via transporter categories 235<br />
C ∗ (•\Id G , k). Furthermore C ∗ (π/•) ≃ C ∗ (P) ⊗ B∗<br />
G <strong>and</strong><br />
C ∗ (•\π) ≃ C ∗ (P) ⊗ B∗ G .<br />
3. The chain map C ∗ (P × Id G /•, k) → C ∗ (Id G /•, k), induced<br />
by P → •, corresponds to the chain map C ∗ (P, k) ⊗<br />
C ∗ (Id G /•, k) → C ∗ (Id G /•, k), induced by the augmentation<br />
map C ∗ (P, k) → k.<br />
Pro<strong>of</strong>. The first statement is a consequence <strong>of</strong> the preceding<br />
corollary.<br />
Since π/• ∼ = P × Id G /•, we have isomorphism C ∗ (π/•, k) ∼ =<br />
C ∗ (P × Id G /•, k). By Theorem 2.2.31, there is a natural<br />
chain homotopy equivalence C ∗ (P × Id G /•, k) → C ∗ (P, k) ⊗<br />
C ∗ (Id G /•, k), called the Alex<strong>and</strong>er-Whitney map. From its<br />
definition, one can see it is a chain map <strong>of</strong> kG-modules.<br />
To see the third statement, we draw a commutative diagram<br />
C ∗ (P × Id G /•, k)<br />
≃ <br />
C ∗ (P, k) ⊗ C ∗ (Id G /•, k)<br />
ɛ⊗1<br />
C ∗ (Id G /•, k) k ⊗ C ∗ (Id G /•, k)<br />
Suppose (x 0 → · · · → x n ) ⊗ (g 0 → · · · → g n ) in a base<br />
element <strong>of</strong> C n (P × Id G /•, k). Then the Alex<strong>and</strong>er-Whitney<br />
map sends it to<br />
n∑<br />
(x 0 → · · · → x i ) ⊗ (g i → · · · → g n ),<br />
i=0<br />
whose image under the right vertical map equals ɛ ⊗ 1(x 0 ⊗<br />
(g 0 → · · · → g n )) = g 0 → · · · → g n . This is exactly the image<br />
<strong>of</strong> (x 0 → · · · → x n ) ⊗ (g 0 → · · · → g n ) under the left vertical<br />
map, which is a projection map.<br />
⊓⊔<br />
The last corollary <strong>of</strong> Proposition 6.3.1 is actually a consequence<br />
<strong>of</strong> Corollary 6.3.2. Since it is closely related to the<br />
second main result in this section, we put it here.
236 6 Connections with group representations <strong>and</strong> cohomology<br />
Corollary 6.3.10. Suppose M ∈ kG-mod <strong>and</strong> n ∈ k(G ∝<br />
P)-mod. Then LK π (κ M ˆ⊗n) ∼ = M⊗LK π n <strong>and</strong> RK π (κ M ˆ⊗n) ∼ =<br />
M ⊗ RK π n as kG-modules. In particular LK π (κ M ) ∼ = M ⊗<br />
LK π k <strong>and</strong> RK π (κ M ) ∼ = M⊗RK π k, where LK π k ∼ = H0(BP, k) ∼ =<br />
H 0 (BP, k) ∼ = RK π k <strong>of</strong> dimension equal to the number <strong>of</strong><br />
connected components <strong>of</strong> P.<br />
Pro<strong>of</strong>. For the left Kan extension we have<br />
LK π (κ M ˆ⊗n) ∼ = lim −→P<br />
(κ M ˆ⊗n) ∼ = M ⊗ lim −→P<br />
n ∼ = M ⊗ LK π n.<br />
The second isomorphism is true because κ M as a kP-module<br />
admits trivial action. The statement for the right Kan extension<br />
is similar.<br />
⊓⊔<br />
The above corollary implies that LK π κ M<br />
∼ = RKπ κ M , suggesting<br />
the existence <strong>of</strong> a transfer map, which we will construct<br />
later on.<br />
The next result is a direct generalization <strong>of</strong> the fact that the<br />
two obvious kG-module structures on P ⊗ M are isomorphic,<br />
for P, M ∈ kG-mod with P projective. It reveals another connection<br />
between representations <strong>of</strong> groups <strong>and</strong> <strong>of</strong> transporter<br />
categories.<br />
Theorem 6.3.11. Let P ∈ k(G ∝ P)-mod be a projective<br />
module <strong>and</strong> κ M = Res π M for some M ∈ kG-mod. Then<br />
P ˆ⊗κ M is a projective k(G ∝ P)-module. Consequently<br />
ˆ⊗κ M → k ˆ⊗κ M = κ M → 0 is a projective resolution.<br />
B G∝P<br />
∗<br />
Pro<strong>of</strong>. We will prove P ˆ⊗κ M<br />
∼ = P⊗M, with k(G ∝ P) acting<br />
on the latter via left multiplication. To this end, we assume<br />
P = kHom G∝P (x, −). The pro<strong>of</strong> is entirely analogues to the<br />
case when P = •, i.e. when G ∝ • = G.<br />
We define a k-linear map ϕ : kHom G∝P (x, −) ⊗ M →<br />
kHom G∝P (x, −) ˆ⊗κ M as follows. On base elements ϕ((g, gx ≤<br />
y) ⊗ m) = (g, gx ≤ y) ⊗ (g, gx ≤ y)m, where the latter m is
6.3 The functor π: group representations via transporter categories 237<br />
considered as an element in κ M (x). For any (h, hy ≤ z), we<br />
readily verify<br />
(h, hy ≤ z)ϕ[(g, gx ≤ y)⊗m] = ϕ[(h, hy ≤ z)((g, gx ≤ y)⊗m)].<br />
Thus ϕ is a homomorphism <strong>of</strong> k(G ∝ P)-modules.<br />
We remind the reader that, following definition, given any<br />
pair <strong>of</strong> x, y ∈ Ob(G ∝ P) with Hom G∝P (x, y) non-empty,<br />
both Aut G∝P (x) <strong>and</strong> Aut G∝P (y) act freely on Hom G∝P (x, y).<br />
This implies that kHom G∝P (x, y) is a free kAut G∝P (x)- or<br />
kAut G∝P (y)-module. Consequently ϕ restricts on each y to<br />
the classical kAut G∝P (y)-isomorphism (see [5, 3.1.5])<br />
kHom G∝P (x, y)⊗M → (kHom G∝P (x, −) ˆ⊗κ M )(y) = kHom G∝P (x, y)⊗<br />
Furthermore because as vector spaces<br />
kHom G∝P (x, −) ˆ⊗κ M =<br />
⊕<br />
kHom G∝P (x, y)⊗M = kHom G∝P (x<br />
y∈Ob(G∝P)<br />
the linear map ϕ is actually one-to-one <strong>and</strong> hence an isomorphism<br />
<strong>of</strong> k(G ∝ P)-modules.<br />
⊓⊔<br />
For an arbitrary category algebra, the tensor product <strong>of</strong> a<br />
projective module with a non-trivial module usually does not<br />
stay projective, as one can find a counter-example in [, 2.5].<br />
The key point here is that the structure <strong>of</strong> G ∝ P allows us<br />
to apply some results on group algebras.<br />
Assume C ∗ is a complex <strong>of</strong> kG-modules. Then we naturally<br />
obtain a complex <strong>of</strong> k(G ∝ P)-modules via restriction, written<br />
as κ C∗ .<br />
Lemma 6.3.12. The functor π : G ∝ P → G induces a<br />
natural chain map Π from B∗ G∝P → k → 0 to κ B<br />
G ∗<br />
→ k → 0.<br />
More generally for any M ∈ kG-mod, it naturally induces<br />
a chain map between exact sequences <strong>of</strong> k(G ∝ P)-modules
238 6 Connections with group representations <strong>and</strong> cohomology<br />
Π M : {B∗ G∝P ˆ⊗κ M → κ M → 0} → {κ B<br />
G ∗<br />
ˆ⊗κ M = κ B<br />
G ∗ ⊗M → κ M → 0}.<br />
Pro<strong>of</strong>. The reason is that π induces a natural map between<br />
the nerves <strong>of</strong> these categories, while the bar resolutions are<br />
constructed from the chain complexes from the nerves. More<br />
explicitly the complexes B∗ G∝P → k → 0 <strong>and</strong> κ B<br />
G ∗<br />
→ k → 0<br />
evaluated at any x ∈ Ob(G ∝ P) are the augmented chain<br />
complexes C ∗ (Id G∝P /x) → k → 0 <strong>and</strong> B∗<br />
G → k → 0, respectively.<br />
The chain map Π is induced by π[(g, ga ≤ b)] = g.<br />
The construction <strong>of</strong> Π M for a fixed M ∈ kG-mod is similar.<br />
⊓⊔<br />
6.3.2 On finite generation <strong>of</strong> cohomology<br />
Subsections 6.4.2 <strong>and</strong> 6.4.3 contains two applications <strong>of</strong> results<br />
from Subsection 6.4.1 to transporter category cohomology<br />
<strong>and</strong> its connection with group cohomology. To study the<br />
functor ρ : G ∝ P → C, we will start a new Section 6.5.<br />
Let G be a finite group <strong>and</strong> P a finite G-poset. Then there<br />
exists a sequence <strong>of</strong> functors<br />
P ι →G ∝ P π →G,<br />
where ι is the natural embedding, <strong>and</strong> whose topological realization<br />
is a fibration<br />
BP<br />
=<br />
Bι <br />
B(G ∝ P) Bπ<br />
≃<br />
<br />
BG<br />
≃<br />
<br />
BP<br />
<br />
EG × G BP<br />
<br />
EG × G •<br />
where • is a point fixed by G. In order to study the finite<br />
generation <strong>of</strong> cohomology rings, we need to recall the<br />
Grothendieck cohomology spectral sequence for a functor<br />
u : D → C
6.3 The functor π: group representations via transporter categories 239<br />
H i (C; H j (\u; N)) ⇒ H i+j (D; N),<br />
where x\u is the undercategory for each x ∈ Ob C <strong>and</strong> N is an<br />
kD-module which can be regarded as a x\u-module through<br />
the forgetful functor x\u → D. Since G only has one object,<br />
the Grothendieck spectral sequence for π : G ∝ P → G reads<br />
as follows<br />
H i (G; H j (•\π; N)) ⇒ H i+j (G ∝ P; N),<br />
or, for being consistent with our Ext notation,<br />
Ext i kG(k, Ext j k(•\π)<br />
(k, N)) ⇒ Exti+j<br />
k(G∝P)<br />
(k, N),<br />
for any N ∈ k(G ∝ P)-mod, in which •\π has its skeleton is<br />
isomorphic to the poset P. By Section 4., the above spectral<br />
sequence is a module over<br />
Ext i kG(k, Ext j k(•\π)<br />
(k, k)) ⇒ Exti+j<br />
k(G∝P)<br />
(k, k).<br />
Meanwhile we have a morphism between the following Grothendieck<br />
spectral sequences, induced by<br />
P<br />
ι <br />
G ∝ P π<br />
π<br />
<br />
G<br />
•<br />
<br />
G ∝ • ∼=<br />
<br />
G,<br />
from which we obtain a diagram <strong>of</strong> spectral sequences<br />
Ext i kG(k, Ext j collapses<br />
<br />
k•<br />
(k, k))<br />
Ext i kG(k, Ext j k(•\π) (k, k)) <br />
Ext i+j<br />
Ext i+j<br />
k(G∝•)<br />
(k, k) .<br />
k(G∝P) (k, k)<br />
More precisely it makes the lower spectral sequence, <strong>and</strong> hence<br />
Ext i kG(k, Ext j k(•\π)<br />
(k, N)) ⇒ Exti+j<br />
k(G∝P)<br />
(k, N),
240 6 Connections with group representations <strong>and</strong> cohomology<br />
modules over Ext ∗ k(G∝•)(k, k) = Ext ∗ kG(k, k). We point out<br />
that the group cohomology ring H ∗ (G, k) ∼ = Ext ∗ kG(k, k) acts<br />
on<br />
H ∗ (G ∝ P; N) ∼ = Ext ∗ k(G∝P)(k, N)<br />
via the algebra homomorphism induced by π (or Bπ),<br />
Ext ∗ kG(k, k) ∼ = H ∗ (G, k) → H ∗ (|G ∝ P|, k) ∼ = H ∗ (G ∝ P; k) ∼ = Ext ∗ k(G<br />
Since P has the property that k is <strong>of</strong> finite projective dimension,<br />
we know H j (•\π; −) ∼ = Ext j k(•\π)<br />
(k, −) vanishes for large<br />
j. Furthermore, the well-known theorem <strong>of</strong> Evens <strong>and</strong> Venkov<br />
says that, for each j, the module Ext ∗ kG(k, Ext j k(•\π)<br />
(k, N)) is<br />
a finitely generated over Ext ∗ kG(k, k). Since E ∞ is a subquotient<br />
<strong>of</strong> E 2 <strong>of</strong> a cohomology spectral sequence, we have the<br />
following statement.<br />
Lemma 6.3.13. For any N ∈ kTr P (G)-mod, Ext ∗ k(G∝P)(k, N)<br />
is a finitely generated Ext ∗ kG(k, k)- <strong>and</strong> Ext ∗ k(G∝P)(k, k)-module.<br />
Theorem 6.3.11 helps us to extend this finite generation<br />
property.<br />
Proposition 6.3.14. For any M ∈ kG-mod <strong>and</strong> n ∈ k(G ∝<br />
P)-mod, Ext ∗ k(G∝P)(κ M , n) is finitely generated over Ext ∗ k(G∝P)(k, k).<br />
Pro<strong>of</strong>. One can easily deduce from Theorem 6.4.11, together<br />
with the internal hom in Section 3.4., an Eckmann-Sharpiro<br />
type isomorphism<br />
Ext ∗ k(G∝P)(κ M , n) ∼ = Ext ∗ k(G∝P)(k, Hom(κ M , n)).<br />
Then we apply the finite generation result that we just quoted.<br />
⊓⊔<br />
The ultimate goal is to establish the finite generation <strong>of</strong><br />
Ext ∗ k(G∝P)(m, n), for any m, n ∈ k(G ∝ P)-mod. However
6.3 The functor π: group representations via transporter categories 241<br />
the existence <strong>of</strong> an internal hom is not enough because, unless<br />
m is induced from a kG-module, Ext ∗ k(G∝P)(m, n) ≁ =<br />
Ext ∗ k(G∝P)(k, Hom(m, n)). We shall establish the finite generation<br />
theorem, Theorem 6.5.3, in the very last section because<br />
we have to rely on the Hochschild cohomology <strong>of</strong> transporter<br />
categories.<br />
Example 6.3.15. Let G = C 2 = {g ∣ ∣ g 2 = e} <strong>and</strong> chark =<br />
2. Then the G-poset S e 2 is {e} → C 2 , <strong>and</strong> the transporter<br />
category C := C 2 ∝ S e 2 is<br />
e <br />
{e}<br />
e<br />
g<br />
g<br />
C 2 .<br />
There are two simple modules S {e} = S {e},k <strong>and</strong> S C2 =<br />
S C2 ,k, both <strong>of</strong> dimension 1. Furthermore there is a short<br />
exact sequence 0 → S C2 → P {e} → S {e} → 0. If we<br />
apply Ext ∗ kC(−, S C2 ) to it, then we get Ext i kC(S {e} , S C2 ) ∼ =<br />
Ext i−1<br />
kC (S {e}, S {e} ) for i > 0 <strong>and</strong> Ext 0 kC(S {e} , S C2 ) = 0. The<br />
cohomology ring is finitely generated because<br />
Ext ∗ kC(k, k) ∼ = Ext ∗ kC(S {e} , S {e} ) ∼ = Ext ∗ kC 2<br />
(k, k) ∼ = k[x]<br />
it is a polynomial ring with one indeterminant x <strong>of</strong> degree 1.<br />
We can see that Ext i kC(S {e} , S C2 ) is <strong>of</strong> dimension 1 for i > 0<br />
<strong>and</strong> Ext ∗ kC(S {e} , S C2 ) is finitely generated over Ext ∗ kC(k, k). By<br />
contrast,<br />
Ext ∗ kC(k, Hom(S {e} , S C2 )) = 0<br />
since Hom(S {e} , S C2 ) = 0 by direct calculation.<br />
Remark 6.3.16. Using Dan Swenson’s definition [72] <strong>of</strong> the<br />
internal hom one verifies that<br />
Consequently,<br />
κ Homk (M,N) ∼ = Hom(κ M , κ N ).<br />
e
242 6 Connections with group representations <strong>and</strong> cohomology<br />
Ext ∗ k(G∝P)(k, κ Homk (M,N)) ∼ = Ext ∗ k(G∝P)(k, Hom(κ M , κ N )) ∼ = Ext ∗ k(G∝P)<br />
Before moving to the next section, we record a connection<br />
between cohomology <strong>of</strong> transporter categories <strong>and</strong> equivariant<br />
cohomology, which is perhaps known to the experts.<br />
Proposition 6.3.17. The left Kan extension induces an<br />
isomorphism<br />
λ M : Ext ∗ k(G∝P)(k, κ M ) ∼ = H ∗ G(BP, M),<br />
where H ∗ G (BP, M) is the equivariant cohomology group for<br />
some M ∈ kG-mod.<br />
Pro<strong>of</strong>. Take the bar resolution B∗<br />
G∝P → k → 0 <strong>and</strong> consider<br />
the complex Hom k(G∝P) (B∗<br />
G∝P , κ M ). The left Kan extension<br />
induces a chain map<br />
Hom k(G∝P) (B∗<br />
G∝P , κ M ) ∼ = Hom kG (LK π B∗ G∝P , M) ∼ = Hom kG (B∗ G ⊗C ∗ (P<br />
But the rightmost term is<br />
Hom kG (B G ∗ ⊗ C ∗ (P), M) ∼ = Hom kG (B G ∗ , Hom k (C ∗ (P), M)),<br />
which gives rise to H ∗ G (BP, M).<br />
⊓⊔<br />
In light <strong>of</strong> the above proposition, we may introduce the Tate<br />
cohomology <strong>of</strong> transporter categories as Tate equivariant cohomology.<br />
With Remark 3 in mind, one can further define<br />
negative degree Ext groups Ext ∗ k(G∝P)(κ M , κ N ).<br />
6.3.3 Transfer for ordinary cohomology<br />
Let u : D → C be a functor between small categories. There<br />
is always a restriction res u : H ∗ (C; k) → H ∗ (D; k). However<br />
usually one cannot construct a map in the opposite direction,<br />
unless the two Kan extensions are connected by a natural<br />
transformation. In this section, based on our knowledge about
6.3 The functor π: group representations via transporter categories 243<br />
representations <strong>of</strong> k(G ∝ P), we establish various transfer<br />
maps, including the Becker-Gottlieb transfer map with respect<br />
to π P : G ∝ P → G. Here we provide an algebraic<br />
alternative to the construction <strong>of</strong> Becker-Gottlieb [4], <strong>and</strong> the<br />
core idea is taken from Dwyer-Wilkerson [21, 9.13]). Essentially<br />
our construction incorporates [21, 9.13] in an entirely<br />
representation-theoretic setting. The upshot is that our construction<br />
is analogues to the classical situation, see for instance<br />
[5, 3.6.17]. Keep in mind that Res π <strong>and</strong> LK π generalize<br />
↓ G H <strong>and</strong> ↑G H , respectively, used in group cohomology.<br />
In Corollary 6.3.9 (3), we described a chain map<br />
LK π B∗<br />
G∝P ∼ = C∗ (π/•, k) ∼ = C ∗ (P×Id G /•, k) −→ C ∗ (Id G /•, k) ∼ = B∗ G ,<br />
induced by P → •. This chain map gives rise to a cochain<br />
map<br />
Hom kG (B∗ G , k) → Hom kG (LK π B∗<br />
G∝P , k) ∼ = Hom k(G∝P) (B∗ G∝P , k)<br />
<strong>and</strong> hence the restriction<br />
res π : Ext ∗ kG(k, k) → Ext ∗ k(G∝P)(k, k).<br />
We want to establish a map, called transfer, on the opposite<br />
direction<br />
Ext ∗ k(G∝P)(k, k) → Ext ∗ kG(k, k).<br />
Example 6.3.18. Before we construct the transfer, we examine<br />
the special case for group cohomology. Suppose H ⊂ G is<br />
a subgroup. We want to define a map tr G H : H ∗ (H, k) →<br />
H ∗ (G, k).<br />
For the functor π G/H : G ∝ (G/H) → G. We get<br />
LK πG/H B G ∗ = C ∗ (π G/H /•, k) ∼ = C ∗ ((G/H) × Id G /•, k)<br />
as complexes <strong>of</strong> kG-modules. The rightmost admits a map<br />
to C ∗ (Id G /•, k), induced by the G-functor pt : G/H → •. If
244 6 Connections with group representations <strong>and</strong> cohomology<br />
we invoke C ∗ ((G/H) × Id G /•, k) ≃ k(G/H) ⊗ B G ∗ , then this<br />
chain map is identified with the chain map k(G/H) ⊗ B G ∗ →<br />
C ∗ (Id G /•, k), induced by the augmentation<br />
ɛ : k(G/H) = k ↑ G H→ k.<br />
By applying Hom kG (−, k), this chain map induces the restriction<br />
res πG/H : H ∗ (G, k) → H ∗ (G ∝ (G/H), k) ∼ = H ∗ (H, k).<br />
To produce another map in the opposite direction, we notice<br />
that although there is no G-functor • → G/H (unless G =<br />
H), we can always build a kG-map after linearization<br />
It induces a chain map<br />
k = k• → k(G/H) = k ↑ G H .<br />
B G ∗ → k ↑ G H ⊗B G ∗ .<br />
When we apply Hom kG (−, k) to the following <strong>and</strong> take cohomology<br />
B∗ G → k ↑ G H ⊗B∗<br />
G ɛ⊗1<br />
−→B∗ G ,<br />
since<br />
k ↑ G H ⊗B∗ G ∼ = (k ⊗ B∗ G ↓ G H) ↑ G H= B∗ G ↓ G H↑ G H,<br />
we get Hom kG (B∗ G ↓ G H ↑G H , k) ∼ = Hom kH (B∗ G ↓ G H , k) <strong>and</strong> consequently<br />
two induced maps<br />
H ∗ (G, k) trG H<br />
←−H ∗ (H, k) resG H<br />
←−H ∗ (G, k).<br />
Since the composite k → k(G/H) = k ↑ G H→ k equals a scalar<br />
multiplication by [G : H], we know tr G H ◦ resG H = [G : H] =<br />
|G/H|. Here the restriction res G H is identified with res π G/H<br />
upon<br />
the isomorphism H ∗ (G ∝ (G/H), k) ∼ = H ∗ (H, k).
6.3 The functor π: group representations via transporter categories 245<br />
Since for arbitrary G ∝ P, we have constructed the restriction<br />
res π : H ∗ (G, k) → H ∗ (G ∝ P, k).<br />
Now we would like to have a generalized transfer map in the<br />
opposite direction. As in the group case, it should be induced<br />
by some B∗ G → LK π B∗<br />
G∝P , or rather<br />
C ∗ (Id G /•, k) → C ∗ (π/•, k) ∼ = C ∗ (P×Id G /•, k) ≃ C ∗ (P, k)⊗C ∗ (Id G /<br />
Indeed, it should be reduced to establishing a map<br />
k = k• = C ∗ (•, k) → C ∗ (P, k).<br />
Note that, although there are many functors from • → P, all<br />
<strong>of</strong> them have the problem that they are not G-functors. Thus<br />
they will not directly produce chain maps between the above<br />
two complexes <strong>of</strong> kG-modules. Now we do have to work on the<br />
chain level. If such a chain map exists, then we immediately<br />
obtain a chain map<br />
C ∗ (•, k)⊗C ∗ (Id G /•, k) → C ∗ (P, k)⊗C ∗ (Id G /•, k) ≃ C ∗ (P×Id G /•, k)<br />
As the first attempt, we record the following observation.<br />
Lemma 6.3.19. For any n ≥ 0 <strong>and</strong> any x 0 → · · · →<br />
x n ∈ NP n , one can construct a chain map c i : C ∗ (•, k) →<br />
C ∗ (P, k), for each 0 ≤ i ≤ n.<br />
Pro<strong>of</strong>. Since C ∗ (•, k) = k, we define the chain maps as<br />
C 0 (•, k) → C 0 (P, k), 1 ↦→ ∑ I<br />
gx i ,<br />
Here I is the G-orbit <strong>of</strong> x i .<br />
Obviously various linear combinations <strong>of</strong> the above maps<br />
are still kG-maps.<br />
⊓⊔
246 6 Connections with group representations <strong>and</strong> cohomology<br />
Example 6.3.20. Suppose x ∈ NP 0 . Then C 0 (•, k) maps isomorphically<br />
to a 1-dimensional submodule <strong>of</strong> C 0 (P, k), that<br />
is, kO + (x) for O + (x) = ∑ y∈O(x)<br />
y where O(x) is the G-orbit<br />
<strong>of</strong> x in G ∝ P. Hence we get<br />
C ∗ (•, k)<br />
<br />
kO + (x) ɛ <br />
<br />
C ∗ (•, k)<br />
C ∗ (•, k)<br />
<br />
kO(x)<br />
<br />
<br />
C ∗ (•, k)<br />
C ∗ (•, k)<br />
<br />
C ∗ (P, k) ɛ<br />
<br />
C ∗ (•, k)<br />
Here kO(x) is isomorphic to k(G/Stab G (x)) ∼ = k ↑ G Stab G (x) .<br />
Tensoring with C ∗ (Id G /•, k) we get<br />
C ∗ (Id G /•, k)<br />
∼ =<br />
<br />
kO + (x) ⊗ C ∗ (Id G /•, k) ɛ<br />
C ∗ (Id G /•, k)<br />
<br />
kO(x) ⊗ C ∗ (Id G /•, k) ɛ <br />
<br />
<br />
C ∗ (Id G /•, k)<br />
C ∗ (Id G /•, k)<br />
C ∗ (Id G /•, k)<br />
<br />
C ∗ (P, k) ⊗ C ∗ (Id G /•, k)<br />
<br />
ɛ C ∗ (Id G /•, k)<br />
In the central column, the top one is isomorphic to C ∗ (Id G /•, k),<br />
<strong>and</strong> the middle is isomorphic to C ∗ (π O(x) /•, k). Thus by applying<br />
Hom kG (−, k) <strong>and</strong> taking cohomology we a commutative<br />
diagram<br />
H ∗ (G, k)<br />
<br />
H ∗ (G, k) H ∗ (G, k)<br />
∼ =<br />
H ∗ (G, k) H ∗ (G ∝ O(x), k)<br />
tr O(x)<br />
tr<br />
res<br />
H ∗ (G, k) H ∗ (G ∝ P, k)<br />
∼ =<br />
res O(x)<br />
res π<br />
<br />
H ∗ (G, k)<br />
H ∗ (G, k)
6.3 The functor π: group representations via transporter categories 247<br />
The transfer map is the usual one for group cohomology, see<br />
Example 6.3.17. The restrictions are induced by the functors<br />
G ∝ O(x) ↩→ G ∝ P → G. The above diagram reduces to a<br />
sequence <strong>of</strong> maps<br />
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (G ∝ O(x), k) → H ∗ (G, k),<br />
or equally<br />
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (Stab G (x), k) → H ∗ (G, k).<br />
We can actually take any G-set sitting inside P. The natural<br />
c<strong>and</strong>idates are all the sets NP n , n ≥ 0. For instance if α =<br />
x 0 < · · · < x n ∈ NP n <strong>and</strong> O(α) denotes the G-orbit <strong>of</strong> α,<br />
then we will get a sequence <strong>of</strong> maps<br />
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (G ∝ O(α), k) → H ∗ (G, k),<br />
or equally<br />
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (Stab G (α), k) → H ∗ (G, k).<br />
Note that Stab G (α) = ⋂ i Stab G(x i ).<br />
The preceding example implies that using naively constructed<br />
chain maps C ∗ (•, k) → C ∗ (P, k) it is unlikely to<br />
obtain a novel H ∗ (G ∝ P, k) → H ∗ (G, k) which does not depend<br />
on the transfer in group cohomology H ∗ (Stab G (α), k) →<br />
H ∗ (G, k). Thus we must try something more complicated.<br />
In the pro<strong>of</strong> <strong>of</strong> the next theorem, we construct the Becker-<br />
Gottlieb transfer, also seen in Dwyer-Wilkerson.<br />
It seems like many important theorems in group cohomology<br />
share the same nature <strong>and</strong> live in the same context. Using<br />
the double complex Hom kG (C † ∗(P) ⊗ B G ∗ , k) to obtain Webb’s<br />
Theorem, Dwyer’s sharp decompositions results
248 6 Connections with group representations <strong>and</strong> cohomology<br />
When P = S p , by Brown’s theorem, χ(P) = 1 in k. By<br />
Quillen’s() Theorem res π : H ∗ (G, k) → H ∗ (G ∝ P, k) is an<br />
isomorphism. What about A p etc<br />
Theorem 6.3.21. Suppose Res π : kG-mod → k(G ∝ P)-<br />
mod is the restriction along π <strong>and</strong> write κ M = Res π M for<br />
any M ∈ kG-mod. Then we have the following two maps,<br />
restriction <strong>and</strong> transfer,<br />
Ext ∗ kG(M, N) res P<br />
−→Ext ∗ k(G∝P)(κ M , κ N ) tr P<br />
−→Ext ∗ kG(M, N),<br />
which compose to χ(P; k) · 1, multiplication by the Euler<br />
characteristic <strong>of</strong> (the order complex <strong>of</strong>) P.<br />
Pro<strong>of</strong>. We shall construct these two maps. Then in the sequel<br />
we can deduce the statement on their composite. The restriction<br />
is a generalized version <strong>of</strong> the one shown at the beginning<br />
<strong>of</strong> this section.<br />
Suppose B∗<br />
G <strong>and</strong> B∗<br />
G∝P are the bar resolutions <strong>of</strong> k ∈ kGmod<br />
<strong>and</strong> k ∈ k(G ∝ P)-mod, respectively. Then B∗<br />
G ⊗ M<br />
is a projective resolution <strong>of</strong> a fixed M ∈ kG-mod. For<br />
another kG-module N, the cochain complex Hom kG (B∗<br />
G ⊗<br />
M, N) computes Ext ∗ kG(M, N). The exact functor Res π sends<br />
this cochain complex to Hom k(G∝P) (κ B<br />
G ∗ ⊗M, κ N ). Applying<br />
Lemma 6.3.12 we obtain the composite <strong>of</strong> two chain maps,<br />
unique up to chain homotopy,<br />
Hom kG (B∗ G ⊗M, N) Res π<br />
→ Hom k(G∝P) (κ B<br />
G ∗ ⊗M, κ N ) Π∗ M<br />
→Hom k(G∝P) (B∗<br />
G∝P<br />
These seemingly abstract maps can be written down explicitly,<br />
but we will leave it to the interested reader. The composite<br />
ΠM ∗ ◦ Res π induces a map on cohomology, which we call<br />
the restriction,<br />
res P : Ext ∗ kG(M, N) → Ext ∗ k(G∝P)(κ M , κ N ).<br />
ˆ⊗κ
6.3 The functor π: group representations via transporter categories 249<br />
It is helpful to have a different characterization <strong>of</strong> the<br />
restriction. In order to do so we use a series <strong>of</strong> obvious<br />
isomorphisms to rewrite the previously mentioned complex<br />
Hom k(G∝P) (B∗<br />
G∝P ˆ⊗κ M , κ N ). Firstly by adjunction, it is isomorphic<br />
to<br />
Hom kG (LK π (B∗<br />
G∝P ˆ⊗κ M ), N).<br />
Here (ΠM ∗ ◦ Res π)α is mapped to (Λ N ◦ LK π )(ΠM ∗ ◦ Res πα),<br />
where Λ N : LK π κ N → N, determined by the counit <strong>of</strong><br />
adjunction, is isomorphic to ¯ɛ ⊗ Id N . The natural map ¯ɛ :<br />
H0(P) → k is induced by the augmentation map ɛ : C 0 (P) →<br />
k (or rather P → •), by Corollary 6.3.10. Secondly from the<br />
same corollary our cochain complex is canonically isomorphic<br />
to<br />
Hom kG (LK π (B∗<br />
G∝P ) ⊗ M, N).<br />
Thirdly by Corollary 6.3.9 LK π B∗ G∝P ≃ B∗ G ⊗ C ∗ (P), the<br />
above complex is<br />
Hom kG (LK π (B G∝P<br />
∗ ) ⊗ M, N) ≃ Hom kG (B G ∗ ⊗ C ∗ (P) ⊗ M, N)<br />
∼ = HomkG (B G ∗ ⊗ M ⊗ C ∗ (P), N).<br />
The observations imply that res P : Ext ∗ kG(M, N) → Ext ∗ k(G∝P)(κ M , κ N<br />
is the same as the map induced by the following chain map<br />
Hom kG (B G ∗ ⊗ M, N) → Hom kG (B G ∗ ⊗ M ⊗ C ∗ (P), N),<br />
which is given by the augmentation ɛ : C ∗ (P) → k. Now we<br />
are ready to build the transfer map. From the cochain complex<br />
Hom kG (B G ∗ ⊗ M ⊗ C ∗ (P), N) we continue to establish a map<br />
which leads back to group cohomology<br />
Θ ∗ : Hom kG (B G ∗ ⊗ M ⊗ C ∗ (P), N) → Hom kG (B G ∗ ⊗ M, N).<br />
The map Θ ∗ is induce by the following chain map Θ : k →<br />
C ∗ (P) <strong>of</strong> Dwyer-Wilkerson [21, 9.13], as the composite <strong>of</strong>
250 6 Connections with group representations <strong>and</strong> cohomology<br />
k a↦→a·Id<br />
→ Hom k (C ∗ (P), C ∗ (P)) ∼= →C ∗ (P) ∧ ⊗C ∗ (P) Id⊗∆<br />
→ C ∗ (P) ∧ ⊗C ∗ (P)⊗C<br />
Here C ∗ (P) ∧ = Hom k (C ∗ (P), k), non-positively graded, is the<br />
k-dual <strong>of</strong> C ∗ (P). The composite <strong>of</strong><br />
Hom k(G∝P) (B∗<br />
G∝P<br />
defines a map<br />
ˆ⊗κ M , κ N ) ∼= →Hom kG (LK π (B G∝P<br />
∗<br />
tr P : Ext ∗ k(G∝P)(κ M , κ N ) → Ext ∗ kG(M, N).<br />
ˆ⊗κ M ), N) →Hom Θ∗<br />
kG<br />
which is called the transfer.<br />
In fact we have the following commutative diagram <strong>of</strong><br />
cochain complexes<br />
Hom kG (B∗ G ⊗ M, N) Hom k(G∝P) (B∗ G∝P ˆ⊗κ M , κ N )<br />
<br />
Hom kG (B∗ G ⊗ M<br />
= <br />
≃ <br />
=<br />
<br />
Hom kG (B∗ G ⊗ M, N) Hom kG (B∗ G ⊗ M ⊗ C ∗ (P), N) Hom kG (B∗ G ⊗ M<br />
Upon passing to cohomology, both rows give rise to<br />
Ext ∗ kG(M, N) res P<br />
→Ext ∗ k(G∝P)(κ M , κ N ) tr P<br />
→Ext ∗ kG(M, N).<br />
In the end we prove that the composite res P ◦tr P = χ(P)·1.<br />
Since the normalization C ∗ (P, k) → C † ∗(P, k) is a G-chain<br />
homotopy equivalence, we can replace C ∗ (P, k) by C † ∗(P, k) in<br />
our calculation. The normalized chain complex is finite so we<br />
can find an integer d such that C † d (P, k) ≠ 0 but C† n(P, k) = 0<br />
for all n > d. The following pro<strong>of</strong> is due to Dwyer-Wilkerson<br />
[21, 9.13] too, which shows by direct calculation that<br />
k → C † ∗(P) → k<br />
is a scalar multiplication by χ(P). Write the natural basis <strong>of</strong><br />
C † n(P) as {c i n} d n<br />
i=1 for d n = dim k C † n(P) (see Section 6.2.2).<br />
The step-by-step images <strong>of</strong> 1 ∈ k under Θ are
6.3 The functor π: group representations via transporter categories 251<br />
1 ↦→ Id C<br />
†<br />
∗ (P)<br />
↦→ ∑ d<br />
n=0 (−1)n { ∑ d n<br />
i=1 (ci n) ∧ ⊗ c i n}<br />
↦→ ∑ d<br />
n=0 (−1)n { ∑ d n<br />
= ∑ d<br />
n=0 (−1)n { ∑ d n<br />
↦→ ∑ d<br />
n=0 (−1)n { ∑ d n<br />
↦→ ∑ d<br />
n=0 (−1)n d n<br />
= χ(P).<br />
i=0 (ci n) ∧ ⊗ [c i n ⊗ t(c i n)]}<br />
i=1 [(ci n) ∧ ⊗ c i n] ⊗ t(c i n)}<br />
i=1 t(ci n)}<br />
Here t(c i n) ∈ C † 0 (P) denotes the last object, i.e. the target, <strong>of</strong><br />
the n-chain <strong>of</strong> morphisms c i n ∈ C † n(P).<br />
⊓⊔<br />
Remark 6.3.22. In fact, by Remark 6.3.15, the above restriction<br />
<strong>and</strong> transfer coincide with<br />
Ext ∗ kG(k, Hom k (M, N)) res P<br />
→Ext ∗ k(G∝P)(k, κ Homk (M,N)) tr P<br />
−→Ext ∗ kG(k, Hom<br />
When M = N = k, our construction is exactly the Becker-<br />
Gottlieb transfer ([4],[21]), because Ext ∗ k(G∝P)(k, κ M ) ∼ = H ∗ G (BP, M).<br />
We emphasize that if either M ∈ kG-mod is not acted trivially<br />
by kG or if G sends a connected component <strong>of</strong> P to a different<br />
one, then the constantly valued κ M ∈ k(G ∝ P)-mod<br />
is not truly constant since H 0 (G ∝ P; κ M ) ∼ = lim ←−G∝P<br />
κ M<br />
∼ =<br />
lim lim κ ∼ ←−G ←−P<br />
M = (M ⊗ H 0 (P)) G .<br />
Corollary 6.3.23. If χ(P; k) is invertible in k, then res P is<br />
an injective homomorphism.<br />
According to Dwyer, a collection <strong>of</strong> subgroups <strong>of</strong> G is a<br />
set <strong>of</strong> subgroups that is closed under conjugation. If P is a<br />
collection then it is naturally a G-poset in which the relations<br />
are inclusions <strong>and</strong> G-acts by conjugation. A collection is called<br />
ample if for a fixed prime p <strong>and</strong> a field k <strong>of</strong> characteristic p<br />
the restriction res P : Ext ∗ k(G∝P)(k, k) → Ext ∗ kG(k, k) is an
252 6 Connections with group representations <strong>and</strong> cohomology<br />
isomorphism. There is an extensive discussion on such posets<br />
in [17] or [18]. When P is ample, we get tr P = χ(P)res −1<br />
P .<br />
6.4 The functor ρ: invariants <strong>and</strong> coinvariants<br />
We have exploited the functor π : G ∝ P → G. Now we<br />
turn to study the other functor ρ : G ∝ P → C, where<br />
C is a quotient category <strong>of</strong> G ∝ P. In general situation it<br />
seems hard to make group theoretic interpretation <strong>of</strong> ↑ kC<br />
k(G∝P)<br />
<strong>and</strong> ⇑ kC<br />
k(G∝P)<br />
. However we can do so when we have certain<br />
quotient categories, which are part <strong>of</strong> some category extension<br />
sequences in the sense <strong>of</strong> H<strong>of</strong>f.<br />
An extension E <strong>of</strong> a category C via a category K is a sequence<br />
<strong>of</strong> functors<br />
K ι<br />
−→E ρ<br />
−→C,<br />
which satisfies certain properties. A sequence K ↩→ E ↠ C<br />
is called an opposite extension if K op ↩→ E op ↠ C op is an<br />
extension.<br />
The advantage <strong>of</strong> considering π : G ∝ P → C which is part<br />
<strong>of</strong> an extension (or opposite extension) is that it enables us<br />
to provide a good characterization <strong>of</strong> the left (or right) Kan<br />
extension. Indeed it is the case for many familiar category<br />
constructions in representation theory <strong>and</strong> homotopy theory.<br />
Remark 6.4.1. We emphasize that for any quotient category<br />
C <strong>of</strong> G ∝ P, P is naturally a subcategory <strong>of</strong> C. Thus for<br />
any kC-module, it makes sense to consider its limits. Indeed<br />
if m ∈ kC-mod then both ←−P<br />
lim m ∼ = lim ←−P<br />
Res ρ m <strong>and</strong> −→P<br />
lim m ∼ =<br />
lim Res −→P ρm are kG-modules.<br />
Lemma 6.4.2. Let K → E → C a sequence <strong>of</strong> three EIcategories<br />
<strong>and</strong> m ∈ kE-mod.
6.4 The functor ρ: invariants <strong>and</strong> coinvariants 253<br />
1. Suppose K → E → C is an extension. Then LK ρ m ∼ = m K ,<br />
where m K as a functor over C is given by m K (x) = m(x) K(x)<br />
(K(x)-coinvariants <strong>of</strong> the kAut E (x)-module m(x)), for any<br />
x ∈ Ob C = Ob E = Ob K.<br />
2. Suppose K → E → C is an opposite extension. Then<br />
RK ρ m ∼ = m K , where m K as a functor over C is given<br />
by m K (x) = m(x) K(x) (K(x)-invariants <strong>of</strong> the kAut E (x)-<br />
module m(x)), for any x ∈ Ob C = Ob E = Ob K.<br />
In the above lemma, there is another way to express the<br />
Kan extensions. Under the same assumptions, they are m K =<br />
H0(K; m) <strong>and</strong> m K = H 0 (K; m) respectively.<br />
In what follows, we shall apply the above statements to various<br />
local categories <strong>of</strong> G, in combination with the Frobenius<br />
reciprocity<br />
(i’) Hom kG (M, lim ←−P<br />
n) ∼ = Hom kC (LK ρ κ M , n);<br />
(ii’) Hom kG (lim −→P<br />
m, N) ∼ = Hom kC (m, LK ρ κ N ).<br />
6.4.1 Orbit categories<br />
Suppose P is a collection <strong>of</strong> subgroups <strong>of</strong> G on which G acts<br />
by conjugation. Then it forms a G-poset <strong>and</strong> we can define<br />
an orbit category O P as the quotient category <strong>of</strong> G ∝ P by<br />
asking<br />
Hom OP (P, Q) = Q\N G (P, Q).<br />
Then we have an extension sequence<br />
S ↩→ G ∝ P ↠ O P ,<br />
where S is the disjoint union <strong>of</strong> all objects in P, regarded as<br />
a subcategory <strong>of</strong> G ∝ P.<br />
When m = κ M for some M ∈ kG-mod, (LK ρ κ M )(P ) =<br />
M P for any P ∈ Ob P. We denote such a kO P -module by
254 6 Connections with group representations <strong>and</strong> cohomology<br />
M S := (κ M ) S = LK ρ κ M . Since giving a morphism (g, g P ≤<br />
Q) is the same as giving a group homomorphism P → Q, the<br />
conjugation induced by g, there is a natural way to construct<br />
a map M P → M Q , identical to the natural map H0(P ; M) =<br />
k ⊗ kP M → k ⊗ kQ M = H0(Q; M). Hence we know how kO P<br />
acts on M S .<br />
Proposition 6.4.3. Let M ∈ kG-mod <strong>and</strong> n ∈ kO P -mod.<br />
Then<br />
Hom kG (M, lim ←−P<br />
n) ∼ = Hom kOP (M S , n),<br />
where M S is as above.<br />
Corollary 6.4.4. lim ←−P<br />
n ∼ = Hom kOP (kG S , n) <strong>and</strong> Hom kG (k, lim ←−P<br />
n) ∼ =<br />
Hom kOP (k, n).<br />
As an example we let H be a subgroup <strong>of</strong> G <strong>and</strong> P the<br />
subgroups that are conjugate to H. The size <strong>of</strong> the discrete<br />
poset P is G/N G (H). Note that both G ∝ P <strong>and</strong> O P are<br />
connected groupoids, the former equivalent to N G (H) <strong>and</strong><br />
the latter N G (H)/H. Thus the above isomorphism can be<br />
interpreted as<br />
Hom kG (M, Hom kNG (H)(kG, N)) ∼ = Hom kNG (H)(M, N)<br />
∼ = Homk(NG (H)/H)(k(N G (H)/H) ⊗ kNG<br />
∼=<br />
Homk(NG (H)/H)(M H , N),<br />
where M ∈ kG-mod <strong>and</strong> N ∈ k(N G (H)/H)-mod.<br />
By looking at the special case we have just mentioned, the<br />
following statement implies that Proposition 5.2 may be useful<br />
in a greater generality.<br />
Lemma 6.4.5. Suppose M is an indecomposable kG-module.<br />
Let P be a p-subgroup <strong>and</strong> N a projective simple k(N G (P )/P )-<br />
module. Then Hom kNG (P )/P(M P , N) ≠ 0 if <strong>and</strong> only if there
6.4 The functor ρ: invariants <strong>and</strong> coinvariants 255<br />
exists a surjective map f : M → N ↑ G N G (P ). In this case<br />
N ↑ G N G (P ) is indecomposable <strong>and</strong> f(M) ∼ = k ↑ G P .<br />
Pro<strong>of</strong>. If Hom kNG (P )/P(M P , N) ≠ 0, then N has to be a direct<br />
summ<strong>and</strong> <strong>of</strong> M P because N is projective simple. By adjunction<br />
Hom kG (M, N ↑ G N G (P ) ) ∼ = Hom kG (M, Hom kNG (P )(kG, N)) ∼ = Hom k(NG (P<br />
we know there is a non-trivial map f : M → N ↑ G N G (P ). If we<br />
restrict this map back to N G (P ), the right side is a semisimple<br />
module <strong>and</strong> thus the image <strong>of</strong> M ↓ NG (P ) contains at least a<br />
copy <strong>of</strong> g ⊗ N for some g ∈ G. It forces the map f : M ↠<br />
N ↑ G N G (P ) to be surjective which implies that N ↑G N G (P ) = f(M)<br />
is indecomposable. Furthermore since as a kN G (P )-module<br />
N has P as a vertex, we get f(M) ∣ (N ↓ P ↑ N G(P ) ) ↑ G . But<br />
N ↓ N G(P )<br />
P<br />
is a direct sum <strong>of</strong> trivial modules. Simultaneously<br />
we obtain f(M) ∼ = k ↑ G P .<br />
The converse is straightforward by the adjunction. ⊓⊔<br />
6.4.2 Brauer categories, fusion <strong>and</strong> linking systems<br />
Suppose b is a p-block <strong>of</strong> the group algebra kG <strong>and</strong> P b is the<br />
set <strong>of</strong> b-Brauer pairs. Then for any G-subposet P ⊂ P b we can<br />
introduce the Brauer category B P as the quotient category <strong>of</strong><br />
G ∝ P such that<br />
Hom BP (P, Q) = Hom G∝P (P, Q)/C G (P ).<br />
This gives us an opposite extension, which means that the<br />
following sequence<br />
C G ↩→ (G ∝ P) op → B op<br />
P
256 6 Connections with group representations <strong>and</strong> cohomology<br />
is an extension sequence, given that C G is the disjoint union<br />
<strong>of</strong> all C G (P ), P ∈ Ob P. Dual to the extension situation we<br />
examined before, now we are able to describe the right Kan<br />
extension <strong>of</strong> modules.<br />
If m = κ M for some M ∈ kG-mod, we denote by M C G the<br />
kF P -module RK ρ κ M . Since a morphism (g, g P ≤ Q) provides<br />
a group homomorphism P → Q <strong>and</strong> thus induces an injection<br />
c g<br />
−1 : C G (Q) → C G (P ), we obtain an injection M C G(P ) →<br />
M CG(Q) . This leads to the kB P -action on M C G .<br />
Proposition 6.4.6. Let m ∈ kB P -mod <strong>and</strong> N ∈ kG-mod.<br />
Then<br />
Hom kG (lim −→P<br />
m, N) ∼ = Hom kBP (m, N C G<br />
),<br />
where N C G<br />
is as above.<br />
As an example we assume b is the principal block b 0 <strong>and</strong><br />
P is the conjugacy class <strong>of</strong> a fixed p-subgroup H. Then the<br />
discrete poset P has |G/N G (H)| objects. Both G ∝ P <strong>and</strong><br />
B P are connected groupoids, the former equivalent to N G (H)<br />
<strong>and</strong> the latter N G (H)/C G (H). Thus the above isomorphism<br />
can be interpreted as<br />
Hom kG (kG ⊗ kNG (H) M, N) ∼ = Hom kNG (H)(M, N)<br />
∼ = Homk(NG (H)/C G (H))(M, Hom kNG (H)(k(N G<br />
∼ = Homk(NG (H)/C G (H))(M, N C G(H) ),<br />
where N ∈ kG-mod <strong>and</strong> M ∈ k(N G (H)/C G (H))-mod.<br />
Corollary 6.4.7. We have Hom kG (lim −→P<br />
m, k) ∼ = Hom kBP (m, k)<br />
for any m ∈ kB P -mod.<br />
Suppose b is a nilpotent block. Then, for every b-Brauer pair<br />
(H, e), N G (H, e)/C G (H) is a p-group.<br />
Let B b = B Pb . If we fix a maximal object (S, e S ) <strong>and</strong> take<br />
all objects (Q, e Q ) with Q ⊂ S, then the full subcategory <strong>of</strong>
6.4 The functor ρ: invariants <strong>and</strong> coinvariants 257<br />
B b , consisting <strong>of</strong> all these objects, is a fusion system, usually<br />
written as F b or F S . Note that the inclusion F b ⊂ B b is an<br />
equivalence. There is a general theory <strong>of</strong> p-local finite groups<br />
introduced by Broto, Levi <strong>and</strong> Oliver. A p-local finite group<br />
consists <strong>of</strong> three categories (S, F, L), where S is a p-group, F<br />
is an (abstract) fusion system–a finite category whose objects<br />
are subgroups <strong>of</strong> S. Let us take the full subcategory F c ⊂ F<br />
consisting <strong>of</strong> F-centric subgroups <strong>of</strong> S. (If F = F b , the F-<br />
centric subgroups correspond to the so-called self-centralizing<br />
b-Brauer pairs in modular representation theory.) A centric<br />
linking system L c , if it exists, situates in the middle <strong>of</strong> a<br />
sequence<br />
Z ↩→ L c ↠ F c<br />
which is an opposite extension. Here Z is the disjoint union<br />
<strong>of</strong> the centers <strong>of</strong> all F-centric subgroups.<br />
Proposition 6.4.8. Let n ∈ kL c -mod <strong>and</strong> m ∈ kF c -mod.<br />
Then<br />
Hom kL c(Res ρ m, n) ∼ = Hom kF c(m, n Z ),<br />
where n Z is defined by n Z (P ) = n(P ) Z(P ) .<br />
Let Bb c be the full subcategory <strong>of</strong> B b for a block b, consisting<br />
<strong>of</strong> self-centralizing b-Brauer pairs. Since Fb c naturally identifies<br />
with a full subcategory <strong>of</strong> Bb c which induces an equivalence,<br />
we similarly can consider an opposite extension<br />
Z ↩→ ˜Lc b ↠ B c b.<br />
If ˜Lc b<br />
exists, then so is L c b<br />
, <strong>and</strong> vice versa. Moreover between<br />
the corresponding extensions there exists a natural embedding<br />
L c b → ˜Lc b<br />
inducing an category equivalence. By taking the<br />
larger category ˜Lc b<br />
(but essentially the same as L c b<br />
), we can<br />
write down<br />
C G /Z ↩→ G ∝ Pb c ↠ ˜Lc b ,
258 6 Connections with group representations <strong>and</strong> cohomology<br />
another opposite extension. Here C G /Z is the disjoint union<br />
<strong>of</strong> C G (P )/Z(P ) in which P runs over all F-centric subgroups.<br />
For the sake <strong>of</strong> convenience, we introduce a notation C G ′ =<br />
C G /Z so that C G ′ (P ) = C G(P )/Z(P ) ∼ = P C G (P )/P for each<br />
P .<br />
When m = κ M for some M ∈ kG-mod, we write the k ˜Lc b<br />
-<br />
module RK ρ κ M as M C′ G . Given a morphism (g, g P ≤ Q)<br />
it induces an injection C G ′ (Q) → C′ G (P ) thus a morphism<br />
M C′ G (P ) → M C′ G (Q) . Hence we get the k ˜Lc b<br />
-action on M C′ G<br />
Proposition 6.4.9. Let m ∈ k(G ∝ Pb c )-mod <strong>and</strong> N ∈ kGmod.<br />
Then<br />
Hom kG (lim −→P<br />
m, N) ∼ = Hom k ˜Lc b<br />
(m, N C′ G ),<br />
where N C′ G is defined as above.<br />
In particular if m = Res ρ m ′ for some m ′ ∈ kBb c -mod then<br />
Hom kG (lim −→P<br />
m ′ , N) ∼ = Hom kB<br />
c<br />
b<br />
(m ′ , N C G<br />
),<br />
a special situation <strong>of</strong> Proposition 6.5.3.<br />
Pro<strong>of</strong>. The first part is a direct consequence <strong>of</strong> Proposition<br />
5.4. As for the special case We need to notice that<br />
lim Res −→P ρm ′ ∼ = lim −→P<br />
m ′ as kG-modules. Then<br />
Hom kG (lim −→P<br />
Res ρ m ′ , N) ∼ = Hom k ˜Lc b<br />
(Res ρ m ′ , N C′ G )<br />
∼ = HomkB c<br />
b<br />
(m ′ , (N C′ G ) Z )<br />
∼ = HomkB c<br />
b<br />
(m ′ , N C G ).<br />
⊓⊔<br />
6.4.3 Puig categories<br />
If we take P A to be the poset <strong>of</strong> pointed subgroups on an interior<br />
G-algebra A, then analogues to the Brauer category for
6.4 The functor ρ: invariants <strong>and</strong> coinvariants 259<br />
any G-subposet P ⊂ P A we can introduce the Puig category<br />
L P as a quotient category <strong>of</strong> G ∝ P such that<br />
Hom LP (P γ , Q δ ) = Hom G∝P (P γ , Q δ )/C G (P ).<br />
Then some results in last section can be obtained accordingly.<br />
If m = κ M for some M ∈ kG-mod, we denote by M C G the<br />
kL P -module RK ρ κ M . Since a morphism (g, g P ≤ Q) provides<br />
a group homomorphism P → Q <strong>and</strong> thus induces an injection<br />
c g<br />
−1 : C G (Q) → C G (P ), we obtain an injection M C G(P ) →<br />
M CG(Q) . This leads to the kL P -action on M C G .<br />
Proposition 6.4.10. Let m ∈ kL P -mod <strong>and</strong> N ∈ kG-mod.<br />
Then<br />
Hom kG (lim −→P<br />
m, N) ∼ = Hom kLP (m, N C G<br />
),<br />
where N C G<br />
is as above.<br />
6.4.4 Orbit categories <strong>of</strong> fusion systems<br />
This method also works for the orbit category <strong>of</strong> a fusion<br />
system. Suppose F is an abstract fusion system. The one may<br />
define the orbit category O F in a similar fashion as above by<br />
Hom OF (P, Q) = Q\Hom F (P, Q).<br />
Again we obtain an extension sequence<br />
S ↩→ F ↠ O F ,<br />
where S is the disjoint union <strong>of</strong> objects in F.<br />
Proposition 6.4.11. If m ∈ kF-mod <strong>and</strong> n ∈ kO F -mod,<br />
then<br />
Hom kF (m, Res ρ n) ∼ = Hom kOF (m S , n).<br />
Moreover for F = Fb c, Hom kG(lim −→P<br />
m, k) ∼ = Hom kOF c(m S , k).<br />
b<br />
Pro<strong>of</strong>. When F = Fb c , we have isomorphisms
260 6 Connections with group representations <strong>and</strong> cohomology<br />
Hom kG (lim −→P<br />
m, k) ∼ = Hom kF (m, k)<br />
∼ = HomkOF c<br />
b<br />
(m S , k).<br />
⊓⊔<br />
6.5 Hochschild cohomology<br />
In this section we continue to demonstrate the similarities between<br />
transporter categories <strong>and</strong> groups. The first main assertion<br />
is the the Hochschild cohomology ring <strong>of</strong> a finite transporter<br />
category is finitely generated. From here we will establish<br />
the finite generation <strong>of</strong> cohomology. The second is that we<br />
can construct a transfer map between Hochschild cohomology.<br />
Both results are established by passing between Hochschild<br />
cohomology <strong>and</strong> ordinary cohomology <strong>of</strong> F (G ∝ P) <strong>and</strong><br />
G ∝ P. Thus computing various over <strong>and</strong> undercategories<br />
for functors from F (G ∝ P) to appropriate categories are the<br />
major auxiliary statements.<br />
6.5.1 <strong>Finite</strong> generation<br />
In Section 4.2.2 we have seen that the ordinary cohomology<br />
ring <strong>of</strong> a finite category can be infinitely generated even after<br />
quotient out nilpotent elements. Based on the Theorem<br />
5.2.2, the Hochschild cohomology ring <strong>of</strong> such a finite category<br />
algebra is not finitely generated either. So the question<br />
reduces to finding out whether or not Ext ∗ kCe(kC, kC) modulo<br />
nilpotents is finitely generated over Ext ∗ kC(k, k) if the latter is<br />
Noetherian. On the first attempt to solve this question, one<br />
may want to check if the Evens-Venkov Theorem on the finite<br />
generation <strong>of</strong> group cohomology could be generalized to category<br />
cohomology. This is not true since Example 4.2.2 (3)<br />
implies that we can not expect a finite generation property
6.5 Hochschild cohomology 261<br />
<strong>of</strong> Ext ∗ kC(M, N) over a finitely generated Ext ∗ kC(k, k). Thus<br />
we have to look at particular families <strong>of</strong> finite categories for<br />
the finite generation property. In what follows, we show finite<br />
transporter categories constructed over a finite group are very<br />
close to what we expect.<br />
To examine the finite generation <strong>of</strong> Hochschild cohomology<br />
ring, we use the isomorphism<br />
Ext ∗ k(G∝P) e(k(G ∝ P), k(G ∝ P)) ∼ = Ext ∗ kF (G∝P)(k, Res ∇ (k(G ∝ P)))<br />
because it allows us to use the Grothendieck spectral sequence.<br />
Let us introduce a functor ˜π = π ◦ t<br />
t<br />
F (G ∝ P)<br />
<br />
G ∝ P<br />
˜π<br />
π<br />
G .<br />
When we investigated the finite generation <strong>of</strong> H ∗ (G ∝ P, k)<br />
we used the Grothendieck spectral sequence. Based on the fact<br />
that •\π ≃ P, we obtained the finite generation. Naturally<br />
we want to look at the Grothendieck spectral sequence for<br />
˜π. In order to underst<strong>and</strong> the spectral sequence, we have to<br />
know the undercategory •\˜π. Since t induces a homotopy<br />
equivalence, the undercategory •\˜π should be similar to •\π.<br />
If we take P = • in the above diagram, then π is a canonical<br />
isomorphism <strong>and</strong> •\π ≃ •. By contrast ˜π can be identified<br />
with t G : F (G) → G <strong>and</strong> we also have •\˜π = •\t G ≃ •.<br />
Lemma 6.5.1. Let t G : F (G) → G be the target functor.<br />
Then we have two isomorphic categories<br />
1. •\˜π ∼ = F (P) × •\t G .<br />
2. ˜π/• ∼ = F (P) × t G /•.<br />
3. •\t G<br />
∼ = tG /• are equivalent to •.<br />
Pro<strong>of</strong>. We only prove Parts 1 <strong>and</strong> 3. First we prove (1).<br />
The objects in •\˜π are (•, [(g, gx ≤ y)]). A morphism from
262 6 Connections with group representations <strong>and</strong> cohomology<br />
(h, [(g, gx ≤ y)]) to (h ′ , [(g ′ , g ′ x ′ ≤ y ′ )]) is a morphism<br />
((l 1 , l 1 y ≤ y ′ ), (l 2 , l 2 x ′ ≤ x)) : [(g, gx ≤ y)] → [(g ′ , g ′ x ′ ≤ y ′ )]<br />
such that l 1 h = h ′ <strong>and</strong> g ′ = l 1 gl 2 . Thus from one object in<br />
•\˜π to another there is at most one morphism. It implies<br />
this finite undercategory is equivalent to a finite poset. Furthermore<br />
we notice that (h, [(g, gx ≤ y)]) is isomorphic to<br />
(e, [(h −1 g, (h −1 g)x ≤ h −1 y)]). But [(h −1 g, (h −1 g)x ≤ h −1 y)]<br />
is isomorphic to [(e, e(h −1 gx) ≤ h −1 y)] in F (G ∝ P). Now<br />
we define a functor •\˜π → F (P) × •\t G by<br />
(h, [(g, gx ≤ y)]) ↦→ [h −1 gx ≤ h −1 y] × (h, [g]).<br />
Its inverse is given by<br />
[x ≤ y] × (h, [g]) ↦→ (h, [(g, g(g −1 hx) ≤ hy)]).<br />
The isomorphism in (3) is easy to write down as (h, [g]) ↦→<br />
([g], h −1 ). Furthermore, any object ([g], h) ∈ Ob(t G /•) is isomorphic<br />
to ([e], e) which has only one endomorphism. ⊓⊔<br />
Now we can state the consequence <strong>of</strong> Lemma 6.5.1 (1). It is<br />
similar to the main result in Section 6.3.2.<br />
Proposition 6.5.2. Let G be a finite group <strong>and</strong> P a finite<br />
G-poset. Then for any M ∈ kF (G ∝ P)-mod, Ext ∗ kF (G∝P)(k, M)<br />
becomes a finitely generated Ext ∗ kG(k, k)- <strong>and</strong> Ext ∗ kF (G∝P)(k, k)-<br />
module. Especially the Hochschild cohomology ring Ext ∗ k(G∝P) e(k(G ∝<br />
P), k(G ∝ P)) is finitely generated.<br />
Pro<strong>of</strong>. We apply the Grothendieck spectral sequence to ˜π.<br />
Since •\˜π is equivalent to a finite poset F (P), H j (•\˜π, M)<br />
vanishes for all j larger than a chosen positive integer. Consequently<br />
the E 2 page <strong>of</strong> the spectral sequence only has finitely<br />
many non-zero rows in the first quadrant, <strong>and</strong> thus we have<br />
the finite generation <strong>of</strong> Ext ∗ kF (G∝P)(k, M).<br />
⊓⊔
6.5 Hochschild cohomology 263<br />
The preceding result enables us to prove a finite generation<br />
theorem. It will be the foundation for developing a support<br />
variety theory over the ring Ext ∗ k(G∝P)(k, k).<br />
Theorem 6.5.3. Suppose M, N are two k(G ∝ P)-modules.<br />
Then the module Ext ∗ k(G∝P)(M, N) is finitely generated over<br />
Ext ∗ k(G∝P)(k, k).<br />
Pro<strong>of</strong>. By Theorem 5.2.11,<br />
Ext ∗ k(G∝P)(M, N) ∼ = Ext ∗ kF (G∝P)(k, Res ∇ Hom k (M, N)).<br />
Hence by Proposition 6.5.2, Ext ∗ kF (G∝P)(k, Res ∇ Hom k (M, N))<br />
is finitely generated over Ext ∗ kF (G∝P)(k, k) ∼ = Ext ∗ k(G∝P)(k, k).<br />
We are done.<br />
⊓⊔<br />
If P = •, we get the usual assertion that Ext ∗ kGe(kG, kG)<br />
is finitely generated over Ext ∗ kG(k, k). If P = S b as in Example<br />
6.1., we have Ext ∗ kG(k, k) acting on Ext ∗ k(G∝S b ) e(k(G ∝<br />
S b ), k(G ∝ S b )) via Ext ∗ k(G∝S b )(k, k). Especially when b = b 0 ,<br />
S b<br />
∼ = Sp <strong>and</strong> Ext ∗ k(G∝S p )(k, k) ∼ = Ext ∗ kG(k, k).<br />
Corollary 6.5.4. If k is a field with positive characteristic<br />
p ∣ |G|, <strong>and</strong> b is a p-block, then, for any full subcategory<br />
Tr ⊂ Tr b (G) whose objects are closed under G-conjugation,<br />
Ext ∗ kTr<br />
e(kTr, kTr) is a finitely generated algebra.<br />
Given the principal block b 0 <strong>of</strong> a group algebra kG, we have<br />
a fusion system F b0 = F p over a fixed Sylow p-subgroup S.<br />
As we mentioned earlier, there exists a centric linking system<br />
L b0 = L c p which is determined by the full subcategory<br />
Tr c p(G) ≤S <strong>of</strong> the transporter category Tr p (G), consisting <strong>of</strong> all<br />
p-centric subgroups contained in S [10]. In fact L c p is a quotient<br />
category <strong>of</strong> Tr c p(G) ≤S by some p ′ -groups. In other words,<br />
if one looks at the canonical functor π : Tr c p(G) ≤S → L c p, each
264 6 Connections with group representations <strong>and</strong> cohomology<br />
undercategory has the property such that it has a minimal object<br />
whose automorphism group is p ′ <strong>and</strong>, if one regards this<br />
p ′ -automorphism group as a subcategory, the left Kan extension<br />
along the inclusion is exact. Furthermore the left Kan<br />
extension <strong>of</strong> the trivial group module is the trivial module <strong>of</strong><br />
the undercategory. This implies, by an Eckmann-Shapiro type<br />
result, the cohomology <strong>of</strong> each undercategory can be reduced<br />
to the cohomology <strong>of</strong> the automorphism group <strong>of</strong> the abovespecified<br />
minimal object in it. Consequently, the mod-p cohomology<br />
<strong>of</strong> each undercategory <strong>of</strong> π with arbitary coefficients<br />
vanishes in positive degrees. To summarize, since Tr c p(G) ≤S is<br />
equivalent to Tr c p(G) <strong>and</strong> the Grothendieck spectral sequence<br />
for π collapses, we have an isomorphism<br />
Ext ∗ kTr c p(G)(k, V ) ∼ = Ext ∗ kTr c p(G) ≤S<br />
(k, V ) ∼ = Ext ∗ kL c p (k, RK πV ),<br />
where RK π V = H 0 (\π; V ) ∼ = lim ←−\π<br />
V is the right Kan extension<br />
along π <strong>of</strong> V . It is similar to [10, Lemma 1.3 (iii)]<br />
in which right modules are considered <strong>and</strong> thus the left Kan<br />
extension is applied. Especially, the functors G ← Tr c p(G) ←↪<br />
Tr c p(G) ≤S → L c p induce isomorphisms <strong>of</strong> mod-p ordinary cohomology<br />
rings.<br />
Proposition 6.5.5. Let L c p be the centric linking system<br />
associated to the principal block <strong>of</strong> a finite group algebra<br />
kG. Then Ext ∗ kL (k, RK πV ) is finitely generated as an<br />
c p<br />
Ext ∗ kL (k, k)-module.<br />
c p<br />
However this is still far from underst<strong>and</strong>ing the finite generation<br />
<strong>of</strong> the Hochschild cohomology ring Ext ∗ k(L c p) e(kLc p, kL c p) ∼ =<br />
Ext ∗ kF (Lp)(k, Res c τ (kL c p)).
6.5 Hochschild cohomology 265<br />
6.5.2 Transfer for Hochschild cohomology<br />
We end this chapter with a transfer map between Hochschild<br />
cohomology. Consider the following commutative diagram <strong>of</strong><br />
functors<br />
(G ∝ P) e πe <br />
G e<br />
F (P)<br />
t <br />
∇<br />
<br />
F (G ∝ P)<br />
t <br />
F (π) <br />
∇<br />
F (G)<br />
P<br />
<br />
G ∝ P π<br />
<br />
G<br />
Recall that in Section 6.3.3, based on the bottom row, we<br />
were able to establish a transfer map between ordinary cohomology.<br />
Since the three target functors all induce homotopy<br />
equivalences, the middle row may as well give a transfer map<br />
Ext ∗ kF (G∝P)(−, −) → Ext ∗ kF (G)(−, −), for suitable modules.<br />
Since furthermore the upper right square demonstrates connections<br />
between the ordinary cohomology over kF (G ∝ P)<br />
<strong>and</strong> kF (G) with the Hochschild cohomology <strong>of</strong> kF (G ∝ P)<br />
<strong>and</strong> kF (G), respectively, we automatically wonder if there<br />
would be a transfer between the Hochschild cohomology <strong>of</strong><br />
kF (G ∝ P) <strong>and</strong> kF (G) The answer is yes, <strong>and</strong> we shall<br />
provide the construction here, which in spirit is similar to the<br />
transfer we built between the ordinary cohomology. A predecessor<br />
<strong>of</strong> this construction is due to Lincklemann [45]. For G<br />
a group <strong>and</strong> H a subgroup, he developed a method over symmetric<br />
algebras <strong>and</strong> defined a transfer map from HH ∗ (kH) to<br />
HH ∗ (kG). Recall that H can be recovered as the transporter<br />
category G ∝ (G/H), our construction generalizes his.<br />
Lemma 6.5.6. Let kG ∈ kG e -mod <strong>and</strong> k(G ∝ P) ∈ k(G ∝<br />
P) e -mod. Then the kF (G ∝ P)-module Res F (π) Res ∇ kG contains<br />
Res ∇ k(G ∝ P) as a submodule.<br />
t
266 6 Connections with group representations <strong>and</strong> cohomology<br />
Pro<strong>of</strong>. Let (x, y) ∈ Ob(G ∝ P) e . Then<br />
k(G ∝ P)(x, y) = kHom G∝P (y, x)<br />
= k{(h, hy ≤ x) ∣ ∣ h ∈ G}<br />
⊂ kG<br />
= (Res π ekG)(x, y).<br />
From here we can verify that k(G ∝ P) is a k(G ∝ P) e -<br />
submodule <strong>of</strong> Res π ekG. Hence the result follows. ⊓⊔<br />
If P = G/H, then G ∝ (G/H) ≃ H <strong>and</strong> F (G ∝ (G/H)) ≃<br />
H. The above inclusion says that the kH-module kG ↓ G H contains<br />
kH as a submodule. Here kG <strong>and</strong> kH are acted by G<br />
<strong>and</strong> H, respectively, by conjugation.<br />
Lemma 6.5.7. Consider the functor F (π) : F (G ∝ P) →<br />
F (G). Then F (π)/− ∼ = F (P) − × Id F (G) /−, where F (P) [g]<br />
denotes a poset, indexed by [g] ∈ Ob F (G), which is canonically<br />
isomorphic to F (P). Consequently<br />
C ∗ (F (π)/−) ∼ = C ∗ (F (P) − × Id F (G) /−)<br />
as complexes <strong>of</strong> kF (G)-modules. Since F (G) ≃ G,<br />
C ∗ (F (π)/[e]) ∼ = C ∗ (F (P) [e] × Id F (G) /[e]) ≃ C ∗ (F (P)) ⊗ B G ∗<br />
is a projective resolution <strong>of</strong> the complex <strong>of</strong> kG-modules<br />
C ∗ (F (P)).<br />
Pro<strong>of</strong>. Suppose [g] is an object in F (G). The the overcategory<br />
F (G)/[g] has objects {([(h, hx ≤ y)], (h 1 , h 2 ))}, in which<br />
[(h, hx ≤ y)] ∈ Ob F (G ∝ P). It implies g = h 1 hh 2 . As a<br />
consequence, hx ≤ y is equivalent to h −1<br />
2 x ≤ g−1 h 1 y. Morphisms<br />
((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x)) are given by
6.5 Hochschild cohomology 267<br />
(v 1 ,v 2 )=F (π)((v 1 ,v 1 y≤y ′ ),(v 2 ,v 2 x ′ ≤x))<br />
[h] = F (π)([(h, hx ≤ y)])<br />
(h 1 ,h 2 )<br />
(h ′ 1 ,h′ 2 ) <br />
[h ′ ] = F (π)([(h ′ , h ′ x ′ ≤ y ′ )])<br />
It implies various identities: g = h 1 hh 2 = h ′ 1h ′ h ′ 2, h 1 = h ′ 1v 1<br />
<strong>and</strong> h 2 = v 2 h ′ 2. We define a functor F (π)/[g] → F (P) ×<br />
Id F (G) /[g] such that on objects<br />
([(h, hx ≤ y)], (h 1 , h 2 )) ↦→ ([h −1<br />
2 x ≤ g−1 h 1 y], ([h], (h 1 , h 2 ))),<br />
<strong>and</strong> on morphisms ((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x)) ↦→ ((e, e), (v 1 , v 2 )),<br />
because v 1 y ≤ y ′ implies g −1 h 1 y ≤ g −1 h ′ 1y ′ while v 2 x ′ ≤ x implies<br />
h ′ −1 2 x ′ ≤ h2 −1 x. The inverse <strong>of</strong> this functor is defined by<br />
([x ≤ y], ([h], (h 1 , h 2 ))) ↦→ ([(h, h(h 2 x) ≤ h −1<br />
1 gy)], (h 1, h 2 ))<br />
on objects, <strong>and</strong> on morphisms is defined by ((e, e), (v 1 , v 2 )) ↦→<br />
((v 1 , v 1 h −1<br />
1 gy ≤ h′ 1−1 gy ′ ), (v 2 , v 2 h ′ 2x ′ ≤ h 2 x)). Thus we obtain<br />
an isomorphism<br />
F (π)/[g] ∼ = F (P) [g] × Id F (G)/[g],<br />
where F (P) [g] denotes a copy <strong>of</strong> F (P) indexed by the object<br />
[g]. If (l 1 , l 2 ) : [g] → [g ′ ] is a morphism in F (G). Then it<br />
induces a functor F (π)/[g] → F (π)/[g ′ ] given by<br />
<strong>and</strong><br />
([(h, hx ≤ y)], (h 1 , h 2 )) ↦→ ([(h, hx ≤ y)], (l 1 h 1 , h 2 l 2 ))<br />
((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x)) ↦→ ((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x))<br />
Using the isomorphisms F (π)/[g] ∼ = F (P) [g] × Id F (G) /[g] <strong>and</strong><br />
F (π)/[g ′ ] ∼ = F (P) [g ′ ] × Id F (G) /[g ′ ], one can see it induces an<br />
isomorphism F (P) [g] → F (P) [g ′ ].<br />
[g]
268 6 Connections with group representations <strong>and</strong> cohomology<br />
Finally since G is isomorphic to the automorphism group <strong>of</strong><br />
[e] in the groupoid F (G), we have an equivalence F (G) ≃ G.<br />
Thus C ∗ (F (P) [e] × Id F (G) /[e]) ≃ C ∗ (F (P)) ⊗ C ∗ (Id F (G) /[e])<br />
is a complex <strong>of</strong> projective kG-modules. Furthermore because<br />
([e]), it has to be a projective resolution<br />
<strong>of</strong> the trivial kG-module k. Hence we get the chain<br />
homotopy equivalence as stated.<br />
⊓⊔<br />
C ∗ (Id F (G) /[e]) = B F (G)<br />
∗<br />
This lemma allows us to describe the transfer map in Theorem<br />
5.2.2 in terms <strong>of</strong> factorization categories:<br />
tr P : Ext ∗ kF (G∝P)(k, k) → Ext ∗ kF (G)(k, k).<br />
Indeed we have a commutative diagram<br />
F (G ∝ P) F (π) <br />
F (G)<br />
t <br />
G ∝ P π<br />
<br />
G<br />
Thus we get a commutative diagram <strong>of</strong> cochain complexes<br />
Hom kF (G∝P) (B F (G∝P)<br />
∗ , k)<br />
∼ = <br />
∼ = <br />
Hom k(G∝P) (LK t B F (G∝P)<br />
∗ , k) ∼=<br />
<br />
≃<br />
<br />
Hom k(G∝P) (B G∝P<br />
∗ , k) ≃<br />
t<br />
Hom kF (G) (LK F (π) B F (G∝P)<br />
∗ , k)<br />
∼ =<br />
Hom kG (LK˜π B∗ F (G∝P) , k)<br />
≃<br />
<br />
Hom kG (C ∗ (P) ⊗ B G ∗ , k)<br />
Hom k(G∝P) (B∗ G∝P , k)<br />
<br />
Hom kG (B∗ G , k)<br />
≃<br />
<br />
Hom kG (LK t B F (G)<br />
∗ , k)<br />
∼ =<br />
<br />
Hom kF (G) (B F (G) , k)
6.5 Hochschild cohomology 269<br />
The isomorphisms are given by adjunctions <strong>and</strong> the chain homotopy<br />
equivalences are induced by changing projective resolutions.<br />
From the upper left corner to the lower right corner<br />
is the transfer that we want to describe. It factors through<br />
the lowest square, including the chain map which gives rise to<br />
the transfer constructed in Theorem 6.3.21. Note that F (G)<br />
is a groupoid <strong>and</strong> G is a skeleton <strong>of</strong> F (G). Comparing with<br />
[45, Section 4], F (G) plays the role <strong>of</strong> ∆G if one identifies G e<br />
with G × G.<br />
Theorem 6.5.8. There exists a map<br />
htr P : Ext ∗ k(G∝P) e(k(G ∝ P), k(G ∝ P)) → Ext∗ kGe(kG, kG).<br />
Pro<strong>of</strong>. Using Lemma 6.5.6 we have cochain maps<br />
Hom kF (G∝P) (B∗<br />
G∝P , Res F (π) Res ∇ kG) ∼ = Hom kF (G) (LK F (π) B∗<br />
G∝P , Res ∇<br />
∼ = HomkF (G) (C ∗ (F (π)/−), Res ∇<br />
∼ = HomkG (C ∗ (F (π)/[e]), kG)<br />
≃ Hom kG (C ∗ (F (P)) ⊗ B∗ G , kG)<br />
→ Hom kG (B∗ G , kG)<br />
Here the module kG in Hom kG (−, kG) is acted by kG via<br />
conjugations. The last map is induced by k → C ∗ (F (P)),<br />
a chain map constructed in the same way as the one in the<br />
pro<strong>of</strong> <strong>of</strong> Theorem 6.3.21 k → C ∗ (P). Indeed since t : F (P) →<br />
P is a G-functor, it induces an chain homotopy equivalence<br />
C ∗ (F (P)) ≃ C ∗ (P).<br />
By Lemma 6.5.5 we also have a chain map<br />
Hom kF (G∝P) (B G∝P<br />
∗<br />
, Res ∇ kF (G ∝ P)) → Hom kF (G∝P) (B∗<br />
G∝P , Res F (π)<br />
induced by the inclusion Res ∇ kF (G ∝ P) → Res F (π) Res ∇ kG.<br />
Hence altogether we obtain a chain map<br />
Hom kF (G∝P) (B∗<br />
G∝P , Res ∇ kF (G ∝ P)) → Hom kG (B∗ G , kG).
270 6 Connections with group representations <strong>and</strong> cohomology<br />
Passing to cohomology we get a map between Hochschild cohomology<br />
by Theorem 5.2.2<br />
Ext ∗ kF (G∝P)(k, Res ∇ kF (G ∝ P)) htr P<br />
∼ =<br />
<br />
Ext ∗ kG(k, kG)<br />
∼ =<br />
Ext ∗ k(G∝P) e(kF (G ∝ P), kF (G ∝ P)) htr P<br />
<br />
Ext ∗ kG e(kG, kG) ⊓⊔<br />
The above map deserves to be called a transfer since<br />
k ∣ ∣ Res ∇ kF (G ∝ P), k ∣ ∣ kG <strong>and</strong> k ∣ ∣ Res ∇ kF (G). In the<br />
construction <strong>of</strong> htr P if we replace the second module in all<br />
Hom − (−, −) <strong>and</strong> Ext ∗ −(−, −) by k then it is exactly the transfer<br />
created in Theorem 6.3.21 <strong>and</strong> reinterpreted before Theorem<br />
6.5.7. In other words, we have a commutative diagram<br />
Ext ∗ k(G∝P)(k, k)<br />
∼ =<br />
<br />
Ext ∗ kF (G∝P)(k, k)<br />
injection<br />
Ext ∗ kF (G∝P)(k, Res ∇ kF (G ∝ P))<br />
∼ =<br />
<br />
tr P<br />
tr P<br />
Ext ∗ kG(k, k)<br />
∼ =<br />
Ext ∗ kF (G)(k, k)<br />
injection<br />
Ext ∗ kF (G)(k, Res ∇ kF (G))<br />
Ext ∗ k(G∝P) e(k(G ∝ P), k(G ∝ P)) <br />
htr P<br />
Ext ∗ kGe(kG, kG)<br />
∼ =
References 271<br />
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