Representations and Cohomology of Finite Categories (DRAFT)

Fei Xu
Representations and Cohomology of Finite
Categories (DRAFT)
Category Algebras & Simplicial Methods
September 1, 2011

Preface
These are expanded lecture notes that I used for a short course
of the same title at Universitat Autònoma de Barcelona in
2010-2011 academic year. The subtitle reveals the main tools
we are using. It means that we shall study representations
and cohomology of category algebras via simplicial modules.
I assume the reader to have good background on homological
algebra from classical books such as “A Course in Homological
Algebra” by Hilton and Stammbach. The main theme of these
notes is to answer the question: what we may do with an
abstract finite category Here we regard a finite category as
a generalization of an abstract finite group and of an abstract
finite partially ordered set. Starting from a finite group and
a base ring, there are well developed group representation
theory and group cohomology theory. Parallel theories have
been introduced to finite partially ordered sets as well. This
book presents a theory that extending all of the above.
Category cohomology theory is a place where simplicial
methods, homotopy theory and representation theory naturally
meet. The objectives of our theory may be summarized
in the following diagram.

vi
C
classifying space
functor category
nerve
category algebra and modules
BC
geo.
bar resolution
NC
RC−mod
realiz.
of the trivial RC−module R
singular cohomology
simplicial cohomology
cohomology of modules
H ∗ (C; N) ∼ ∗
= lim ←−
C N ∼ = Ext ∗ RC(R, N)
Ext ∗ RC(M, N)
Preface
functor cohomology
In the diagram, C is a finite category, NC is its nerve, a combinatorial
construction, and BC is the geometric realization of
NC (or C) which is a CW-complex. Also R is a commutative
ring with identity and M, N ∈ (R-mod) C are two covariant
functors. (A functor in (R-mod) C should be considered as a
diagram of R-modules.) In this picture, RC is the so-called
category algebra. It was shown by B. Mitchell [56] that there
exists an equivalence between (R-mod) C and RC-mod (the
category of left RC-modules). Thus a functor from C to R-
mod is always an RC-module. Notably when C is a group, the
theory becomes the usual group cohomology theory. When C
is a transporter category (see Chapter 6) defined over a group
G, we recover the equivariant cohomology theory.
We can similarly study category homology by replacing
the last two rows in the picture by H∗(C; N) ∼ ∗
= lim −→ N ∼ =
C
Tor RC
∗ (R, N) and Tor RC
∗ (M ′ , N) (M ′ is a right RC-module).
In these notes we shall focus on cohomology theory because
homology theory can be developed parallel to it.
Much of the general theory is indeed established for all small
categories. To make this book as useful as we can, we shall
state results in their general forms, for small, not just finite,
categories whenever it is the case. It is perhaps a good point
to explain why small categories are of particular interests. In
fact this is a set-theoretic issue. In the above diagram, if we
(R−mod

Preface
consider a small category C then we can still construct a simplicial
set NC, a topological space BC, an algebra RC and
a functor category (R-mod) C . In all these constructions, explicitly
defined in the main text, the class of morphisms in C,
Mor C, has to be a set, which is precisely the smallness condition
on C. It lies in the applications of category cohomology
and representations that we have in mind that C is frequently
finite (Mor C is a finite set). The chief reason is that we understand
quite well the representations of RC, which becomes
an associative algebra with identity and is of finite R-rank.
There exists another functor (co)homology theory [25], used
in Steenrod algebras and cohomology of finite group schemes.
In that (co)homology theory, one mainly investigates representations
and cohomology of the concrete (essentially small
but infinite) category of finite-dimensional vector spaces over
a field. The relationship between these two theories is comparable
to that between the cohomology theories of abstract
finite groups and of general linear groups. They start off the
same foundation but are of different flavors and use quite different
methods, even though they supply important ideas and
results to each another. Thus although there are something in
common between [25] and this book, the main bulks of these
two are different and to some extent complementary to each
other.
The ingredients shown in the previous diagram have been
studied since the early stage of homological algebra in one
way or another. In this book we collect many existing materials
to form a source for self-learning as well as a reference
for researchers. The major inputs from the author are firstly
to provide a systematic introduction to the uses of simplicial
methods, and secondly to develop tools for comparing
cohomology of two small categories connected by a functor.
vii

viii
Preface
Simplicial methods help us to construct some (complexes of)
modules combinatorially while the tools we have mentioned
tell us how to compare these (complexes of) modules (and
their cohomology). We shall explain these two features next.
Simplicial methods should be considered as natural tools
for algebraists. If A is a ring, then a simplicial A-module M
is a collection of A-modules M n , n ≥ 0, equipped with certain
maps among them. By Dold-Kan Correspondence, the
category of non-negatively (or non-positively) graded chain
complexes of A-modules is equivalent to the category of simplicial
(or cosimplicial) A-modules. It also implies the category
of first quadrant double complexes is equivalent to the
category of bisimplicial A-modules. The complexes of modules
have long been used in (at least the homological aspects
of) the representation theory of algebras, so there is no reason
why we do not consider their combinatorial predecessors.
In this book we shall focus on the case for A = RC, where
simplicial methods seem to be truly powerful.
Let u : D → C be a functor between small categories. We
construct a simplicial RC-module R[N(u/−)] (a bar construction).
It has the properties that, for each n ≥ 0, R[N(u/−)] n
is a left projective RC-module, and that, for each x ∈ Ob C,
R[N(u/x)] is a simplicial R-module. Here u/x is the category
over x and N(u/x) is its nerve, a simplicial set. Much of category
(co)homology is developed based on this construction.
For instance, R[N(Id C /−)] gives rise to the bar resolution
of the trivial RC-module R such that R[N(Id C /−)] = RC.
This is the reason why simplicial methods are indispensable
to us. Due to the existence of these constructions, the Kan
extensions are used, as generalizations of the usual induction
and co-induction on which one relies to investigate group
(co)homology, to compare (co)homology of C and D. In fact,

Preface
let LK u : RD-mod → RC-mod be the left Kan extension
along u. Then LK u R[N(Id D /−)] ∼ = R[N(u/−)]. Accordingly
understanding various overcategories becomes an important
issue as they show up in the constructions of simplicial modules
as well as in the definitions of Kan extensions. An illuminating
example is as follows. Suppose G is a finite group
and H is a subgroup. Consider them as categories with one
object •, equipped with the inclusion i : H → G. Then the
only overcategory i/• is equivalent to the set of left cosets
G/H. The left Kan extension of the bar resolution of the
trivial RH-module is a projective resolution of the induced
module R(G/H) = R ↑ G H . As in algebraic topology where
they originate and their applications confine, we shall show
that simplicial methods now become crucial theoretical and
computational tools in algebra too.
What we do here in this book is really to take up the algebraic
and combinatorial tools invented by topologists and
apply them to algebra. We shall see that we get satisfactory results.
Simplicial methods were introduced by algebraic topologists
and lie in the center of homotopy theory. In homotopy
theory, one often considers diagrams of spaces, which are simply
functors from an index category C to T op, the category
of topological spaces. When u : D → C and F : C → T op are
two functors, one can naturally obtain a new functor F ◦ u.
This is called the restriction along u, written as Res u F. Indeed
Res u is a functor from the category T op C to T op D . On the
other hand if G : D → T op is a functor, we can construct
two functors C → T op. These two functors are called the left
and right Kan extensions of G along u. It is intuitive if we
record topologists’ notations
C ⊗ D G and Hom D (C, G).
ix

x
Preface
These two functors C ⊗ D −, Hom D (C, −) are the left and right
adjoints of Res u . In Hollender-Vogt [37], one can see that
these two functors enjoy many properties that an algebraist
would expect for similar functors, induced by a ring homomorphism
R 1 → R 2 , between R 1 -mod and R 2 -mod. In fact
in [37], a covariant functor in T op C is called a left C-module
while a contravariant functor is called a right C-module. I
would like to point out here that since a functor u : D → C
does not lead to an algebra homomorphism from RD to RC,
one has to be very careful when constructing RC ⊗ RD − and
Hom RD (RC, −). This is the reason why I refrain from using
topologists’ notations of the Kan extensions. Plus the intuitive
notations do not seems to be convenient in computations
as they are in other places such as group (co)homology.
However the Kan extensions are truly generalizations of well
known functors. When i : H → G is the inclusion functor at
the end of last paragraph, the left and right Kan extensions
along i are isomorphic to the induction RG ⊗ RH − and coinduction
Hom RH (RG, −) (which happen to be isomorphic
to each other).
Let us now recall the history of (co)homology theory of small
categories and related works. Homology theory of small categories,
H∗(C; N), can be found in Gabriel-Zisman [28] (1967),
as a special case of homology of simplicial sets with coefficients,
in the appendices. There they also showed H∗(C; N) ∼ =
lim
−→ ∗ N, where lim
C −→ ∗ are the derived functors of direct limit
C
functor −→C
lim . By contrast, although the special case of partially
ordered sets was studied by J. E. Roos [66] (1961), cohomology
of small categories began with Baues-Wirsching’s
theory [3] (1985). They actually studied H ∗ (C; N ), where N
is not a functor from C. Instead, it is a functor from another
category F (C), the category of factorizations in C. In their

Preface
situation cohomology theory of small category can also be
∗
introduced by ←−
lim , derived functors of inverse limit functor
C
lim . Simplicial methods are used to define various chain or
←−C
cochain complexes whose (co)homology is the (co)homology
of C.
In these theories the coefficients of (co)homology are provide
by functors. Thus analyzing structures of functors should be
of great importance. A significant achievement in this direction
was obtained by Lück [51] and tom Dieck [] (1987),
where they classified simple and projective functors under
reasonable assumptions on the small category. An influential
approach to functor categories interestingly began with P.
Gabriel’s thesis [27] (1962). Given a small category, he made
it into an additive category by linearizing all morphism sets.
Then he showed that the category of addictive functors from
such an additive category to a module category is equivalent
to another module category, of modules over an algebra
that now we may call the additive category algebra of a small
category. Later on B. Mitchell [56] (1972) introduced a different
construction and defined an associative ring over a small
category. Consequently he established connections between
(R-mod) C and RC-mod. This is the result we pictured in the
diagram.
At this point, one perhaps does not see a connection between
Mitchell’s work and category (co)homology. These
two seemingly unrelated subjects were pieced up together
by P. J. Webb (around 2000) who named these rings appeared
in Mitchell’s paper, as “category algebras”. Representation
theory of categories generalizes both representation
theory of groups and of quivers, and thus is of great interest.
Representations of categories are important to category
(co)homology in the same way as group representations to
xi

xii
Preface
group (co)homology. From category algebra point of view, we
can put all necessary ingredients of category (co)homology
under one framework, as shown in the diagram. Furthermore
from the intrinsic structure of a category algebra, one can
see the similarities with and differences from a groups algebra.
Hence it explains why some classical results in group
(co)homology may be generalized to category (co)homology,
while others may not.
At the early stage of category (co)homology, it seemed
like a purely theoretic construction with few calculations.
The thrust of recent development was brought in by representation
and homotopy theorist working on locally determined
structures. Their work completely reshaped the whole
(co)homology theory and provided many interesting concrete
categories to work with. We shall comment on it in Chapter
6. In the last decade many interesting results on both abstract
and concrete small categories have been obtained. For
instance, a pivotal discovery is that the category of factorizations,
F (C), first used by Quillen to show homotopy equivalence
between BC and BC op , and then by Baues-Wirsching
to introduce their cohomology theory, possesses the property
that all (co)homology theories we consider here are indeed
(co)homology of F (C) with appropriate coefficients. However
when one tries to teach oneself about category (co)homology,
one finds materials scattered in the literature and different
writers have different background and styles. As an example
the existing introduction to this subject by Webb [80]
emphasizes its module-theoretic aspects. Lack of a comprehensive
treatment makes the theory daunting for whoever
wants to learn, and even for a researcher who uses category
(co)homology theory it may cause inconvenience. Thus
a book, introducing basic ideas of category (co)homology the-

Preface
ory, presenting standard techniques and addressing the interactions
between representation and homotopy theories, seems
necessary. Such a book should provide a clear view of basic
ideas, key methods, known results and unsolved conjectures,
being a handy introduction that can be used to foster further
investigations and to search for future applications.
Finally we turn to the structure of this book. The first
two chapters consists of preliminaries needed in category
(co)homology. It means one can find them in various classical
books. The reason why I collect them here are, firstly it
is convenient for the reader, secondly I try to provide some
concrete examples to illustrate many abstract constructions
which are in the center of this book. The first chapter recalls
some basic definitions from category theory. The main
focuses are limits of functors and their generalizations, that
is, the Kan extensions. The purpose of the second chapter
is to equip the reader with necessary knowledge about simplicial
methods. We begin with a review of chain complexes.
It is followed by an introduction to simplicial sets and the
nerve of a small category where many combinatorially constructed
chain complexes appear. Simplicial (co)homology is
defined and several examples are given. To tell how to compare
(co)homology of small categories, we have to inform the
reader how to compare small categories and their nerves as
well as classifying spaces. Thus some important categorical
constructions are provided, which are needed throughout this
book, for example to state Quillen’s Theorem A. For future
references, and for the interested reader, we end Chapter 2
with bisimplicial sets and several key results. Although only
the statements will be used in future, we nonetheless present
their proofs.
xiii

xiv
Preface
The third chapter introduces category algebras and their
representations. Examples are served at the beginning to motivate
the reader. We shall classify projective and injective
modules under mild assumptions. Moreover we give an intrinsic
characterization of category algebras so that they are
comparable with cocommutative bialgebras. It explains why
category algebras possess interesting homological properties.
The fourth chapter studies (ordinary) (co)homology of category
algebras. We begin with the most economical way by
using derived functors to define category (co)homology. Then
we recall Baues-Wirsching’s construction on the way to introduce
the bar resolution. The bar resolution is simplicially
constructed and it leads to various important modules by applying
Kan extensions on it. We will see they are the most
powerful tools for us. In this chapter we also define the extension
of a category by a group and the Grothendieck spectral
sequences.
The fifth chapter discusses Hochschild (co)homology of category
algebras. The key result here is a theorem to interpret
Hochschild (co)homology of category algebras by their ordinary
(co)homology, and vice versa. A theorem we prove here
shows that all the previously mentioned (co)homology theories
are just (co)homology of F (C) with coefficients. Some
examples are given to demonstrate explicit calculations.
The sixth chapter talks about connections between category
and group cohomology. It contains mostly unpublished
results. We bring up the notion of a local category of a finite
group. Local categories are the motivating cases for research
in category representations and cohomology. We put
it in the end because we do need techniques developed earlier.
This chapter contains many concrete categories and we
shall see clearly how one can compute using various abstract

Preface
machineries introduced before. A key concept in this chapter
is a transporter category. We show how closely related
are the transporter categories to the groups on which they
are defined. The most important result is perhaps the finite
generation of cohomology of modules of finite transporter categories.
Also we will construct transfer maps for ordinary and
Hochschild cohomology of transporter categories. This chapter
should help the reader to carry on further readings in
advanced research papers.
There may be many errors or even mistakes in this unfinished
manuscript, all of which are my responsibility.
xv
Universitat Autònoma de Barcelona,
June 2011
Fei Xu
xu@mat.uab.cat

Contents
1 Functors and their Kan extensions . . . . . . . . . . . 1
1.1Functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1Basic category theory . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2Restriction and Kan extensions . . . . . . . . . . . . . . . . . 21
1.2.1Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.2Overcategories and undercategories . . . . . . . . . . 23
1.2.3Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Simplicial methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1Complexes and homology . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1Chain complexes, homology and chain homotopy 32
2.1.2Double complexes and operations on chain
complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2Nerves, classifying spaces and cohomology . . . . . . . . 39
2.2.1Simplicial sets and nerves of small categories . . 39
2.2.2Classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2.3Cup product and cohomology ring . . . . . . . . . . . 62
2.3Quillen’s work on classifying spaces. . . . . . . . . . . . . . 67
2.3.1Quillen’s Theorem A . . . . . . . . . . . . . . . . . . . . . . . 67
2.3.2Constructions over categories and relevant
functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4Further categorical and simplicial constructions . . . 74

xviii
Contents
2.4.1Grothendieck constructions . . . . . . . . . . . . . . . . . 75
2.4.2Bisimplicial sets and homotopy colimits . . . . . . . 80
2.4.3Proofs of Quillen’s Theorem A and
Thomason’s theorem . . . . . . . . . . . . . . . . . . . . . . . 83
3 Category algebras and their representations . 89
3.1Basic concepts and examples . . . . . . . . . . . . . . . . . . . 89
3.1.1Category algebras . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.1.2Representations of categories and Mitchell’s
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.1.3Three examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.2A closed symmetric monoidal category . . . . . . . . . . . 96
3.2.1Tensor structure and an intrinsic
characterization of category algebras . . . . . . . . . 96
3.2.2The internal hom . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3Functors between module categories . . . . . . . . . . . . . 105
3.3.1Restriction on algebras and modules . . . . . . . . . 105
3.3.2Kan extensions of modules . . . . . . . . . . . . . . . . . . 108
3.3.3Dual modules and Kan extensions . . . . . . . . . . . 114
3.4EI categories, projectives and simples . . . . . . . . . . . . 116
3.4.1EI condition and its implications . . . . . . . . . . . . . 116
3.4.2Some representation theory . . . . . . . . . . . . . . . . . 118
3.4.3Classifications of projectives and simples . . . . . . 125
3.4.4Projective covers, injective hulls and their
restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4 Cohomology of categories and modules. . . . . . . 133
4.1General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.1.1Cohomology of modules . . . . . . . . . . . . . . . . . . . . 133
4.1.2Cohomology of a small category with
coefficients in a functor . . . . . . . . . . . . . . . . . . . . . 140

Contents
4.1.3Extensions of categories and low dimension
cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.2Classical methods for computation . . . . . . . . . . . . . . 157
4.2.1Minimal resolutions and reduction . . . . . . . . . . . 157
4.2.2Examples using classifying spaces . . . . . . . . . . . . 160
4.3Computation via adjoint functors . . . . . . . . . . . . . . . 162
4.3.1Adjoint restrictions . . . . . . . . . . . . . . . . . . . . . . . . 162
4.3.2Kan extensions of resolutions . . . . . . . . . . . . . . . . 164
4.4Grothendieck spectral sequences . . . . . . . . . . . . . . . . 170
4.4.1Grothendieck spectral sequences for a functor . . 170
4.4.2Spectral sequences of category extensions . . . . . 175
5 Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . 185
5.1Hochschild homology and cohomology . . . . . . . . . . . 185
5.1.1Definition and general properties. . . . . . . . . . . . . 185
5.1.2Ring homomorphisms from the Hochschild
cohomology ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.2Hochschild (co)homology of category algebras . . . . . 193
5.2.1Basic ideas and examples . . . . . . . . . . . . . . . . . . . 193
5.2.2Hochschild (co)homology as ordinary
(co)homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.2.3EI categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.3Examples of the Hochschild cohomology rings of
categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.3.1The category E 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.3.2The category E 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.3.3The category E 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.3.4The category E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6 Connections with group representations and
cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.1Local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
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Contents
6.1.1G-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.1.2Homology representations of kG . . . . . . . . . . . . . 220
6.1.3Transporter categories as Grothendieck
constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.1.4Local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.2Properties of local categories . . . . . . . . . . . . . . . . . . . 226
6.2.1Two diagrams of categories . . . . . . . . . . . . . . . . . 226
6.2.2Frobenius Reciprocity . . . . . . . . . . . . . . . . . . . . . . 227
6.3The functor π: group representations via
transporter categories . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.3.1Homology representations via transporter
categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.3.2On finite generation of cohomology . . . . . . . . . . . 238
6.3.3Transfer for ordinary cohomology . . . . . . . . . . . . 242
6.4The functor ρ: invariants and coinvariants . . . . . . . . 252
6.4.1Orbit categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.4.2Brauer categories, fusion and linking systems . . 255
6.4.3Puig categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
6.4.4Orbit categories of fusion systems . . . . . . . . . . . . 259
6.5Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . . . 260
6.5.1Finite generation . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6.5.2Transfer for Hochschild cohomology . . . . . . . . . . 265
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

List of symbols
Conventions: Categories are denoted by C, D in general unless
otherwise specified; x, y are objects; α, β are morphisms
in a category; F, G are general functors; u, v are functors between
small categories; natural transformations between two
functors are Φ, Ψ etc. Groups are written as G, H even when
they are regarded as categories with one object.
In a functor category we normally use M, N to denote the
objects.
• the trivial category of only one object and one morphism
Set the category of sets
T op the category of topological spaces
Cat the category of small categories
T C the category of all (if not specified, covariant) functors
from a small category C to a category T
Res u the restriction along a functor u
LK u , RK u the left and right Kan extensions along u
u/x, x\u categories over and under x ∈ Ob C, for a given
u : D → C
△ n the standard n-simplex
n the totally ordered set 0 < 1 < · · · < n, a combinatorial
construction of △ n
△ the category of standard simplicies

xxii
List of symbols
∆ the diagonal functor C → C × C and various maps
induced by it
C e = C × C op the enveloping category of C
F (C) the category of factorizations in C
∇ : F (C) → C e the skew diagonal functor
NC ∗ the nerve of C, a simplicial set
BC the classifying space of C, geometric realization of
NC ∗ , a CW complex
C ∗ , D ∗ , ... chain complexes
C ∗,∗ , D ∗,∗ , ... double complexes
Gr C M Grothendieck construction over M : C → Cat
G ∝ P transporter category over a G-poset P (a Grothendieck
construction)
R a commutative ring with identity
R-Mod the category of all R-modules
R-mod the category of finitely generated R-modules
k a field, usually algebraically closed
Vect k the category of all k-vector spaces
V ect k the category of finite-dimensional k-vector spaces
RC the R-category algebra of C
R the trivial RC-module, a constant functor
H n (C, R), Hn(C, R) the nth simplicial (co)homology of C with
coefficients in R
H n (C; M), Hn(C; M) the nth (co)homology of C with coefficients
in a functor/module M
H n (BC, R), Hn(BC, R) the nth singular (co)homology of BC
with coefficients in R
H ∗ (C, R) the simplicial cohomology ring
H ∗ (BC, R) the singular cohomology ring
Ext ∗ RC(R, R) the ordinary cohomology ring of RC
HH ∗ (RC) = Ext ∗ RCe(RC, RC) the Hochschild cohomology ring
of RC

Chapter 1
Functors and their Kan extensions
Abstract We recall some basics from category theory and homological
algebra. In this way we set conventions. The main
constructions we discuss here are limits and their generalizations,
the Kan extensions. We shall include some examples to
illustrate these abstract concepts.
1.1 Functors and limits
1.1.1 Basic category theory
In this section we recall some standard definitions and constructions.
One purpose is to settle the conventions and terminologies
that we shall follow in these notes.
Definition 1.1.1. A category C consists of a class of objects
Ob C and a class of morphisms Mor C such that
1. each morphism α ∈ Mor C has a source and a target, written
as s(α) and t(α), which are two objects in Ob C.
2. for any pair of object x, y ∈ Ob C,
Hom C (x, y) := {α ∣ ∣ s(α) = x and t(α) = y}
is a set.
3. for any α ∈ Hom C (x, y), β ∈ Hom C (y, z), their composite
exists and is written as β ◦ α ∈ Hom C (x, z);
4. for each object x there is an identity morphism 1 x ∈
Hom C (x, x) satisfying the condition that 1 x ◦ α = α and

2 1 Functors and their Kan extensions
β ◦ 1 x = β, if α ∈ Hom C (y, x) and β ∈ Hom C (x, z) for some
y, z ∈ Ob C;
5. if α ∈ Hom C (w, x), β ∈ Hom C (x, y) and γ ∈ Hom C (y, z),
then (γ ◦ β) ◦ α = γ ◦ (β ◦ α).
If α ∈ Hom C (x, y) is a morphism in C, very often we will picture
it as α : x → y. A morphism α ∈ Mor C is a monomorphism
if αβ = αγ for two morphisms β, γ ∈ Mor C, then
β = γ. A morphism α ∈ Mor C is an epimorphism if βα = γα
for two morphisms β, γ ∈ Mor C, then β = γ. A morphism
α ∈ Hom C (x, y) is an isomorphism if there exists a morphism
β ∈ Hom C (y, x) such that αβ = 1 y and βα = 1 x . An
isomorphism is always both monomorphic and epimorphic.
An isomorphism is also called an invertible morphism.
We say two objects x and y are isomorphic, written as x ∼ =
y, if there exists an isomorphism α ∈ Hom C (x, y). Let x be
an object in a category C. Then the isomorphism class of x
consists of all y ∈ Ob C that are isomorphic to x. The class
of objects isomorphic to x is denoted by [x] ⊂ Ob C.
A category is called a groupoid if every morphism is an
isomorphism.
An object x in C is initial if to any other object y, there
exists a unique morphism x → y. An object x in C is terminal
if to any other object y, there exists a unique morphism y →
x. An object is a zero object if it is both initial and terminal.
All initial (or terminal or zero) objects are isomorphic.
Suppose C has a zero object 0, then for any two objects
x, y we call the composite x → 0 → y the zero morphism
in Hom C (x, y), written as 0 x,y . Note that the composite of
a zero morphism with any other morphism is a zero morphism.
Let α ∈ Hom C (x, y) be a morphism. A kernel of α is
a morphism β ∈ Hom C (w, x) satisfying αβ = 0 w,y such that
if β ′ ∈ Hom C (w ′ , x) satisfies αβ ′ = 0 w ′ ,x then there exists a

1.1 Functors and limits 3
unique morphism µ : w ′ → w such that β ′ = βµ. A cokernel
of α is a morphism γ ∈ Hom C (y, z) satisfying γα = 0 x,z such
that if γ ′ ∈ Hom C (x, z ′ ) satisfies γ ′ α = 0 x,z
′ then there exists
a unique morphism ν : z → z ′ such that γ ′ = νγ.
Let I ⊂ Ob C be a subset. Then the coproduct of objects
in I, written as ⊕ I x i, is an object X ∈ Ob C such that each
x i ∈ I is equipped with a morphism α i : x i → X and if
Y is another object equipped with β i : x i → Y then there
exists a unique morphism γ : X → Y such that β i = γα i .
The product of objects in I, written as ∏ I x i, is an object
X ′ ∈ Ob C such that each x i ∈ I is equipped with a morphism
α ′ i : X → x i and if Y ′ is another object equipped with β ′ i :
Y → x i then there exists a unique morphism γ ′ : Y ′ → X ′
such that α ′ i = β ′ i γ′ .
Definition 1.1.2. A category C is called preadditive if every
Hom C (x, y) is an abelian group and compositions are bilinear.
A preadditive category is additive if furthermore all finite
coproducts and all products exists in C.
An additive category C is said to be preabelian if
1. it has a zero object;
2. every morphism has a kernel and a cokernel.
A preabelian category is abelian if every monomorphism is a
kernel of some morphism and every epimorphism is a cokernel
of some morphism.
There are abundance of categories in mathematics, many of
them are very natural while one can also cook up all kinds of
abstract categories.
Example 1.1.3.1. The trivial category • has exactly one object
• and one morphism 1 • ;
2. A group G gives rise to two categories: the first one has
only one object • and its morphism set is G; the sec-

4 1 Functors and their Kan extensions
ond, denoted by EG, has Ob EG = G and Mor EG =
⊎ g1 ,g 2 ∈Ob EGHom EG (g 1 , g 2 ) with Hom EG (g 1 , g 2 ) = {g 2 g1 −1 }.
The former will usually be written just as G while the latter
is called the Cayley graph of G;
3. A partially ordered set (poset in short) P is naturally a category,
still named P, if we let Ob P be the set of elements in
the poset and Mor P = ⊎ x,y∈Ob C Hom C (x, y) in which the set
Hom P (x, y) = {x ≤ y} if the two objects are comparable, or
empty otherwise. Posets can be characterized as categories
such that there exists at most one morphism between any
two objects.
4. The category of sets and set maps is denoted by Set, ;
5. Let R be a commutative ring with identity and A an associative
R-algebra. Then A-Mod and A-mod are the categories
of all left A-modules and finitely generated left A-modules;
6. The category Z-Mod is often written as Ab, the category of
abelian groups;
7. The category of topological spaces and continuous maps is
written as T op.
In these notes, for an R-algebra A, if we do not specify, any
A-module will be a left A-module.
Definition 1.1.4. A category C is a small category if Ob C
is a set, and is a finite category if Mor C is a finite set.
A finite category is necessarily small by Definition 1.1.1 (4).
In Example 1.1.3, the first three are small while the rest are
abelian and not small.
Definition 1.1.5. Let C be a category. Then a subcategory
D ⊂ C consists of a subclass of objects Ob D ⊂ Ob C and a
subclass of morphisms Mor D ⊂ Mor C, satisfying the axioms
of a category with composition laws inherited from C.

1.1 Functors and limits 5
A subcategory D ⊂ C if full, if for any pair of objects x, y ∈
Ob D, we always have Hom D (x, y) = Hom C (x, y).
Definition 1.1.6. A covariant functor F from a category
D to another category C assigns to each x ∈ Ob D an object
F(x) ∈ Ob C and to each α ∈ Hom D (x, y) a morphism F(α) ∈
Hom C (F(x), F(y)) satisfying the conditions that
1. F(1 x ) = 1 F(x) for every x ∈ Ob D;
2. if α, β ∈ Mor D and β◦α exists, then F(β◦α) = F(β)◦F(α).
A contravariant functor F ′ from a category D to another
category C assigns to each x ∈ Ob D an object F ′ (x) ∈
Ob C and to each α ∈ Hom D (x, y) a morphism F ′ (α) ∈
Hom C (F ′ (x), F ′ (y)) satisfying the conditions that
1 ′ . F ′ (1 x ) = 1 F ′ (x) for every x ∈ Ob D;
2 ′ . if α, β ∈ Mor C and β◦α exists, then F ′ (β◦α) = F ′ (α)◦F ′ (β).
Example 1.1.7. Let C be a category and x ∈ Ob C an object.
Then
1. there exists a covariant functor Hom C (x, −) : C → Set such
that for any y ∈ Ob C, Hom C (x, −)(y) := Hom C (x, y) and
for any α ∈ Hom C (y, y ′ ) the composition Hom C (x, −)(α) =
α ◦ − : Hom C (x, y) → Hom C (x, y ′ ); and
2. there exists a contravariant functor Hom C (−, x) : C → Set
such that for any y ∈ Ob C, Hom C (−, x)(y) := Hom C (y, x)
and for any α ∈ Hom C (y, y ′ ) the composition Hom C (−, x)(α) =
− ◦ α : Hom C (y ′ , x) → Hom C (y, x).
An object x ∈ Ob C is projective if Hom C (x, −) : C → Set
preserves epimorphisms. An object x ∈ Ob C is injective if
Hom C (−, x) : C → Set preserves monomorphisms. If C is an
abelian category, we say that C has enough projective objects
if for any y ∈ Ob C there exists a projective object x along
with an epimorphism x → y and that C has enough injective

6 1 Functors and their Kan extensions
objects if for any y ∈ Ob C there exists an injective object x
along with a monomorphism y → x.
Definition 1.1.8. Suppose C is a category. Its opposite category,
named C op , share the same objects with C. Each α ∈
Hom C (x, y) defines a unique morphism α op ∈ Hom C op(y, x). If
α op ∈ Hom C op(y, x) and β op ∈ Hom C op(z, y), their composite
is α op ◦ β op := (β ◦ α) op ∈ Hom C op(z, x).
Passing from a category to its opposite has a dualizing effect
on many categorical concepts, constructions and properties.
For instance, it interchanges initial objects with terminal objects,
monomorphisms with epimorphisms, projective objects
with injective objects, products with coproducts etc. Moreover
one can readily verify that a covariant functor F : C → D
naturally determines a contravariant functor, the dual functor,
F ∧ : C op → D, and vice versa. Thus what we learn about
covariant functors can be directly translated to contravariant
functors. From now on, if not specified, all functors will be
covariant in these notes.
Definition 1.1.9. A functor F : D → C is full if F(D) ⊂ C
is a full subcategory.
A functor F : D → C is faithful if for any α, β ∈ Mor D
such that F(α) = F(β) ∈ Mor C, then α = β.
If D and C are two preadditive categories, then a functor F :
D → C is called additive if Hom D (x, y) → Hom C (F(x), F(y))
is a homomorphism between abelian groups.
Definition 1.1.10. Suppose F, G : D → C are two functors.
A natural transformation Φ : F → G assigns to each object
x ∈ Ob D a morphism Φ x : F(x) → G(x) so that we have a
commutative diagram

1.1 Functors and limits 7
F(x) Φ x
G(x)
F(α)
G(α)
F(y) Φy
G(y)
for any given α ∈ Hom D (x, y). If every Φ x is an isomorphism
in C, we call such Φ a natural equivalence and write F ∼ = G
in this situation.
Definition 1.1.11. Two categories C and D are equivalent
, written as C ≃ D, if there exist functors F : C → D and
G : D → C such that FG ∼ = Id D and GF ∼ = Id C .
Definition 1.1.12. Let C be a category. If we take exactly
one object from each isomorphism class of objects in C, then
we can form a full subcategory consisting of these chosen objects.
This subcategory is called a skeleton of C.
Since skeletons of a category C are naturally equivalent to
each other, we can speak about the skeleton of C. We shall
always denote by [C] the skeleton of a category C. Then one
can verify that [C] ≃ C.
It is a fact that when D is small all the covariant functors
from D to C form a category whose objects are these functors
and morphisms are the natural transformations. We call it a
functor category, written as C D . Sometimes we call such a
D an index category. Functor categories are of pivotal importance
in these notes. We often consider functor categories
whose index categories are finite (such as a finite group G)
and whose target categories are large (such as R-mod), in order
to understand finite categories via their representations
(see Chapter 3) in certain large categories.
The functors in Example 1.1.7 are very important to us.
Here we present a crucial property of those functors.

8 1 Functors and their Kan extensions
Definition 1.1.13. A covariant (respectively, contravariant)
functor F : C → Set is called representable if it is naturally
equivalent to Hom C (x, −) (respectively, Hom C (−, x)) for some
x ∈ Ob C.
Lemma 1.1.14 (Yoneda Lemma). Let F : C → Set be a
functor. Then we have
for any x ∈ Ob C.
Hom Set
C(Hom C (x, −), F) ∼ = F(x),
Proof. Each natural transformation Φ in the left side is uniquely
determined by the morphism
Φ x : Hom C (x, x) → F(x)
which is uniquely determined by the image of Φ x (1 x ) ∈ F(x).
Hence we can define a bijection Φ → Φ x (1 x ) and then the
statement follows.
⊓⊔
In the end, we record a construction that we will use in the
next section.
Definition 1.1.15. Suppose C and D are two categories.
Then we can define their product category C × D, whose objects
are {(x, y) ∣ x ∈ Ob C, y ∈ Ob D} and whose morphisms
are {(α, β) ∣ α ∈ Mor C, β ∈ Mor D}.
A functor from a product category C × D to some target
category sometimes is called a bifunctor.
1.1.2 Adjoint functors
We take the opportunity to record the definition of an adjoint
functor and its basic properties for future reference.
Definition 1.1.16. Let F : C → D and G : D → C be two
functors such that there is a natural equivalence

1.1 Functors and limits 9
Ω : Hom D (F(−), −) ∼= →Hom C (−, G(−))
of functors C op ×D → Set. We say that F is left adjoint to G,
G is right adjoint to F and write Ω : F ⊣ G for the adjunct.
Example 1.1.17. If F : C → D and G : D → C give rise to
category equivalences as in Definition 1.1.11, then F is both
a left and a right adjoint of G.
Another example is for U, V, W ∈ V ect k , there is an isomorphism
Hom k (U ⊗ k V, W ) ∼ = Hom k (U, Hom k (V, W )).
Next we deduce some fundamental properties of the adjunction.
1. Naturality of Ω: for any α : x ′ → x and β : y → y ′ the
following diagram commutes
Hom D (F(x), y) Ω x,y
Hom C (x, G(y))
β◦(−)◦F(α)
G(β)◦(−)◦α
Hom D (F(x ′ ), y ′ )
Ωx
Hom
′ ,y ′ C (x ′ , G(y ′ )).
Equivalently, we have for any ϕ : F(x) → y
Ω x ′ ,y ′ (β ◦ ϕ ◦ F(α)) = G(β) ◦ Ω x,y (ϕ) ◦ α.
We shall repeatedly use the above formula. Very often, the
subscript of Ω will be omitted for the sake of convenience.
For instance, when x ′ = x and α = 1 x , we have
Ω(β ◦ ϕ) = G(β) ◦ Ω(ϕ),
and when y = F(x) and ϕ = 1 F(x) , we get
Ω(β ◦ F(α)) = G(β) ◦ α.

10 1 Functors and their Kan extensions
One should try to deduce similar formulas for future applications.
2. Unit and counit: we define, for each x ∈ Ob C, Σ x =
Ω(1 F(x) ) : x → GF(x) and, for each y ∈ Ob D, Λ y =
Ω −1 (1 G(y) ) : FG(y) → y. By naturality of Ω, we actually
obtain two natural transformations
Σ : Id C → GF and Λ : FG → Id D .
For example, for any α : x ′ → x, using various formulas we
obtain from the naturality of Ω, we get
GF(α) ◦ Σ x
′ = GF(α) ◦ Ω(1 F(x ′ ))
= Ω(F(α) ◦ 1 F(x ′ ))
= Ω(1 F(x) ◦ F(α))
= Ω(1 F(x) ) ◦ α
= Σ x ◦ α.
Thus we have a commutative diagram
x ′
α
x
Σ x ′
GF(x ′ )
Σx
GF(α)
GF(x).
We can deduce that the following
F FΣ
−→FGF ΛF
−→F,
G ΣG
−→GFG GΛ
−→G
compose to identities. For example, given some x ∈ Ob C,
one can verify Λ F(x) ◦ F(Σ x ) = 1 F(x) by showing
Ω(Λ F(x) ◦ F(Σ x )) = Ω(Λ F(x) ) ◦ Σ x = Σ x = Ω(1 F(x) ).
Moreover one can recover the adjunct by unit and counit:
for any ϕ : F(x) → y and ψ : x → G(y)
Ω(ϕ) = G(ϕ) ◦ Σ x
and Ω −1 (ψ) = Λ y ◦ F(ψ).

1.1 Functors and limits 11
Theorem 1.1.18. Suppose F : C → D and G : D → C
are two functors. If there exist two natural transformations
Σ : Id C → GF and Λ : FG → Id D such that the following
two composites
F FΣ
−→FGF ΛF
−→F,
G ΣG
−→GFG GΛ
−→G
are identities, then F is a left adjoint of G. Moreover
Ω : Hom D (F(x), y) → Hom C (x, G(y)) defined by Ω(ϕ) =
G(ϕ) ◦ Σ x is the adjunct, with Σ, Λ the unit and counit of
the adjunction.
Proof. For x ∈ Ob C, y ∈ Ob D, φ ∈ Hom D (F(x), y) and
ψ ∈ Hom C (x, G(y)), we define
Ω x,y (φ) = G(φ) ◦ Σ x and ¯Ωx,y (ψ) = Λ y ◦ F(ψ).
Then we can verify that they define two natural transformations
of functors C op × D → Sets. In fact for any α ∈
Hom C (x, x ′ ), β ∈ Hom D (y, y ′ ) and φ ∈ Hom D (F(x), y) we
have
G(β) ◦ Ω(φ) ◦ α = G(β) ◦ G(φ) ◦ Σ x ◦ α
= G(β ◦ φ) ◦ GF(α) ◦ Σ x
′
= G(β ◦ φ ◦ F(α)) ◦ Σ x
′
= Ω(β ◦ φ ◦ F(α)),
which implies Ω is a natural transformation. The assertion
for ¯Ω is proved in a similar way.
Moreover Ω and ¯Ω provide natural equivalences because we
can show
( ¯Ω ◦ Ω)(φ) = ¯Ω(G(φ) ◦ Σx )
= Λ y ◦ F(G(φ) ◦ Σ x )
= Λ y ◦ FG(φ) ◦ F(Σ x )
= φ ◦ Λ F(x) F(Σ x )
= φ,

12 1 Functors and their Kan extensions
and similarly (Ω ◦ ¯Ω)(ψ) = ψ.
This theorem also implies that the adjoint of a functor is
unique up to natural equivalence.
Corollary 1.1.19. If G and G ′ are right adjoints of F, then
they are naturally equivalent. Similarly if F and F ′ are two
left adjoints of G, then they are naturally equivalent.
Proof. We only prove the first assertion. For any x ∈ Ob C
and y ∈ Ob D we have two isomorphisms
Hom C (x, G(y)) Ω
←−Hom D (F(x), y) Ω′
−→Hom C (x, G ′ (y)).
Then (Ω◦Ω ′−1 )(1 G (y)) : ′ G ′ (y) → G(y) and (Ω ′ ◦Ω −1 )(1 G(y) ) :
G(y) → G ′ (y) induce a natural equivalence between G and
G ′ .
⊓⊔
⊓⊔
1.1.3 Limits
In these notes, direct and inverse limits of functors are key
concepts so we introduce their definitions here. Suppose M ∈
Ob(T C ) is an object in a functor category. Then it can be
identified with a commutative diagram of objects {M(x) ∣ ∣
x ∈ Ob C} and morphisms {M(α) ∣ ∣ α ∈ Mor C} in T . For
convenience and future reference, we call this the diagram of
M.
Definition 1.1.20. Let C be a small category and T an arbitrary
category. Consider the functor category T C . Suppose
M ∈ Ob(T C ). Then it has an inverse limit, denoted by
lim M ∈ Ob T , if for each x ∈ Ob C there exists a morphism
θ x : ←−C
←−C
lim M → M(x) such that, by adding the object
lim M and the morphisms {θ ∣
←−C
x x ∈ Ob C} to the diagram
of M, we obtain an enlarged commutative diagram, and this

1.1 Functors and limits 13
object lim ←−C
M is universal in the sense that if t is another object
in Ob T that enjoys the same properties as lim ←−C
M then
we must have a unique morphism Θ t : t → lim ←−C
M making
the whole diagram commutative as pictured as follows
t
θ ′ y
Θ t
θ ′ x
lim M θ x
←−C
θ y
M(α)
M(x)
M(y) .
Dually the functor M has a direct limit in T , denoted by
lim M ∈ Ob T , if for each x ∈ Ob C there exists a morphism
−→C
τ x : M(x) → lim ←−C
M such that, by adding the object −→C
lim M
∣
and the morphisms {τ x x ∈ Ob C} to the diagram of M,
we obtain an enlarged commutative diagram, and this object
lim M is universal in the sense that if t is another object in
−→C
Ob T that enjoys the same properties as lim −→C
M then we must
have a unique morphism Ξ t : −→C
lim M → t making the whole
diagram commutative as pictured as follows
M(α)
M(y) τy
M(x)
τ ′ y
τ x
τ ′ x
lim
−→C M Ξ t
t
It follows directly from the definition that, when a limit
exists, it is unique up to isomorphism. If either the index
category or the functor itself is structurally simple, we may
explicitly compute the limits.

14 1 Functors and their Kan extensions
Example 1.1.21.1. Let C = Z and T = Set. Then a functor
M : Z → Set is represented by a chain of set maps
· · · → M(−1) → M(0) → M(1) → M(2) → · · · .
In particular if M 0 is given by the following diagram in which
each map is an inclusion
· · · ⊂ M 0 (−1) ⊂ M 0 (0) ⊂ M 0 (1) ⊂ M 0 (2) ⊂ · · · ,
then lim ←−Z
M 0 = ⋂ i∈Z M 0(i) and −→Z
lim M 0 = ⋃ i∈Z M 0(i).
2. If C = Z and T = V ect k , the category of finite-dimensional
k-vector spaces. Suppose N is represented by the following
· · · → 0 N(1) N(2) N(3) · · · .
Then −→Z
lim N does not exist in V ect k , the category of finitedimensional
k-vector spaces, but lim ←−Z
N = {0}.
3. Let T be a category and I a set considered as a discrete
category. Then a functor M : I → T is simply an I-indexed
set of objects in T . The the coproduct of these objects is
defined as ⊕ I x i = −→I
lim M and the product of these objects
is ∏ I x i = ←−I
lim M.
4. Let C be the following poset
x
α
y β
z
and T = Vect k , the category of all k-vector spaces. Each
functor M ∈ Ob T C is represented by
M(y) M(β)
M(x)
M(α)
M(z).

1.1 Functors and limits 15
Then ←−C
lim M is called the pullback of the latter diagram and
lim M = M(z).
−→C
Dually if D is the following poset
a f
g
c
and N ∈ Ob T D then −→D
lim N is called the pushout of the
diagram
N(a)
N(g)
N(c)
b
N(f)
N(b)
and lim ←−D
N = N(a).
The definition of limits can be rewritten by using a simple,
yet very important, construction.
Definition 1.1.22. There is a constant functor
K : T → T C
such that, for any t ∈ Ob T , K(t) is defined by K(t)(x) = t
and K(α) = 1 t for any x ∈ Ob C and α ∈ Mor C.
In the literature, the functor K : T → T C is often named the
“diagonal functor”. Since the terminology is used for another
purpose, see Definition 2.2.34, we shall stick with our notion
which seems to be more appropriate.
Now we present alternative characterizations of limits. The
definition of an inverse limit can be rephrased as saying that
there exists lim ←−C
M ∈ Ob C with a natural transformation
Γ : K(lim ←−C
M) → M which is universal in the sense that if
there is another object t, along with a natural transformation
Υ : K(t) → M, then there exists a unique morphism θ t :

16 1 Functors and their Kan extensions
t → lim ←−C
M hence a natural transformation K(θ t ) : K(t) →
K(lim ←−C
M) such that Γ ◦ K(θ t ) = Υ .
K(θ t )
K(t)
K(lim ←−C
M) Γ
Especially we obtain a morphism
Ω : Hom T C(K(t), M) → Hom T (t, lim ←−C
M),
given by Ω(Υ ) = θ t . It is an isomorphism because each t →
lim M extends to a functor K(t) → M. Thus if every M ∈
←−C
Ob(T C ) has a limit in T , then lim ←−
: T C → T is the right
adjoint of K and Ω becomes the corresponding adjunct. We
characterize the direct limit lim −→C
in a similar way but we leave
it to the reader. When T is “large” enough with respect to C,
we can introduce the limits in an economical way.
Proposition 1.1.23. If K has a right adjoint R : T C → T ,
then for each M ∈ Ob(T C ), R(M) is the inverse limit of
M. If K has a left adjoint L : T C → T , then for each
M ∈ Ob(T C ), L(M) is the direct limit of M.
For convenience, we introduce the following concepts.
Definition 1.1.24. A category T is called complete if for
any small category C and any functor M ∈ T C the inverse
limit lim ←−C
M exists. A category T is called cocomplete if for
any small category C and any functor M ∈ T C the direct
limit lim −→C
M exists.
A category T is called finitely complete (respectively finitely
cocomplete) if for any finite category C and any functor
Υ
M

1.1 Functors and limits 17
M ∈ T C the inverse limit ←−C
lim M (respectively the direct limit
lim M) exists.
−→C
The categories Set and R-Mod are both complete and cocomplete.
By Example 1.1.21, a coproduct is a direct limit and a product
in an inverse limit. In abelian categories, the existence of
coproducts, respectively product, is equivalent to the cocompleteness,
respectively completeness, condition on a category
T .
Proposition 1.1.25. Let T be an abelian category. Then
1. it is complete (or finitely complete) if and only if all products
(or all finite products) exist; and
2. it is cocomplete (or finitely cocomplete) if and only if all
coproducts (or all finite coproducts) exist.
Proof. We only prove (1). Since a product is an inverse limit,
we just have to demonstrate the opposite direction. Suppose
C is a small (or finite) category and M : C → T is a functor.
Let
C 0 (C; M) = {f : Ob C →
∏
M(x) ∣ f(x) ∈ M(x)}
and
C 1 (C; M) = {f : Mor C →
x∈Ob C
∏
α∈Mor C
We define δ : C 0 (C; M) → C 1 (C; M) by
M(y) ∣ ∣ f(x α →y) ∈ M(y)}.
δ(f)(x α →y) = f(y) − M(α)[f(x)].
Then we can verify that the kernel of δ is ←−C
lim M. ⊓⊔
For example the category of finitely generated R-modules,
R-mod, is both finitely complete and finitely cocomplete.

18 1 Functors and their Kan extensions
Completeness and cocompleteness pass to functor categories.
We record this and another result that shall be useful in the
next chapter.
Proposition 1.1.26. Let C be a small category. If T is complete
(resp. cocomplete), then T C is complete (resp. cocomplete).
Proof. Let D be another small category and M : D → T C
a functor. Since (T C ) D ∼ = T C×D , M can be identified with a
bifunctor from ˜M(−, −) : C × D → T .
We may define a functor N : C → T by N(x) = ←−D
lim ˜M(x, −)
for every x ∈ Ob C. Since [lim ←−D ˜M(−, −)](x) ∼ = lim ←−D
[ ˜M(x, −)]
for all x ∈ Ob D, we can show N is the limit ←−D
lim M. It means
T C is complete. The cocompleteness of T C can be proved in
the same way.
⊓⊔
The above result tells us that Set C and (R-Mod) C are both
complete and cocomplete if C is small. Furthermore (R-mod) C
is finitely complete and cocomplete if C is finite.
We record two results on preserving limits.
Proposition 1.1.27. Let T be both complete and cocomplete
and C a small category. Given a functor M ∈ Ob T C
and T ∈ Ob T , we have
Hom T (T, lim ←−C
M) ∼ = lim ←−C
Hom T (T, M(−))
and
Hom T (lim −→C
M, T ) ∼ = lim ←−C
Hom T (M(−), T )
Proof. To prove the first isomorphism, we take the diagram of
M, which determines lim ←−C
M. Applying the covariant functor
Hom T (T, −) to the diagram it results in the diagram of the
functor Hom T (T, M(−)). From here one may complete the

1.1 Functors and limits 19
proof based by using the universal property of an inverse limit.
The second isomorphism may be proved in the same way. ⊓⊔
Proposition 1.1.28. Let F : T → T ′ be the left adjoint of
G : T ′ → T .
1. Suppose M : C → T has a direct limit. Then F(lim −→C
M) ∼ =
lim F ◦ M. −→C
2. Suppose N : D → T ′ has an inverse limit. Then G(lim ←−D
N) ∼ =
lim G ◦ N. ←−D
Proof. We only prove (1). It is obvious that F(lim −→C
M) fits into
the defining diagram of −→C
lim F◦M. We need to show F(lim −→C
M)
is universal. Suppose t ′ ∈ Ob T ′ is another object satisfying
the colimit defining diagram. It follows from the adjunction
Hom T ′(F(lim −→C
M), t ′ ) ∼ = Hom T (lim −→C
M, G(t ′ )),
along with the unit Id → F◦G that G(t ′ ) satisfies the defining
diagram of −→C
lim M. From the universal property of −→C
lim M,
there exists a unique morphism lim −→C
M → G(t ′ ) which gives
rise to a morphism F(lim −→C
M) → t ′ . The universal property
of F(lim −→C
M) follows from it.
⊓⊔
In the end, we give a result that is used in the next section.
Theorem 1.1.29. Let C be a small category. Then any
functor M : C → Set is canonically a colimit of a diagram
of representable functors.
Proof. We shall establish this result by firstly constructing a
small category D from M : C → Set and secondly showing
that M induces a functor ˜M : D → Set C satisfying −→D
lim ˜M ∼ =
M.
The objects in the category D are pairs (x, a) such that x
is an object of C and a is an element in the set M(x). A

20 1 Functors and their Kan extensions
morphism f : (x, a) → (y, b) is a morphism f ∈ Hom C (x, y)
satisfying M(f)(a) = b.
We define a functor ˜M : D → Set C by ˜M(x, a) =
Hom C (x, −). From the Yoneda Lemma, there exists for each
x ∈ Ob C an isomorphism of sets Hom Set
C(Hom C (x, −), M) ∼ =
M(x). It implies that each m x ∈ M(x) determines uniquely
a functor f (x,mx ) : ˜M(x, mx ) → M making the enlarged diagram
of ˜M commutative
Hom C (x, −) = ˜M(x, mx )
f◦−
˜M(y, m y ) = Hom C (y, −)
f (x,m x)
M
f (y,m y)
Note that the commutativity of the diagram forces M(f)(m x ) =
m y .
In order to prove that M is the direct limit we need to
show it is universal. Suppose L is another functor fitting
into the above diagram, and for each (x, m x ) ∈ Ob D it
comes with a functor g (x,mx ) : Hom C (x, −) → L. Again by
the Yoneda Lemma, g (x,mx ) determines uniquely an element
l x ∈ L(x). From the commutativity of the diagram, we must
have L(f)(l x ) = l y . Now we define a functor h : M → L by
h z (m z ) = l z
if z ∈ Ob C. We may readily verify that h z ′M(f) = L(f)h z
for any f : z → z ′ . It implies that h is well defined. Since by
definition hf (x,mx ) = g (x,mx ) for all objects (x, m x ) ∈ Ob D, M
is universal.
⊓⊔
The construction of D from a functor to Set in the proof is
the predecessor of the Grothendieck constructions introduced
in Section 2.4.

1.2 Restriction and Kan extensions 21
1.2 Restriction and Kan extensions
In last section we mentioned that, for a functor category T C ,
if the target category are complete and cocomplete, then we
may define the limits using adjoints of the constant functor
K : T → T C . When we examine closely, we realize that K itself
is induced by another functor pt : C → •. This observation
generates new ideas for constructing some extremely powerful
functors, called Kan extensions.
1.2.1 Restriction
Definition 1.2.1. Suppose u : D → C is a functor between
two small categories. For any target category T , u induces a
functor Res u : T C → T D via precomposition with u, called
the restriction along u.
Example 1.2.2. The canonical functor pt : C → • induces a
restriction T ∼ = T • → T C , which is identical to K of Definition
1.1.22.
In last section, we know explicitly the left and right adjoints
of K. As a generalization, we shall describe the adjoints of
an arbitrary restriction. But before doing that, we provide
several simple properties of the restriction.
Lemma 1.2.3. Suppose u : D → C is a functor. Then for
any M ∈ Ob(T C ), there are canonical morphisms lim ←−C
M →
lim Res ←−D uM and lim −→D
Res u M → lim −→C
M.
Proof. The map between inverse limits follows from the universal
property

22 1 Functors and their Kan extensions
lim M ←−C α x
Θ
lim Res φ x
α y ←−D uM
Res u M(a) = M(u(a))
φ y
Res u M(φ)=M(u(φ))
Res u M(b) = M(u(b)) ,
in which φ : a → b is a morphism in D. The map between
direct limits can be established similarly.
⊓⊔
Proposition 1.2.4. Suppose u : D → C has a left adjoint
v : C → D. Then for any T , Res u has a right adjoint Res v .
In particular, if u and v are category equivalences, Res u and
Res v are equivalences.
If T is complete and cocomplete, then for any M ∈
Ob(T C ), lim −→C
M ∼ = lim −→D
Res u M, and for any N ∈ Ob(T ) D ,
lim N ∼ = lim
←−D ←−C
Res v N.
Proof. We shall prove that for M ∈ Ob(T C ) and N ∈ Ob(T D )
there exists an isomorphism
Hom T D(Res u M, N) ∼ = Hom T C(M, Res v N).
Let Σ : Id C → uv and Λ : vu → Id D be the unit and counit
for the adjunction between u and v. We define two natural
transformations Σ ′ : Id T
C → Res v Res u and Λ ′ : Res u Res v →
Id T
D as follows. Given M ∈ Ob(T C ), x ∈ Ob C, N ∈ Ob(T D )
and a ∈ Ob D,
(Σ ′ M) x := M(Σ x ) : M(x) → Res v Res u M(x) = M(uv(x)),
and
(Λ ′ N) a := N(Λ a ) : Res u Res v N(a) = N(vu(a)) → N(a).
One can verify that Σ ′ and Λ ′ provide the unit and counit of
an adjunction.

1.2 Restriction and Kan extensions 23
As for the second statement, we have for all t ∈ Ob T
and
Hom T (lim −→D
Res u M, t) ∼ = Hom T D(Res u M, K(t))
∼ = HomT C(M, Res v K(t))
∼ = HomT C(M, K(t))
∼ = HomT (lim −→C
M, t).
Hom T (t, lim ←−C
Res v N) ∼ = Hom T C(K(t), Res v N)
∼ = HomT D(Res u K(t), N)
∼ = HomT D(K(t), N)
∼ = HomT (t, lim ←−D
N).
Thus −→D
lim Res u M ∼ = lim −→C
M and ←−C
lim Res v N ∼ = lim ←−D
N. ⊓⊔
In many places we will have to consider the adjoint functors
of some restriction Res u . When u has an adjoint, we get an
adjoint of Res u which is also a restriction, by Proposition 1.2.4.
The truth is that even if u does not admit an adjoint, we can
still construct the adjoints of Res u , and this is the main result
of the upcoming two sections.
1.2.2 Overcategories and undercategories
In order to introduce the adjoints of a restriction, we have
to provide some important categorical constructions. These
categorical constructions are of great importance in both homological
algebra and homotopy theory of classifying spaces.
We shall be familiar with them as they appear almost everywhere
throughout these notes.
Definition 1.2.5. Let u : D → C be a functor between
(small) categories and x ∈ Ob C. The category over x, u/x,
consists of objects {(a, α) ∣ ∣ a ∈ Ob D, α ∈ Hom C (u(a), x)}.
For any two objects (a, α), (b, β), a morphism from (a, α) to

24 1 Functors and their Kan extensions
(b, β) is a morphism µ ∈ Hom D (a, b) making the following
diagram commutative
u(a)
u(µ)
u(b)
The category under x, written as x\u, is defined in a dual
fashion. It consists of objects {(α, a) ∣ ∣ a ∈ Ob D, α ∈
Hom C (x, u(a))}. For any two objects (α, a), (β, b), a morphism
from (α, a) to (β, b) is a morphism µ ∈ Hom D (a, b)
making the following diagram commutative
x
α
β
α
β
u(a)
x,
u(µ)
u(b)
The categories defined above are customarily called overcategories
and undercategories, associated with u : D → C. We
will see later on that Id C /x and x\Id C , for any x ∈ Ob C, are
already very interesting.
Remark 1.2.6.1. From definition, an object in the overcategory
u/x, (a, α), can be pictured as u(a) →x, α and consequently
a morphism µ : (a, α) → (b, β) can be equivalently
interpreted as a sequence u(a) u(µ)
→ u(b) →x. β This kind
of rewritings will be useful for us when dealing with chains
of morphisms in u/x and we shall come back to this point
in later chapters. Similar reinterpretation can be made for

1.2 Restriction and Kan extensions 25
objects and morphisms in undercategories too but we leave
it to the reader.
2. There is a canonical functor P x : u/x → D (resp. P x :
x\u → D given on objects as projection to the first (resp.
the second) component and on morphisms as the identity.
For simplicity, we shall denote such functors just as P.
3. If γ : x → y is a morphism in C, then it naturally induces a
functor γ ∗ : u/x → u/y and a functor γ ∗ : y\u → x\u.
Example 1.2.7.1. Let pt : C → • be the canonical functor.
Then pt/• ∼ = •\pt ∼ = C.
2. Let G be a group and H a subgroup. Then the inclusion
functor i H : H ↩→ G gives exactly one overcategory i H /•.
By direct calculation, the objects are {(•, g)|g ∈ G}, and
biject with the elements of G. There is a morphism from
one object (•, g 1 ) to another (•, g 2 ) if there exists a h ∈ H
such that g 1 = g 2 h. Obviously h = g2 −1 g 1. Thus there is at
most one morphism from an object to another. Since h is
invertible, there exists a morphism between two objects if
and only if their are isomorphic in i H /•. In other words, two
objects (•, g 1 ) to another (•, g 2 ) are isomorphic if and only
if g 1 H = g 2 H. Because the category i H /• consists of [G : H]
many groupoids, each of which is equivalent to the trivial
category, i H /• is equivalent to the discrete set G/H of left
cosets (regarded as a category).
Similarly the undercategory •\i H has objects {(g, •)|g ∈
G}. There is a morphism from (g 1 , •) to (g 2 , •) if and only
if there exists a (unique) h ∈ H such that hg 1 = g 2 or
equivalently h = g 2 g1 −1 . The undercategory is equivalent
to H\G, the set of right cosets. We have an isomorphism
i H /• → •\i H given by
(•, g) ↦→ (g −1 , •) and (•, g 1 ) g−1 2 g 1
→ (•, g 2 ) ↦→ (g −1
1 , •)g−1
2 g 1
→ (g2 −1 , •).

26 1 Functors and their Kan extensions
When G = H, i G /• = Id G /• ∼ = •\Id G = •\i G is the Cayley
graph of G.
Based on our observations, by Proposition 1.2.4, if M ∈
Ob(V ect H k
) (commonly called a k-representation of H),
lim M ∼ −→i H
= lim
/• −→G/H
M ∼ = ⊕ g∈G/H gH ⊗ k M. One can see that
G permutes these direct summands and the limit lim M −→iH /•
is isomorphic to kG ⊗ kH M.
3. Suppose u : D → C is a functor between two posets. Then
for any x ∈ Ob C, u/x is isomorphic to the subposet of
D consisting of objects {a ∈ Ob D ∣ Hom C (u(a), x) ≠ ∅},
while x\u is isomorphic to the subposet of D consisting of
objects {b ∈ Ob D ∣ Hom C (x, u(b)) ≠ ∅}.
The following observation follows directly from definitions
and will be useful to us. For any functor u : D → C one can
define a (covariant) opposite functor u op : D op → C op such
that u op (x) = x and u op (α op ) = u(α) op . Be aware that it is
different from the dual functor given before Definition 1.1.9.
Lemma 1.2.8. Suppose u : D → C is a functor. Consider
its opposite functor u op : D op → C op . Then for any
x ∈ Ob C = Ob C op we have (u/x) op ∼ = x\u op and (x\u) op ∼ =
u op /x.
1.2.3 Kan extensions
In this section, we assume T to be a complete and cocomplete
abelian category. The reader should bear Example 1.2.7 (1)
in mind in order to see that Kan extensions generalize direct
and inverse limits.
Theorem 1.2.9. Let u : D → C be a functor between small
categories. Then the restriction Res u : T C → T D admits a
left adjoint LK u , called the left Kan extension along u, as

1.2 Restriction and Kan extensions 27
well as a right adjoint RK u , called the right Kan extension
along u.
Proof. We only sketch the constructions of the Kan extensions
and leave details to be filled by the reader.
Given M ∈ Ob(T D ) we define its left and right Kan extensions
along u as
LK u M = lim −→u/−
Res P M
and RK u M = lim ←−−\u
Res P M,
where P is the functor in Remark 1.2.6 (2). Here we only
prove the statement for the left Kan extension because the
proof for the right Kan extension follows the same pattern.
Step 1, we show LK u M is a functor from C to T . If
γ ∈ Hom C (x, y), then we have a functor γ ∗ : u/x → u/y. Since
Res γ∗ Res P M = Res P M as functors over u/x, by Lemma
1.2.3, it determines a canonical morphism
lim
−→ γ ∗ : LK u M(x) = −→u/x
lim Res P M → LK u M(y) = −→u/y
lim Res P M.
Hence LK u M = lim −→u/−
Res P M ∈ Ob(T C ).
Step 2, we state that LK u is a functor from T D to T C . For
any natural transformation Ψ : M → M ′ between two objects
of T D . We can use the universal property to build a canonical
natural transformation LK u Ψ : LK τ M → LK u M ′ .
Step 3, we construct an adjunct
Ω : Hom T C(LK u M, N) → Hom T D(M, Res u N).
For each M ∈ Ob(T D ) we define the counit
by the defining map of a limit
Σ M : M → Res u LK u M
(Σ M ) a : M(a) = Res P M[(a, 1 u(a) )] → lim −→u/u(a)
Res P M,

28 1 Functors and their Kan extensions
for any a ∈ Ob D. For each N ∈ Ob(T C ) we put the counit
Λ N : LK u Res u N → N
such that, for every x ∈ Ob C, lim −→u/x
Res P Res u N → N(x)
comes from the universal property of direct limit. We need to
prove that the following are identities
Res u
Res u Σ
−→ Res u LK u Res u
ΛRes u
−→Resu and LK u
ΣLK u
−→ LKu Res u LK u
LK u Λ
−→LK u
For the first we compute for any N ∈ Ob(T C ) and a ∈ Ob D
the following composite
(Res u N)(a) [Res u(Σ N )] a
−→ (Resu LK u Res u N)(a) (Λ ResuN) a
−→ (Res u N)(a).
This composite is
(Res u N)(a) = (Res P Res u N)[(a, 1 a )] → lim −→u/u(a)
Res P Res u N → (Res u N
which is really an identity by the universal property of lim −→u/u(a)
Res P Re
For the second we compute for any M ∈ Ob(T D ) and x ∈
Ob C that
(LK u M)(x) [LK uΛ M ] x
−→ (LKu Res u LK u M)(x) (Λ LKuM) x
−→ (LK u M)(x)
gives the identity. But it is rewritten as
lim Res −→u/x
PM → lim −→u/x
(Res P Res u LK u M) → lim −→u/x
Res P M,
which in turn equals −→u/x
lim Res P applying to
M → Res u LK u M → M.
However the preceding morphisms compose to the identity
because the following composite
M(a) → lim −→u/u(a)
Res P M → M(a)

1.2 Restriction and Kan extensions 29
is an identity for every a ∈ Ob D, due to the fact that (a, 1 u(x) )
is a terminal object and that Res P M(a, 1 u(a) ) = M(a). ⊓⊔
Remark 1.2.10.1. One may choose T to be Ab, or R-Mod etc,
for practical applications. When the index categories are
finite, we can even use R-mod, the category of finitely generated
R-modules.
2. When u : D → C is a full embedding and M ∈ Ob(Ab D ),
then LK u M ∈ Ob(Ab C ) restricted on u(D) is identified with
M, which means LK u M(u(d)) = M(d) for any d ∈ Ob D.
This is why we call such functors discovered by D. M. Kan,
“the Kan extensions”.
Example 1.2.11. In Example 1.2.7 (2) where the over- and
under-categories associated with i : H ↩→ G are computed,
we can continue to verify that Res i is the usual restriction
↓ G H , LK i is equivalent to the induction ↑ G H = kG ⊗ kH − and
RK i is equivalent to the coinduction ⇑ G H = Hom kH(kG, −).
If G is finite, the two Kan extensions are well known to be
equivalent.
Corollary 1.2.12. Let u : D → C and v : E → D be two
functors between small categories. Suppose T is a complete
and cocomplete category, M ∈ Ob(T E ) and N ∈ Ob(T C ).
Then Res v Res u = Res uv and consequently LK u LK v
∼ = LKuv ,
RK u RK v
∼ = RKuv .
Proof. The equality between restrictions follows directly from
definition. Then we have
Hom T E(M, Res v Res u N) ∼ = Hom T D(LK v M, Res u N)
∼ = HomT C(LK u LK v M, N).
Hence LK u LK v
∼ = LKuv . The isomorphism between right
Kan extensions can be proved similarly.
⊓⊔

Chapter 2
Simplicial methods
Abstract We begin with chain complexes and their homology,
the fundamental concepts of homological algebra. Then
we review simplicial constructions in algebra and topology
as they provide concrete examples and motivating ideas. Out
main interest lies in the nerve of a small category. This particular
simplicial set allows us to define simplicial homology
and cohomology of a category with coefficients in a commutative
ring. The nerve of a small category has a geometric
realization, called the classifying space. Hence we can also
consider the singular homology and cohomology of a classifying
space. We shall show these two theories agree. We will
introduce various important categorical constructions. Meanwhile
we develop techniques for comparing categoires, their
nerves and classifying spaces. A major theorem is Quillen’s
Theorem A. For future references and better understanding
of simplicial methods, we also include a description of bisimplicial
sets and several relevant results.
2.1 Complexes and homology
Here we recall basics about chain and cochain complexes as
well as operations on them.

32 2 Simplicial methods
2.1.1 Chain complexes, homology and chain homotopy
Suppose A is an associative ring with identity and Z is the
totally ordered set of integers. From Example 1.1.3 (3) we
may deem Z as a category.
Definition 2.1.1. A chain complex of A-modules is an object
C ∈ Ob(A-Mod) Zop such that C(n → n + 2) = 0 for
all n ∈ Z. In other words it consists of a collection of objects
in A-Mod, {C n = C(n)} n∈Z , together with morphisms
∂ n = C((n − 1) → n) : C n → C n−1 , called the differentials,
such that ∂ n ◦ ∂ n+1 = 0.
A cochain complex of objects in A-Mod is an object C ∈
Ob(A-Mod) Z such that C(n → n + 2) = 0, for all n ∈ Z.
Alternatively it consists of a collection of objects in A, {C n =
C(n)} n∈Z , together with morphisms ∂ n = C(n → (n + 1)) :
C n → C n+1 , called the differentials, such that ∂ n ◦ ∂ n−1 = 0.
If {C n , ∂ n } n∈Z (respectively, {C n , ∂ n } n∈Z ) is a chain complex
(respectively, cochain complex) of A-modules, and x ∈ C n
(respectively x ∈ C n ), then we say x has degree n, written as
deg x = n.
We call a complex bounded below if C n (or C n ) vanishes
when n is sufficiently small. A complex bounded above if C n
(or C n ) vanishes when n is sufficiently large. A complex if
bounded if it is both bounded below and bounded above (that
is, there are only finitely many non-zero terms). A complex is
called a stalk complex if it has exactly one non-zero term.
Suppose C = {C n , ∂ n } is a chain complex. Then we usually
write Z n (C) = ker ∂ n ⊂ C n . Any element x ∈ Z n (C) is called
an n-cycle. We also write B n (C) = Im∂ n+1 ⊂ C n . Any element
x ∈ B n (C) is called an n-boundary. Similarly suppose
C = {C n , ∂ n } is a cochain complex. Then we usually write
Z n (C) = ker ∂ n ⊂ C n . Any element x ∈ Z n (C) is called an n-

2.1 Complexes and homology 33
cocycle. We also write B n (C) = Im∂ n−1 ⊂ C n . Any element
x ∈ B n (C) is called an n-coboundary.
Definition 2.1.2. The i-th homology of a chain complex
{C n , ∂ n } n∈Z is
Hi(C) = Z i (C)/B i (C).
The i-th cohomology of a cochain complex {C n , ∂ n } n∈Z is
H i (C) = Z i (C)/B i (C).
Definition 2.1.3. Let C and D be two chain complexes (respctively,
cochain complexes). A chain map (respectively,
cochain map) φ : D → C is a natural transformation. In
other words, it consists of maps φ n : D n → C n (respectively,
φ n : D n → C n ) such that the following diagram commutes
D n
φ n
∂ D n
D n−1
φ n−1
respectively, D n ∂n D
φ n
C n ∂
C
n
C n−1 C n ∂ n C
D n+1
φ n+1
C n+1
The chain map φ is a chain isomorphism if it is a natural
equivalence.
Clearly a chain map between chain complexes φ : D → C
induces maps between homology groups φ ∗ : Hn(D) → Hn(C)
and a cochain map between cochain complexes φ : D → C
induces maps between cohomology groups φ ∗ : H n (D) →
H n (C).
For convenience, we shall call both chain maps and cochain
maps just chain maps. Given a chain complex {C n , ∂ n } n∈Z of
objects in an abelian category A, we can obtain a cochain
complex {C n , ∂ n } n∈Z , simply by asking C n = C −n and ∂ n =
∂ −n . Thus in some sense, chain and cochain complexes are
the same. In practice, we will regard a complex as a chain

34 2 Simplicial methods
complex or a cochain complex depending on where it comes
from.
Definition 2.1.4. If φ, ψ : D → C are two chain maps, we
say φ and ψ are chain homotopic, written as φ ≃ ψ, if there
are maps h n : D n → C n+1 such that Φ n − Ψ n = h n−1 ∂ D n +
∂ C n+1h n· · ·
D n+1
∂ D n+1
D n
∂ D n
D n−1
· · ·
φ n+1 −ψ n+1
h n
φ n −ψ n
h n−1
φ n−1 −ψ n−1
· · ·
C n+1 ∂
C
n+1
C n
∂ C n
C n−1
· · ·
We say two complexes D and C are chain homotopy equivalent,
written as D ≃ C if there are chain maps φ : D → C
and ψ : C → D such that φ ◦ ψ ≃ Id C and ψ ◦ φ ≃ Id D .
We say a complex C is contractible, if it is chain homotopy
equivalent to the zero complex. We say a complex C is acyclic
if Hn(C) = 0 for all n. Acyclicity is strictly weaker than contractibility.
Proposition 2.1.5. If φ, ψ : D → C are chain homotopic,
and if F : A-Mod → B-Mod is an additive functor, then
1. φ ∗ = ψ ∗ : Hn(D) → Hn(C). Thus a chain homotopy equivalence
D ≃ C induces isomorphisms Hn(D) ∼ = Hn(C).
2. F(φ) ≃ F(ψ) : F(D) → F(C), and F(φ) ∗ = F(ψ) ∗ :
Hn(F(D)) → Hn(F(C)).
Let C be a bounded below chain complex of A-modules.
A projective resolution of C is a (bounded below) chain
complex P of projective A-modules, along with a chain map
P → C which induces isomorphisms on homology groups. Any
two projective resolutions of C are chain homotopy equiva-

2.1 Complexes and homology 35
lent. Similarly we can consider an injective resolution I of a
bounded above complex of A-modules D.
2.1.2 Double complexes and operations on chain complexes
Definition 2.1.6. A double complex of A-modules is an object
C ∈ Ob(A-Mod) Zop ×Z op such that C(p → (p + 2), 1 q ) =
C(1 p , q → (q + 2)) = 0 and
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
for all p, g ∈ Z.
Alternatively it consists of a collection of objects {C p,q =
C(p, q)} p,q∈Z , together with maps
∂ 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
such that ∂ h ◦ ∂ h = ∂ v ◦ ∂ v = ∂ v ◦ ∂ h + ∂ h ◦ ∂ v = 0.
A map between two double complexes D → C is just a
natural transformation. It is an isomorphism if it is a natural
equivalence.
Double complexes are also called bicomplexes in the literature.
A double complex C = {C p,q } p,q∈Z is called bounded if
it has only finitely many non-zero terms along each diagonal
line p + q = n. An example will be the first quadrant double
complex where C p,q = 0 unless both p and q are non-negative.
Definition 2.1.7. The total complexes of a double complex
C are given by
Tot ⊕ (C) n = ⊕
and Tot ∏ (C) n = ∏
p+q=n
C p,q
with differentials ∂ = ∂ h + ∂ v .
p+q=n
C p,q
These two constructions are usually different, but when C is
bounded they are equal. Now we present a classical construction.

36 2 Simplicial methods
Let A be an associative ring. Suppose D is a chain complex of
right A-modules and C is a chain complex of left R-modules.
We define their product complex D ⊗ A C by
(D ⊗ A C) n = ⊕
D p ⊗ A C q .
p+q=n
It is a double complex with ∂ h p,q = ∂ D p ⊗1 and ∂ v p,q = (−1) p (1⊗
∂ C q ), pictured as
D p−1 ⊗ A C q
∂ h p,q
D p ⊗ A C q
∂ v p−1,q
D p−1 ⊗ A C q−1
∂ h p,q−1
∂ v p,q
D p ⊗ A C q−1
Hence the differential of the total complex Tot ⊕ (D ⊗ A C) is
given by
∂(x ⊗ y) = ∂ D p x ⊗ y + (−1) p x ⊗ ∂ C q y,
for x ∈ D p and y ∈ C q .
Suppose both E and C are chain complexes of left A-
modules. Then we introduce another double complex Hom A (E, C)
by (there are different conventions in the literature)
Hom A (E, C) n = ∏
Hom A (E −p , C q ),
pictured as
q+p=n
Hom A (E −p+1 , C q )
∂ h p,q
Hom A (E −p , C q )
∂ v p−1,q
∂ v p,q
Hom A (E −p+1 , C q−1 ) Hom A (E −p , C q−1 )
∂ h p,q−1
with ∂ h p,q(f) = f∂ E −p+1 and ∂ v p,q(f) = (−1) p+q ∂ C q f.

2.1 Complexes and homology 37
The differential of the total complex Tot ∏ (Hom A (E, C)) is
∂ n = ∂ h p,q + ∂ v p,q : Hom A (E, C) n → Hom A (E, C) n−1 given by
for any f ∈ Hom A (E −p , C q ).
∂ n f = f∂ E −p+1 + (−1) n ∂ C q f
Theorem 2.1.8. Suppose R is a commutative ring. Let C,
D and E be complexes of R-modules. We have an isomorphism
of double complexes
Hom R (C ⊗ R D, E) ∼ = Hom R (C, Hom R (D, E))
The following theorem is well known and its proof can be
found in many places. Here we record it for future references.
Theorem 2.1.9 (Künneth formula). Suppose A is a ring.
Let D be a chain complex of right A-modules and C, E chain
complexes of left A-modules. Then
1. if D n and ∂(D n ) are flat for all n, there is a short exact
sequence
0 → ⊕ p+q=n Hp(D) ⊗ A Hq(C) → Hn(D ⊗ A C)
→ ⊕ p+q=n−1 TorA 1 (Hp(D), Hq(C))
2. if E n and ∂(E n ) are projective for all n, there is a short
exact sequence
0 → ∏ q−p=n+1 Ext1 A(Hp(E), Hq(C)) → Hn(Hom A (E, C))
→
∏
q−p=n Hom A(Hp(E), Hq(C)) →
Corollary 2.1.10 (Universal Coefficient Theorem). Let
D be a chain complex of right A-modules over a ring A, E
a chain complex of left A-modules and M a left A-module
considered as a complex concentrated in degree zero. Then

38 2 Simplicial methods
1. if D n and ∂(D n ) are flat for all n, there is a short exact
sequence
0 → Hp(D) ⊗ A M → Hn(D ⊗ A M) → Tor A 1 (Hn−1(D), M) → 0,
2. if E n and ∂(E n ) are projective for all n, there is a short
exact sequence
0 → Ext 1 A(Hn−1(E), M) → H−n(Hom A (E, M)) → Hom A (Hn(E), M)
In Corollary 2.1.10 (2), the middle term is often written as
cohomology H n (Hom A (E, M)).
The dual complex of C, denoted by C ∧ , is defined to be
Hom A (C, A), where the second A is regarded as a complex
concentrated in degree zero. This is a complex of right
A-modules. Note that if P is a projective A-module then
Hom A (P, A) is a projective right module. We can define an
evaluation map
ev : C ∧ ⊗ A C → A.
It is non-zero only at degree zero. More explicitly it is given
by f ⊗ x ↦→ f(x) for any base elements f ⊗ x ∈ (C ∧ ⊗ C) 0 .
Here x ∈ C n and f ∈ C ∧ −n.
Proposition 2.1.11. Suppose k is a field. Let D and C be
two complexes of finite-dimensional k-vector spaces. If for
every n, Hom k (D, C) n is finite-dimensional, then we have
an isomorphism of complexes
D ∧ ⊗ k C ∼ = Hom k (D, C).
Proof. Note that if M, N are two stalk complexes of k-vector
spaces concentrated in degrees m and n respectively, then we
have M ∧ ⊗ k N ∼ = Hom k (M, N) as stalk complexes concentrated
in degree n − m. Consequently we have an isomorphism
for every integer n, (D ∧ ⊗ k C) n
∼ = Homk (D, C) n since
Hom k (D, C) n is finite-dimensional and then Hom k (D, C) n =

2.2 Nerves, classifying spaces and cohomology 39
⊕ q+p=n Hom k (D −p , C q ). Hence in order to finish the proof, we
only need to verify that the two differentials are identified under
the vector space isomorphisms.
⊓⊔
In the next section, we will see many combinatorially constructed
complexes and double complexes.
2.2 Nerves, classifying spaces and cohomology
By Dold-Kan Correspondence, the category of non-negatively
graded complexes of A-modules is equivalent to the category
of simplicial A-modules. It implies also that the category
of first quadrant double complexes of A-modules is
equivalent to the category of bisimplicial A-modules. Since in
these notes, we are mainly interested in various non-negatively
graded complexes such as projective resolutions of modules,
and moreover these complexes come from corresponding simplicial
sets, it is useful to introduce simplicial sets and modules.
2.2.1 Simplicial sets and nerves of small categories
We recall the fundamental idea in algebraic topology of singular
(co)homology theory. From here we shall see how algebraic
and combinatorial methods enter the study of spaces. Then
we will apply the same methods, called simplicial methods, to
investigate small categories (instead of spaces). Our first definition
of category (co)homology is given soon after the basic
definitions are recorded.
Let us start with the topological simplicies. For every integer
n ≥ 0, a standard n-simplex is defined as a subspace of the
(n + 1)-dimensional real vector space R n+1

40 2 Simplicial methods
△ n = {(x 0 , x 1 , · · · , x n ) ∈ R n+1 ∣ ∣ xi ≥ 0 and
n∑
x i = 1}.
An i-face of △ n for 0 ≤ i ≤ n is a subspace such that there are
exactly i+1 chosen entries that are not constantly zero. Each
i-face is isomorphic to the standard i-simplex. For example
in the following picture, △ 2 has exactly one 2-face (e.g. △ 2 ),
three 1-faces (isomorphic to △ 1 ) and three 0-faces (isomorphic
to △ 0 which is a point).
i=0
X 3
(0,0,1)
△ 2 ⊂ R 3
X 2
(1,0,0) (0,1,0)
X 1
Fix an integer n, there are many natural maps among all
the faces of △ n . Since each face is isomorphic to a standard
simplex, these maps can be described as maps among all standard
simplicies. The most distinguished are the face maps
d i : △ n−1 → △ n (inserting a zero) and the degeneracy maps
s i : △ n+1 → △ n (adding up two adjacent entries), given explicitly
by
d i (x 0 , x 1 , · · · , x n−1 ) = (x 0 , · · · , x i , 0, x i+1 , · · · , x n−1 ), 0 ≤ i ≤ n;
s i (x 0 , x 1 , · · · , x n+1 ) = (x 0 , · · · , x i + x i+1 , · · · , x n+1 ), 0 ≤ i ≤ n.
Given a topological space X, in order to define and compute
its singular homology H∗(X, Z) we first form the sets of
continuous maps S(X) n = Hom T op (△ n , X) and then produce
free abelian groups ZS(X) n on top of them. Each face map

2.2 Nerves, classifying spaces and cohomology 41
d i : △ n−1 → △ n induces a map d i : S(X) n → S(X) n−1 , and
moreover ∂ n = ∑ n
i=0 (−1)i d i : ZS(X) n → ZS(X) n−1 satisfies
∂ n+1 ∂ n = 0. In this way we obtain a non-negatively graded
chain complex
{ZS(X) ∗ ; ∂ ∗ } ∗≥0 ,
in which every differential ∂ n is determined by an alternating
sum of face maps. The homology of this complex is defined
to be the singular homology of X, written as H∗(X, Z). The
degeneracy maps are important to us as well, and we shall
discuss their roles shortly.
In order to allow further applications of such a fundamental
construction, we propose some alternative descriptions of the
standard simplices based on the following two observations.
Firstly, to specify a standard complex, it suffices to provide its
vertices because △ n is the convex hull of the set of its vertices.
Moreover since, in R n+1 , one can give the lexicographic order
to its elements by asking (x 0 , · · · , x n ) < (y 0 , · · · , y n ) if there
exists an integer 0 ≤ k ≤ n such that x i = y i for i < k and
x k < y k , particularly there is an order on the set of vertices of
△ n . This totally ordered set of vertices uniquely determines
△ n . For example in R 3 we have (0, 0, 1) < (0, 1, 0) < (1, 0, 0),
and we can certainly identify this totally ordered set with △ 2 .
Thus giving △ n is equivalent to giving the totally ordered set
(0, · · · , 0, 1) < (0, · · · , 1, 0) < · · · < (1, 0, · · · , 0) of points in
R n+1 . Moreover there exists a natural one-to-one correspondence
between the set of all totally ordered subsets of vertices
and the set of faces of △ n .
Secondly, because the face and degeneracy maps on △ n
are completely determined by their values on the vertices,
we can translate the face and degeneracy maps accordingly.
For example, we can write out values of d 0 : △ 2 → △ 3 and
s 0 : △ 2 → △ 1 on the vertices

42 2 Simplicial methods
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
s 0 (0, 0, 1) = (0, 1), s 0 (0, 1, 0) = (1, 0), s 0 (1, 0, 0) = (1, 0).
In general it is easy to see that d i and s i always send vertices
to vertices. Furthermore they (weakly) preserve the order, in
the sense that if a ≤ b then d i (a) ≤ d i (b) and s i (a) ≤ s i (b).
We shall illustrate it by an example. Let us examine △ 1 =
{(0, 1) < (1, 0)} and △ 2 = {(0, 0, 1) < (0, 1, 0) < (1, 0, 0)}.
The face maps △ 1 → △ 2 , adapted to our new combinatorial
expression, are given by embeddings
d 0 : (0, 1) < (1, 0) ↦→ (0, 0, 1) < (0, 1, 0)
d 1 : (0, 1) < (1, 0) ↦→ (0, 0, 1) < (1, 0, 0)
d 2 : (0, 1) < (1, 0) ↦→ (0, 1, 0) < (1, 0, 0).
The degeneracy maps △ 2 → △ 1 are the same as projecting
the vertex (0, 1, 0) to one of the other two, upon identifying
△ 1 with the line segment (0, 0, 1) − (1, 0, 0)
s 0 : (0, 0, 1) < (0, 1, 0) < (1, 0, 0) ↦→ (0, 1) < (1, 0) = (1, 0) (0, 1) <
s 1 : (0, 0, 1) < (0, 1, 0) < (1, 0, 0) ↦→ (0, 1) = (0, 1) < (1, 0) (0, 1) <
In summary, the face and degeneracy maps are indeed weakly
monotonic maps among those totally ordered sets corresponding
to the standard simplices. As a matter of fact, all weakly
monotonic functions among those totally ordered sets, coming
from standard simplices, are composites of these face and
degeneracy maps.
For various good reasons we continue to work on the combinatorial
characterizations of standard simplices. Previously
we have identified △ n with the poset (0, · · · , 0, 1) < (0, · · · , 1, 0) <
· · · < (1, 0, · · · , 0), and have rewritten the face and degeneracy
maps. Now we abstract the totally ordered set as
0 < 1 < · · · < n. Consequently the face and degeneracy

2.2 Nerves, classifying spaces and cohomology 43
maps can be reformulated and will be denoted by d i and s i ,
respectively. This reformulation allows us to forget the geometric
definition of △ n , and thus all relevant constructions
can be made in an entirely combinatorial fashion. For future
applications we write out the ith face map d i (corresponding
to d i ) for 0 ≤ i ≤ n
{0 < 1 < · · · < n − 1} → {0 < 1 < · · · < n}
{
d i j, if j < i
(j) =
j + 1 , if j ≥ i
and the ith degeneracy map s i (corresponding to s i ) for 0 ≤
i ≤ n
{0 < 1 < · · · < n + 1} → {0 < 1 < · · · < n}
{
s i j, if j ≤ i
(j) =
j − 1 , if j > i
One can verify that these maps satisfy the relations (the
cosimplicial identities)
d j d i = d i d j−1 i < j
s j d i = d i s j−1 i < j
s j d i = 1 i = j or j + 1
s j d i = d i−1 s j i > j + 1
s j s i = s i s j+1 i ≤ j .
For each n ∈ {0} ∪ N, we define an ordered set (which happens
to be a finite category) n = 0 < 1 < 2 < · · · < n
(or rather 0 → 1 → 2 → · · · → n). We denote by △ the
category consisting of all such n, in which morphisms are
weakly monotonic functions (equivalently, functors) among
these ordered sets. Recall the motivating example at the
very beginning of this section. Given a space X, in order

44 2 Simplicial methods
to make use of △, we associated to each n, the combinatorial
model of △ n , a set Hom T op (△ n , X), as well as an abelian
group ZHom T op (△ n , X). If we put all pieces together, we realize
that we actually constructed a contravariant functor
Hom T op (−, X) : △ → Set, as well as a ZHom T op (−, X) :
△ → Ab. Since these functors give us the singular homology
of X, contravariant functors from △ to other categories may
lead to interesting constructions too.
Definition 2.2.1. A simplicial object in a category T is a
contravariant functor X : △ → T . Two simplicial objects
X, Y in T are said to be isomorphic, written as X ∼ = Y , if
as functors they are naturally equivalent.
A simplicial set is a simplicial object in Set.
Equivalently, since monotonic maps are compositions of d i
and s i , a simplicial object X in T consists of a set of objects
X n := X([n]) ∈ Ob T (an element of X n is called an n-
simplex) and morphisms among these simplicies which are
composites of two special kinds of morphisms: d i := X(d i ) :
X n → X n−1 , 0 ≤ i ≤ n, and s i := X(s i ) : X n → X n+1 ,
0 ≤ i ≤ n, satisfying (the simplicial identities)
d i d j = d j−1 d i i < j
d i s j = s j−1 d i i < j
d i s j = 1 i = j or j + 1
d i s j = s j d i−1 i > j + 1
s i s j = s j+1 s i i ≤ j .
In order to define a simplicial object X, it suffices to specify
{X n } n≥0 ⊂ Ob T , together with some maps d i and s i satisfying
the above relations.
We shall be interested in the cases where T = Set or R-Mod
for some commutative ring R with identity. By Proposition

2.2 Nerves, classifying spaces and cohomology 45
1.1.27, the two categories SimpSet and Simp(R-Mod) are
both complete and cocomplete.
Example 2.2.2.1. Let C be a small category. We define the
nerve of C to be a simplicial set NC = Hom Cat (−, C) such
that NC n = C n (indeed Ob(C n ) since we only need the underlying
set). Alternatively NC n can be identified with the
set of n-chains of morphisms in C if n > 0 and Ob C if n = 0.
When n > 0, the ith face map d i : NC n+1 → NC n is
α
d i (x 0 → · · · → x
i α i+1
i−1→xi → xi+1 → · · · → x n+1 )
α
= x 0 → · · · → x
i α i+1
i−1→ ̂xi → xi+1 → · · · → x n+1 ,
where ∧ means removing an object. For instance when n = 1
α
the three face maps NC 2 → NC 1 are d 0 (x
1 α 2
0→x1→x2 ) =
α
x
2
α
1→x2 , d 1 (x
1 α 2
α
0→x1→x2 ) = x
2 α 1
α
0 → x2 and d 0 (x
1 α 2
0→x1→x2 ) =
α
x
1
0→x1 . The ith degeneracy map s i : NC n → NC n+1 is given
by
s i (x 0 → · · · → x i−1 → x i → x i+1 → · · · → x n )
= x 0 → · · · → x i−1 → x i
1 xi
→xi → x i+1 → · · · → x n .
One can verify that these maps satisfy the simplicial identities.
2. We consider a special situation of the first example. It
shall fill the gap which may have occurred during transition
from the geometric definition of standard n-simplicies to
the abstract category-theoretic reformulation. We fix a nonnegative
integer m. The totally ordered set m is indeed a
category 0 → 1 → · · · → m. Thus we can consider its nerve
Nm. By the preceding example we have Nm n = Ob(m n ).
In other words the combinatorial m-simplex m corresponds
to the functor Hom △ (−, m) : △ → Set. This correspondence
actually extends to a (covariant) functor, given by

46 2 Simplicial methods
ι(m) = Hom △ (−, m),
By Yondeda Lemma we have
ι : △ → SimpSet = Set △ .
Hom SimpSet (Hom △ (−, m), Hom △ (−, n)) ∼ = Hom △ (m, n).
It means ι is fully faithful.
In the next section we shall talk about the geometric realization
of simplicial sets. Then we will see that the simplicial
set Hom △ (−, m) precisely gives rise to △ m .
3. Let A be an associative ring and A-Mod the category of all
A-modules. A simplicial object in A-Mod is called a simplicial
A-module. From Set to A-Mod, there exists a natural
covariant functor, given by constructing free A-modules.
Suppose X : △ → Set is a simplicial set. Then we naturally
obtain a simplicial A-module
A[X] : △ → Set → A-Mod.
Given a simplicial A-module Y , one can construct a (nonnegatively
graded) chain complex of A-modules by defining
C n (Y ) = Y n and ∂ n = ∑ n
i=0 (−1)i d i : C n (Y ) → C n−1 (Y ). If
Y comes from a simplicial set X, that is, Y = A[X], then
we write the chain complex as C ∗ (X, A) instead of C ∗ (Y )
or C ∗ (A[X]) for various good reasons.
The following characteristic statement is enlightening. Suppose
X is a simplicial set. Then we define the corresponding
simplex category to be ι/X, where ι : △ → SimpSet is the
functor in Example 2.2.2 (2).
Theorem 2.2.3. Let X be a simplicial set. Then we have
an isomorphism
X ∼ = lim −→ι/X
P,

2.2 Nerves, classifying spaces and cohomology 47
where P : ι/X → SimpSet is given by P(n, Φ) = Hom △ (−, n).
Proof. From the contravariant version of Theorem 1.1.28 we
know for the functor X : △ op → Set we can define a category
D and a functor ˜X : D → Set
△ op = SimpSet such that
X ∼ = lim −→D ˜X.
We show the category D is exactly ι/X and subsequently ˜X
can be identified with P.
The objects in the overcategory ι/X are of the form (n, Φ),
where Φ : ι(n) = Hom △ (−, n) → X is a simplicial map. By
the Yoneda Lemma Hom SimpSet (Hom △ (−, n), X) ∼ = X(n) =
X n . Thus Φ can be regarded as an element of the set X(n).
This provides a bijection between Ob(ι/X) and Ob D. From
here one can continue to finish the proof. We leave the details
for the reader.
⊓⊔
Definition 2.2.4. Let C be a small category and NC its
nerve. Suppose R is a commutative ring with identity. Then
the simplicial R-module R[NC] gives rise to a complex of R-
modules, written as C ∗ (C, R). The n-th homology of C ∗ (C, R),
denoted by Hn(C, R), is called the n-th homology of C with
coefficients in R. The n-th cohomology of the cochain complex
C ∗ (C, R) := C ∗ (C, R) ∧ = Hom R (C ∗ (C, R), R),
denoted by H n (C, R), is called the n-th cohomology of C with
coefficients in R.
By direct calculation, one can see that H0(C, R) ∼ = H 0 (C, R)
is a free R-module with rank equal to the number of connected
components of C. For any chain complex C ∗ (C, R), define as
above, we can insert the base ring R at degree -1 and obtain
the so-called reduced chain complex ˜C∗ (C, R) = C ∗ (C, R) →

48 2 Simplicial methods
R → 0. Then the homology of the reduced chain complex is
called the reduced homology of C, written as ˜H∗ (C, R). We
see ˜H−1 (C, R) = 0 and ˜H0 (C, R) is a free R-module with rank
equal to rk RH0(C, R) − 1. A small category C is called R-
acyclic if ˜H∗ (C, R) vanishes.
Now we address the issue of degeneracy maps. Suppose X is
a simplicial object in an abelian category T . Then we have a
complex of objects in T , denoted by X ∗ . It has a subcomplex
X ′ ∗ such that X ′ n = ∑ i s i(X n−1 ) if n > 0 and X ′ 0 = 0. Then
we continue to define a quotient complex X † ∗ = X ∗ /X ′ ∗. We
can also define the normalized complex of X, N ∗ (X), by
N n (X) =
n−1
⋂
i=0
Ker(d i )
and ∂ n = (−1) n d n . This is a subcomplex of X ∗ .
Theorem 2.2.5.1. We have X ∗ = N ∗ (X)⊕X ∗ ′ and N ∗ (X) ∼ =
X ∗. † Furthermore the quotient map X ∗ ↠ X ∗
† induces a
chain homotopy equivalence and X ∗ ′ is contractible.
2. (Dold-Kan Correspondence) Suppose T is an abelian
category and Ch ≥0 (T ) is the category of non-negatively
graded chain complexes in T . Then the normalized chain
complex functor functor
is a category equivalence.
N : SimpT → Ch ≥0 (T ),
Proof. For the first part, see [53, Chapter VIII, Section] and
[84, Section 8.3], and for the second part, the reader is referred
to [84, Section 8.4].
⊓⊔
Because of the above theorem, we also call X † ∗ the normalized
complex of X ∗ .

2.2 Nerves, classifying spaces and cohomology 49
When T = A-Mod, and X is a simplicial A-module, then
X ′ n, for any n > 0, consists of degenerate elements of the form
s i (x ′ ) for some x ′ ∈ X n−1 . When X = R[NC], a simplicial R-
module defined over a small category C and a commutative
ring R, the degenerate elements in R[NC] n are all the linear
combinations of the degenerate elements in NC n which are
exactly those n-chains containing an identity morphism. In
practice, when we compute (co-)homology of a category, we
only need to use the normalized complex C † ∗(C, R).
Example 2.2.6.1. Let C be the following category with two
objects, two identity and non-identity morphisms
x
α
β
To compute H∗(C, Z), we only have to write down the normalized
chain complex 0 → C † 1 (C, Z) → C† 0 (C, Z) → 0
y
0 → Z{α, β} → Z{x, y} → 0.
(By comparison, C ∗ (C, Z) is infinite.) The non-trivial differential
is given by α ↦→ y − x and β ↦→ y − x. Then
the only non-trivial homology groups are H0(C, Z) ∼ = Z and
H1(C, Z) ∼ = Z. The calculation of cohomology groups is left
to the reader.
2. The second category D is slightly different from the first.
One can easily verify that the normalized complex C † ∗(D, Z)
is infinite, of dimension two at each degree. By direct computation
H0(D, Z) ∼ = Z is the only non-trivial homology
group.
α
x y
α −1

50 2 Simplicial methods
However there is an easy way to see it, if one notices that
D ≃ • and knows that a category equivalence induces isomorphism
on homology (see Proposition 2.2.19).
Definition 2.2.7. Let D and C be two small categories.
Then the join of D with C, denoted by D ∗ C, is a category
whose objects are Ob D ∪ Ob C, and whose morphisms
are Mor D ∪ Mor C, plus exactly one extra morphism γ a,x ∈
Hom D∗C (a, x) introduced for every pair of objects a ∈ Ob D
and x ∈ Ob C. The composition laws in D ∗ C are determined
by the composition laws in C and D, plus the equalities
αγ a,x = γ a,y , γ a,x β = γ b,x for any α ∈ Hom C (x, y) and
β ∈ Hom D (b, a).
By definition D ∗ C and C ∗ D are two different categories.
One can easily construct N(D ∗ C) from ND and NC.
Proposition 2.2.8. Suppose both D and C are connected
small categories. If R = k is a field, then
˜Hn(D ∗ C, k) ∼ =
⊕ ˜Hi(D, k) ⊗ ˜Hj (C, k)
i+j=n−1
if n ≥ 0. Particularly H∗(D ∗ C, k) ∼ = H∗(C ∗ D, k).
Proof. For n ≥ 0, C n (D ∗ C, k) = k[N(D ∗ C)] n has a basis
consisting of the following elements
γx i ,y i+1
x 0 → · · · → x i −→ yi+1 → · · · → y n
where x 1 , · · · , x i ∈ Ob D and y i+1 , · · · , y n ∈ Ob C, for 0 ≤
i ≤ n. Here γ xi ,y i+1
is the unique morphism in Hom D∗C (x i , y i+1 ).
Its differential is

2.2 Nerves, classifying spaces and cohomology 51
∑ i−1
j=0 (−1)j γx i ,y i+1
x 0 → · · · → ˆx j → · · · → x i −→ yi+1 → · · · → y n
+ (−1) i γx i−1 ,y i+1
x 0 → · · · → x i−1 −→ yi+1 → · · · → y n
+ (−1) i+1 γx i ,y i+2
x 0 → · · · → x i −→ yi+2 → · · · → y n
+ ∑ n
l=i+2 (−1)l γx i ,y i+1
x 0 → · · · → x i −→ yi+1 → · · · → ŷ l → · · · → y n .
Suppose ˜C∗ (D ∗ C, k), ˜C∗ (D, k) and ˜C∗ (C, k) are the reduced
chain complexes with k inserted in degree -1. Then we can
define a degree -1 chain map
˜C ∗ (D ∗ C, k) → ˜C∗ (D, k) ⊗ ˜C∗ (C, k)
such that, at degree −1 it is identity and at degree n ≥ 0,
˜C n (D ∗ C, k) → [˜C∗ (D, k) ⊗ ˜C∗ (C, k)] n−1 is given by
γx i ,y i+1
x 0 → · · · → x i −→ yi+1 → · · · → y n ↦→ [x 0 → · · · → x i ]⊗[y i+1 → · · ·
along with
and
x 0 → · · · → x n ↦→ [x 0 → · · · → x n ] ⊗ 1,
y 0 → · · · → y n ↦→ 1 ⊗ [x 0 → · · · → x n ].
Here 1 ∈ k = ˜C−1 (D, k) = ˜C−1 (C, k). This chain map is an
isomorphism and thus by Künneth formula we get
˜Hn(D ∗ C, k) ∼ = ⊕ i+j=n−1 Hi(˜C∗ (D, k)) ⊗ Hj(˜C∗ (C, k))
∼ =
⊕i+j=n−1 ˜Hi(D, k) ⊗ ˜Hj (C, k).
Example 2.2.6 (2) tells us that it is important to know how to
compare various categories or more generally simplicial sets.
Definition 2.2.9. A natural transformation Φ : X → Y between
simplicial objects in T is called a simplicial map. The
simplicial objects in T form a category SimpT .
⊓⊔

52 2 Simplicial methods
By definition a simplicial map Φ consists of a sequence of
morphisms in T , Φ n : X n → Y n , commuting with the relevant
face and degeneracy maps.
Lemma 2.2.10. Let R be a commutative ring with identity.
A simplicial map Φ : X → Y between simplicial sets induces
a chain map CΦ : C ∗ (X, R) → C ∗ (Y, R).
Proof. Because every Φ n : X n → Y n commutes with d i , it commutes
with the differentials of these chain complexes which
are alternating sums of d i ’s.
⊓⊔
Lemma 2.2.11. Suppose D and C are two small categories
and Φ : ND → NC is a simplicial map. Then Φ determines
a functor u : D → C which in turn gives rise to Φ as Nu.
Proof. The map Φ 0 : ND 0 → NC 0 gives an assignment
Ob D → Ob C while Φ 1 : ND 1 → NC 1 gives an assignment
Mor D → Mor C. We show that these define a functor
u : D → C such that u(d) = Φ 0 (d) if d ∈ Ob D and
u(f) = Φ 1 (f) if f ∈ Mor D.
First of all, Φ 0 and Φ 1 are compatible in the sense that if
c f →d ∈ ND 1 then u(f) = Φ 1 (f) is a morphism from u(c) =
Φ 0 (c) to u(d) = Φ 0 (d). This follows from the commutative
diagram by choosing i = 0 or 1
Φ
ND
1
1 NC 1
d i d i
ND 0 Φ0
NC 0
Second of all, we have to demonstrate u(fg) = u(f)u(g) for
any two composable morphisms in Mor D. This time we just
use a similar commutative diagram involving d 1 , Φ 1 and Φ 2 .
At last we can show u(1 d ) = 1 u(d) by invoking a degeneracy
map

2.2 Nerves, classifying spaces and cohomology 53
ND 1
Φ 1
NC 1
s 0
s 0
ND
0 Φ0
NC 0
Since Φ is completely determined by Φ 0 and Φ 1 , one can prove
that Nu is exactly Φ.
⊓⊔
Example 2.2.12. Suppose u : D → C is a functor between
small categories and R is a commutative ring with identity.
Then it induces a simplicial map Nu : ND → NC. Furthermore
it induces a chain map Cu : C ∗ (D, R) → C ∗ (C, R).
Corollary 2.2.13. A functor u : D → C induces a map u ∗ :
H∗(D, R) → H∗(C, R) as well as u ∗ : H ∗ (C, R) → H ∗ (D, R).
Proof. The map u ∗ is induced by Cu while u ∗ is induced by
Hom R (Cu, R).
⊓⊔
We shall illustrate the previous comparison results by an
example. Note that if there are two functors v : E → D
and u : D → C, we can easily verify that (uv) ∗ = u ∗ v ∗ and
(uv) ∗ = v ∗ u ∗ .
Example 2.2.14. Let us consider the following category C with
four non-identity morphisms
µ
z
Its opposite category is pictured as
x
α
β
γ
y

54 2 Simplicial methods
y op α op
β op
γ x op
op
z op µ op
In order not to make confusion in the opposite category we
write x op for x etc. By direct calculation we can find that
H0(C, Z) ∼ = H0(C op , Z) ∼ = Z and H1(C, Z) ∼ = H1(C op , Z) ∼ =
Z ⊕ Z. These are the only non-trivial homology groups.
We can list all (covariant) functors from C → C op , since
there are not many. Suppose u : C → C op is a functor. We
write u(C) to be the image of C, a subcategory of C op .
1. If u(x) = x op , then both u(y) and u(z) have to be x op . In this
case u(C) is a trivial category and has the only non-trivial
homology at degree zero, which is Z.
2. If u(x) = y op , then there are several possibilities. In any case
we must have u(y) = u(z). Firstly, we may have u(C) =
{y op }. Then Hi(u(C), Z) ∼ = Z if i = 0 or zero otherwise.
Secondly we may have u(C) equals
or
y op α op x op
β op
y op
x op .
In the former situation we will have H0(u(C), Z) ∼ = H1(u(C), Z) ∼ =
Z and zero otherwise. In the latter situation we have only
non-trivial homology H0(u(C), Z) ∼ = Z.
3. If u(x) = z op , then it is similar to 2.
Since u can be decomposed into a sequence C ↠ u(C) ↩→
C op , u ∗ is also the composite of H∗(C, Z) → H∗(u(C), Z) →
H∗(C op , Z). Under the circumstance we understand u ∗ com-

2.2 Nerves, classifying spaces and cohomology 55
pletely. A crucial fact is that, our previous calculations assert
that there exists no such functor u : C → C op that induces the
isomorphism of graded abelian groups H∗(C, Z) ∼ = H∗(C op , Z).
Simplicial maps are used to compare two simplicial sets (and
resulting complexes). Now we introduce a way to compare two
simplicial maps.
Definition 2.2.15. For any two simplicial sets X, Y , we can
define the Cartesian product X × Y by (X × Y ) n = X n × Y n
with face and degeneracy maps d i = (d X i , dY i ), s i = (s X i , sY i ).
Example 2.2.16. If D and C are two small categories, then
N(D × C) ∼ = ND × NC.
For brevity we denote by {0, · · · , 0, 1, · · · , 1} the element
0 = · · · = 0 < 1 = · · · = 1 of the set N1 n with exactly i
copies of 1 for 0 ≤ i ≤ n + 1.
Definition 2.2.17. Let Φ, Ψ : X → Y be two simplicial
maps between simplicial sets. We say Φ is simplicially
homotopic to Ψ if there exists a natural transformation
: X × N1 → Y such that n | Xn ×{(0,··· ,0)} = Φ n and
n | Xn ×{(1,··· ,1)} = Ψ n for all n ≥ 0. We call a simplicial
homotopy from Φ to Ψ, written as : Φ→Ψ ≃ or simply Φ ≃ Ψ.
We say two simplicial sets X and Y are homotopic, written
as X ≃ Y , if there exist natural transformations Φ : X → Y
and Ψ : Y → X such that ΨΦ ≃ Id X and ΦΨ ≃ Id Y .
There are two canonical simplicial maps ι 0 , ι 1 : N0 → N1
by sending {0, · · · , 0} to {0, · · · , 0} or {1, · · · , 1}, respectively.
Based on this observation, being a simplicial homotopy
is equivalent to having the following commutative diagram

56 2 Simplicial methods
X ∼ = X × N0
Id X ×ι 0
X × N1
Id X ×ι 1
X × N0 ∼ = X
Φ
Ψ
Y .
Since (X × N1) n consists of n + 2 copies of X n in the
form of X n × {0, · · · , 1, · · · , 1}, combinatorially, the above
definition is equivalent to saying that there exist maps ′ i =
s i n | Xn ×{0,··· ,1,··· ,1} : X n → Y n+1 for 0 ≤ i ≤ n such that
d 0 ′ 0 = Φ n
d n+1 ′ n = Ψ n
d i ′ j = ′ j−1 d i i < j
d j+1 ′ j+1 = d j+1 ′ j
d i ′ j = ′ j d i−1 i > j + 1
s i ′ j = ′ j+1 s i i ≤ j
s i ′ j = ′ j s i−1 i > j.
Remember in the definition of ′ i, {0, · · · , 1, · · · , 1} denotes
the element 0 = · · · = 0 < 1 = · · · = 1 in N1 with i copies
of 1.
Definition 2.2.18. If the induced simplicial map Npt : NC →
N• of the canonical functor pt : C → • gives rise to a homotopy
NC ≃ N•, then we say C is contractible.
Note that there always exist various functors • → C. If
C is contractible, then any functor ι : • → C will induce
N• ≃ NC.
Proposition 2.2.19. If Φ : u → u ′ is a natural transformation
between two functors u, u ′ : D → C, then Nu is
homotopic to Nu ′ . Consequently Cu and Cu ′ are chain homotopic.
In particular, if ND ≃ NC, then C ∗ (D, R) and C ∗ (C, R)
are chain homotopy equivalent. Hence H∗(D, R) ∼ = H∗(C, R)
and H ∗ (D, R) ∼ = H ∗ (C, R).

2.2 Nerves, classifying spaces and cohomology 57
Proof. We can define a functor ˜Φ : D × 1 → C by ˜Φ(a, 0) =
u(a), ˜Φ(a, 1) = u ′ (a) and ˜Φ(α, 1{0} ) = u(α), ˜Φ(α, 1{1} ) =
u ′ (α) and ˜Φ(1a ,

58 2 Simplicial methods
when T possesses all limits. This is the historical context for
the Kan extensions.
2.2.2 Classifying spaces
It is often very useful to have some knowledge about classifying
spaces when computing the (co-)homology of a small
category. In this section we recall several important results as
facts, in order to obtain a balanced view towards key results
in the preceding section.
Definition 2.2.22. The geometric realization |X|, of a simplicial
set X, is the quotient of
⋃
X n × △ n
n≥0
by the equivalence relation given by (d i x, y) ∼ (x, d i y) and
(s i x, y) ∼ (x, s i y).
Definition 2.2.23. The geometric realization of NC, the
nerve of a small category C, is called the classifying space
of C, customarily denoted by BC.
In the literature the classifying space of a small category is
often written as |C|. However in these notes we try to avoid
this notation since when C is a finite group, it has been used
to denote the order of the group.
In the following examples, the first two are not hard to verify
while the other two requires further knowledge from algebraic
topology so we point out places where the reader may find
proofs.
Example 2.2.24.1. The classifying space of n is the standard
n-simplex △ n .
2. The classifying space of the following category is S 1 . Recall
its integral homology and compare with Example 2.2.6 (1).

2.2 Nerves, classifying spaces and cohomology 59
x
3. The classifying space of Z 2 is RP ∞ . One can see this by using
Milnor’s construction EZ 2 . Then up to homotopy BZ 2 ≃
EZ 2 /Z 2 .
4. Let D and C be two small categories. There is a natural way
to define the join of ND and NC. Then N(D∗C) ∼ = ND∗NC
and consequently B(D ∗ C) ≃ BD ∗ BC, see [26].
The following theorem asserts that the classifying spaces are
“good”.
Theorem 2.2.25. The space |X| is a CW-complex having
one n-cell for each non-degenerate n-simplex of X.
It explains the structures of classifying spaces in Example
2.2.24.
Proposition 2.2.26. The singular homology and cohomology
of BC with coefficients in R, namely H∗(BC, R) and
H ∗ (BC, R), are identified with H∗(C, R) and H ∗ (C, R).
Proof. The normalized complex C † ∗(C, R) is a cellular complex
for computing the singular homology of H∗(BC, R).
The essential connection between simplicial sets and topological
spaces is given by the following result.
Theorem 2.2.27 (Kan). Let X be a simplicial set and Y
a topological space. Then we have an adjunction
Hom T op (|X|, Y ) ∼ = Hom SimpSet (X, SY ).
Moreover the adjunct preserves homotopies.
Proof. Let f : |X| → Y be a continuous map. Then we define
a simplicial map Φ f by [(Φ f ) n (x n )](a n ) = f(x n , a n ) for n ≥ 0,
x n ∈ X n and a n ∈ △ n . Conversely if Φ : X → SY is a
α
β
y

60 2 Simplicial methods
simplicial map, we define f Φ : |X| → Y by f Φ (x n , a n ) =
[Φ n (x n )](a n ) for n ≥ 0, x n ∈ X n and a n ∈ △ n . These two
assignments give the adjunction.
⊓⊔
As an example, every CW-complex is homotopy equivalent
to the classifying space of the poset of its singular simplices.
Directly from Theorem 2.2.3 we have an alternative characterization
of the geometric realization of a simplicial set X
as
|X| ∼ = lim −→ι/X
|P|,
where |P| sends each object (n, Φ) to △ n , because a left adjoint
functor preserves direct limits. Given this homeomorphism
we can prove Theorem 2.2.27 alternatively as follows.
Hom T op (|X|, Y ) ∼ = Hom T op (lim −→ι/X
|P|, Y )
∼ = lim ←−ι/X
Hom T op (|P|, Y )
∼ = lim ←−ι/X
Hom SimpSet (P, SY )
∼ = HomSimpSet (lim −→ι/X
P, SY )
∼ = HomSimpSet (X, SY ).
The third isomorphism does not depend on Theorem 2.2.27
and it is true because
Hom T op (△ n , Y ) = SY n
∼ = HomSimpSet (Hom △ (−, n), SY ).
Theorem 2.2.28. Let X, Y be two simplicial sets. Then
there exists a natural homeomorphism
|X × Y | ∼ = |X| × |Y |,
if |X| × |Y | is a CW-complex.
The space |X| × |Y | is a CW-complex if both X and Y are
countable (that is, both ∪ n≥0 X n and ∪ n≥0 Y n are countable)

2.2 Nerves, classifying spaces and cohomology 61
or if either |X| or |Y | is locally finite (that is, every point is
an inner point of a finite subcomplex).
We collect some useful statements from last section which
are adapted to classifying spaces. The upshot is that a homotopy
between two simplicial maps or sets does give rise to a
homotopy between continuous maps or spaces, in the usual
topological sense. A small category C is contractible if BC has
the same homotopy type of a point. By Theorem 2.2.27 it is
equivalent to saying that NC ≃ N•.
Corollary 2.2.29. A functor u : D → C induces a continuous
map Bu : BD → BC.
1. If u ′ : D → C is another functor and : ND × N1 →
NC is a homotopy between Nu and Nu ′ , then we obtain a
homotopy B : BD × △ 1 → BC between Bu and Bu ′ .
2. A natural transformation Φ between u, u ′ : D → C induces
a homotopy B Φ : BD × △ 1 → BC between Bu and Bu ′ .
3. If C has an initial or a terminal object, then BC is contractible.
The topological map Bu : BD → BC induces the maps u ∗
and u ∗ as in Corollary 2.2.13.
Remark 2.2.30. We must emphasize that continuous maps between
classifying spaces are not always realized by functors. In
other words, not all topological maps BD → BC come from
a simplicial map ND → NC by Lemma 2.2.11. For instance,
there is a natural homeomorphism between BC and BC op , but
one cannot construct a functor between C and C op realizing
this homeomorphism, except in very special situations (e.g. C
is a group). See Example 2.2.14.
We shall come back to comparing classifying spaces via functors
by introducing Quillen’s work.

62 2 Simplicial methods
2.2.3 Cup product and cohomology ring
In this section, we introduce cross product (or external product),
cup product (or internal product) and the resulting multiplicative
structure on simplicial cohomology.
Theorem 2.2.31 (Eilenberg-Zilber). Let X, Y be two
simplicial sets. There is a natural chain homotopy equivalence
ξ : C ∗ (X × Y, R) → C ∗ (X, R) ⊗ C ∗ (Y, R).
Proof. The chain map, called the Alexander-Whitney map,
can be explicitly constructed as follows
n∑
ξ(a, b) = d i+1 · · · d n (a) ⊗ d i 0(b),
i=0
where (a, b) ∈ C n (X × Y, R) = R[X n ] ⊗ R[Y n ].
The inverse of ξ (up to chain equivalence), called the Eilenber-
Zilber map
is defined by
η(c ⊗ d) =
η : C ∗ (X, R) ⊗ C ∗ (Y, R) → C ∗ (X × Y, R),
∑
(p,q)-shuffles
σ
(−1) |σ| (s σ(1) · · · s σ(p) c, s σ(p+1) · · · s σ(n) d)
for c ∈ C p (X, R) and d ∈ C q (Y, R) with p + q = n. Here a
(p, q)-shuffle σ is a permutation of n letters such that
σ(1) < · · · < σ(p) and σ(p + 1) < · · · < σ(n).
⊓⊔
Remark 2.2.32. The Alexander-Whitney and Eilenberg-Zilber
maps lead to chain homotopy equivalences between normalized
chain complexes, in light of Theorem 2.2.5.

2.2 Nerves, classifying spaces and cohomology 63
Let C be a small category and R a commutative ring with
identity. For C ∗ (−, R) = C ∗ (−, R) ∧ , we have a cross product
induced by the Alexander-Whitney map
× : C i (X, R)⊗C j (Y, R) → Hom R (C i (X, R)⊗C j (Y, R), R) ≃ →C i+j (X×
with the left map given by
(f × g)(a, b) = f(d i+1 · · · d n (a))g(d i 0(b)),
if f ∈ C i (X, R), g ∈ C j (Y, R), a ∈ C n (X, R) and b ∈
C n (Y, R). One can check that ∂ ∗ (f ×g) = ∂ ∗ (f)×g+(−1) i f ×
∂ ∗ (g). Thus it induces a cross product on cohomology
× : H i (X, R) ⊗ H j (Y, R) → H i+j (X × Y, R)
When R is a field, the map × is an isomorphism by Künneth
formula.
Proposition 2.2.33. Let X, Y be two simplicial sets and
τ : X ×Y → Y ×X the twist map, defined by τ(a, b) = (b, a)
for any a ∈ X n and b ∈ Y n . Then τ induces an isomorphism
τ ∗ : H ∗ (X × Y, R) → H ∗ (Y × X, R).
Suppose f ∈ H i (X, R) and g ∈ H j (Y, R). Then τ ∗ (f ×g) =
(−1) ij g × f.
Proof. Since τ is an isomorphism of simplicial sets, it certainly
induces an isomorphism on (co-)chain complexes and
(co-)homology. Now we calculate the value of τ ∗ .
Define τ ∗
′ = ξτ ∗ η : C ∗ (Y, R) ⊗ C ∗ (X, R) → C ∗ (X, R) ⊗
C ∗ (Y, R), as shown in the following commutative diagram
C ∗ (Y × X, R)
η ≃
τ ∗
C ∗ (X × Y, R)
C ∗ (Y, R) ⊗ C ∗ (X, R)
τ
′ C ∗ (X, R) ⊗ C ∗ (Y, R).
∗
≃
ξ

64 2 Simplicial methods
Then by direct calculation, it follows τ ∗ ′ is chain homotopy
equivalent to the canonical chain equivalence τ ∗ ′′ : C ∗ (Y, R) ⊗
C ∗ (X, R) → C ∗ (X, R) ⊗ C ∗ (Y, R) determined by b ⊗ a ↦→
(−1) ij a ⊗ b for any a ∈ X i and b ∈ Y j . Hence τ ∗ : H ∗ (X ×
Y, R) → H ∗ (Y × X, R) is given by the chain equivalence
induced by τ ∗
′′
Hom R (C ∗ (X, R)⊗C ∗ (Y, R), R) → Hom R (C ∗ (Y, R)⊗C ∗ (X, R), R)
On the other hand, there is a canonical chain equivalence
C ∗ (X, R) ⊗ C ∗ (Y, R) → C ∗ (Y, R) ⊗ C ∗ (X, R)
defined in the same fashion as τ ′′
∗ . Thus there is a commutative
diagram of cochain complexes
C ∗ (X, R) ⊗ C ∗ (Y, R)
∼ =
C ∗ (Y, R) ⊗ C ∗ (X, R)
Hom R (C ∗ (X, R) ⊗ C ∗ (Y, R), R) ∼=
Hom R (C ∗ (Y, R) ⊗ C ∗ (X, R), R.
Since the vertical maps give rise to the cross products by
sending f ⊗g to f ×g for any f ∈ C i (X, R) and g ∈ C j (Y, R),
while the horizontal maps map f ⊗g and f ×g to (−1) ij g ⊗f
and (−1) ij g × f, respectively, our formula for τ ∗ follows from
it.
⊓⊔
Definition 2.2.34. There exists a diagonal functor ∆ : C →
C × C such that ∆(x) = (x, x) and ∆(α) = (α, α) for any
x ∈ Ob C and α ∈ Mor C, respectively.
The diagonal functor induces a natural chain map, the simplicial
diagonal map, also denoted by
∆ : C ∗ (C, R) → C ∗ (C × C, R).
Composing with the Alexander-Whitney map
C ∗ (C × C, R) → C ∗ (C, R) ⊗ C ∗ (C, R),

2.2 Nerves, classifying spaces and cohomology 65
we get the diagonal map,
∆ : C ∗ (C, R) → C ∗ (C, R) ⊗ C ∗ (C, R).
More explicitly for any x 0 → x 1 → · · · → x n , a base element
of C n (C, R),
n∑
∆(x 0 → · · · → x n ) = (x 0 → · · · → x i )⊗(x i → · · · → x n ).
i=0
By dualizing the diagonal map, we obtain the cup product on
C ∗ (C, R) = C ∗ (C, R) ∧ as
∪ : C i (C, R) ⊗ C j (C, R) ×
−→C i+j (C × C, R) ∆
−→C i+j (C, R).
Suppose f ∈ C i (C, R) and g ∈ C j (C, R) with i + j = n. Then
(f ∪ g)(x 0 → · · · → x n ) = ∑ n
k=0 f(x 0 → · · · → x k )g(x k → · · · → x n )
= f(x 0 → · · · → x i )g(x i → · · · → x n ).
From direct calculation we get ∂ ∗ (f ∪ g) = ∂ ∗ (f) ∪ g +
(−1) i f ∪ ∂ ∗ (g) for any f ∈ C i (C, R) and g ∈ C j (C, R). Consequently
the cup product passes to cohomology
∪ : H i (C, R) ⊗ H j (C, R) ×
−→H ∗ (C × C, R) ∆
−→H i+j (C, R),
which makes the graded R-module H ∗ (C, R) = ⊕ i≥0 H i (C, R)
a graded ring. This is the ordinary cohomology ring of C with
coefficients in R. We may readily verify the following equality.
Proposition 2.2.35. f ∪ g = (−1) ij g ∪ f for any f ∈
H i (C, R) and g ∈ H j (C, R).
Proof. It follows from Proposition 2.2.33 because the cup
product is induced by a cross product.
⊓⊔
It means that the ordinary cohomology ring of a small category
is graded commutative. Since the cup product is exactly
the same as the one for the topological cohomology ring

66 2 Simplicial methods
H ∗ (BC, R) = ⊕ i≥0 H i (BC, R). We actually have a ring isomorphism
H ∗ (C, R) ∼ = H ∗ (BC, R).
Remark 2.2.36. When C is a (discrete) group, H ∗ (C, R) is the
usual group cohomology ring.
Corollary 2.2.37. Let u : D → C be a functor. Then u ∗ :
H ∗ (C, R) → H ∗ (D, R) is a ring homomorphism.
This ring homomorphism is usually called the restriction.
Proposition 2.2.38. Let D and C be small categories. Suppose
k is a field. Then the cup product of any two positive
degree elements in H ∗ (D ∗ C, k) is constantly zero.
Proof. In Proposition 2.2.8 we computed homology groups of
the join of two small categories. Accordingly there are formulas
for cochain complexes and cohomology as well, i.e. for
each n > 0
˜C n (D ∗ C, k) ∼ = [˜C∗ (D, k) ⊗ ˜C∗ (C, k)] n−1 ,
and
H n (D ∗ C, k) =
⊕
i+j=n−1
˜H i (D, k) ⊗ ˜Hj (C, k).
It implies that H n (D ∗ C, k) is spanned over a set of cohomology
classes represented by functions of the form η α such
that η α (β) = 1 if β = α and zero otherwise, and such that
α ∈ N n (D ∗ C) consists of exactly one morphism γ x,y where
x ∈ Ob D and y ∈ Ob C, as introduced in Definition 2.2.7.
By direct computation, the cup product of any two such functions
is constantly zero. Hence we are done.
⊓⊔
There is a topological interpretation to the above result.
In fact the cohomology ring of a join of two spaces always
have trivial cup product. See Example 2.2.24 (4). There is a
topological interpretation. Suppose X and Y are two spaces.

2.3 Quillen’s work on classifying spaces 67
Then their join X∗Y is a suspension Σ(X×Y/X∨Y ). For any
suspended space, its cohomology ring has trivial cup product.
The structure of X ∗ Y also explains Proposition 2.2.8. In
Example 2.2.6 (1), the cup product of any two elements of
positive degree in H ∗ (C, R) is zero.
2.3 Quillen’s work on classifying spaces
In order to assure a functor u : D → C inducing a simplicial
homotopy equivalence ND ≃ NC, one must have a functor on
the opposite direction. However one may very often obtain a
(topological) homotopy equivalence Bu : BD → BC without
the existence of such v : C → D that Bv is also a homotopy
equivalence. Such homotopies certainly are not simplicial and
will be investigated now.
2.3.1 Quillen’s Theorem A
Quillen’s Theorem A (also called Quillen’s Fibre Theorem)
provides sufficient conditions on a functor between two small
categories which guarantee the functor inducing a homotopy
equivalence between classifying spaces.
Theorem 2.3.1 (Quillen’s Theorem A). Let u : D →
C be a functor between two small categories. If either all
the overcategories or all the undercategories are contractible.
Then Bu is a homotopy equivalence.
Proof. The proof is given in Section 2.4.3.
Under certain circumstances, one may replace the over- or
undercategories with some simpler constructions.
Definition 2.3.2. Let u : D → C be a functor between small
categories. For each x ∈ Ob C, we define the fibre of u over x
⊓⊔

68 2 Simplicial methods
to be the category u −1 (x) whose objects are preimages of x
and whose morphisms are preimages of 1 x .
Apart from intuition, the fibres are usually not good for
comparing the homotopy types of two classifying spaces. For
instance if G is a group and H is a subgroup, then the inclusion
functor has fibre at • equal to the trivial category. But
there is no reason to say that BH would be homotopy equivalent
to BG, especially when we think about the extreme case
where H is the trivial subgroup of G.
Definition 2.3.3. Let u : D → C be a functor between small
categories. We call D prefibred over C if the natural functor i x :
u −1 (x) → x\u has a right adjoint h x for each x ∈ Ob C. We
call D precofibred over C if the natural functor i x : u −1 (x) →
u/x has a left adjoint h x for each x ∈ Ob C.
Corollary 2.3.4. Suppose u : D → C such that D is either
prefibred or precofibred over C. If the fibre u −1 (x) over every
x ∈ Ob C is contractible, then Bu is a homotopy equivalence.
Proof. Since there are adjoint functors between u −1 (x) and
x\u (or u/x) for every x ∈ Ob C. The condition implies the
contractibility of all the undercategories (or overcategories).
Then we apply Quillen’s Theorem A.
⊓⊔
We will see some prefibred and precofibred examples in the
next section. Note that, for G a group and H a subgroup,
i : H ↩→ G is a simple example that H is neither prefibred
nor precofibred over G, unless H = G.
Suppose u : D → C makes D precofibred over C. Then we
have
Hom u −1 (x)(h x (a, α), b) ∼ = Hom u/x ((a, α), i x (b))
for any b ∈ Ob u −1 (x) and (a, α) ∈ Ob u/x. Let us try
to understand h x . The set Hom u/x ((a, α), b) consists of ex-

2.3 Quillen’s work on classifying spaces 69
actly those f ∈ Hom D (a, b) such that u(f) = α : u(a) →
x = u(b). The identity 1 hx (a,α) corresponds to a morphism
in Hom u/x ((a, α), i x h x (a, α)), which is some morphism ˜f ∈
Hom D (a, h x (a, α)) such that u( ˜f) = α. Then each f ∈
Hom D (a, b), satisfying u(f) = α, must uniquely factor through
˜f. If f = g ˜f for g ∈ HomD (h x (a, α), b) such that u(g) = 1 x ,
then g corresponds to f ∈ Hom u/x ((a, α), i x (b)) in the isomorphism.
We have alternative descriptions of h x and h x .
Lemma 2.3.5. Suppose u : D → C is as above and γ : x →
y is a morphism in C.
1. If D is prefibred over C, then it induces a functor
γ ⋆ : u −1 (y) → y\u → x\u → u −1 (x).
Furthermore the right adjoint of u −1 (x) → x\u is given by
(α, a) ↦→ α ⋆ (a).
2. If D is precofibred over C, then it induces a functor
γ ⋆ : u −1 (x) → u/x → u/y → u −1 (y).
Furthermore the left adjoint of u −1 (y) → u/y is given by
(b, β) ↦→ β ⋆ (b).
Proof. We shall only prove (2) because the proof for (1) is
similar.
Let (b, β) ∈ Ob u/y. Then
Hence β ⋆ (b) = h y (b, β).
β ⋆ : b ↦→ (b, 1 y ) ↦→ (b, β) ↦→ h y (b, β).
Definition 2.3.6. Let u : D → C be a functor between small
categories. We call D fibred over C if it is prefibred over C and
moreover (βα) ⋆ = α ⋆ β ⋆ for any composable α, β ∈ Mor C.
⊓⊔

70 2 Simplicial methods
We call D cofibred over C if it is precofibred over C and
moreover (βα) ⋆ = β ⋆ α ⋆ for any composable α, β ∈ Mor C.
2.3.2 Constructions over categories and relevant functors
We describe several important constructions here, which also
serve an examples to illustrate Quillen’s Theorem A.
Definition 2.3.7. Suppose C is a small category. The (first)
category of factorizations in C, denoted by F (C), has Ob F (C) =
Mor C. An object admits a morphism to another object if and
only if as morphisms in C, the first object factors through the
second.
Iterating the construction, for non-negative integer n, we
can define F n (C) = F (F n−1 (C)) as the n-th category of factorizations
in C. Here F 0 (C) is defined to be C.
We shall see that in order to understand C very often we
have to utilize F (C).
In order not to cause confusions, when a morphism α ∈
Mor C is considered as an object in F (C), we write it as [α].
In the above definition, a morphism in F (C) is a pair (α ′ , α ′′ ) :
[α] → [β] and is customarily pictured by
y
α ′
α
x
y ′ x ′ α ′′
β
Example 2.3.8.1. When C is a group, F (C) has all group elements
as objects. The category F (C) is a groupoid whose
skeleton is isomorphic to the group itself.
2. When C is a poset, then F (C) is another poset. For 1, we
have F (1) as follows

2.3 Quillen’s work on classifying spaces 71
(α,1 0 )
(1 1 ,1 0 )
[1 ← 0]
(1 1 ,α)
[0 ← 0]
[1 ← 1]
(1 0 ,1 0 )
(1 1 ,1 1 )
For convenience we denote the non-identity morphism 1 ← 0
by α.
One can see that there are two natural covariant functors
t : F (C) → C and s : F (C) → C op , where t stands for target
while s means source.
Proposition 2.3.9. Suppose D ≃ C. Then F (D) ≃ F (C).
Proof. Let u : D → C and v : C → D be such that vu ∼ = Id D
and uv ∼ = Id C . We construct two functors between F (C) and
F (D) which induce an equivalence.
We define F (u) : F (D) → F (C) by F (u)([α]) = [u(α)],
and F (u)(α ′ , α ′′ ) = (u(α ′ ), u(α ′′ )). One can verify this gives a
functor. Similarly there exists a functor F (v) : F (C) → F (D).
Since F (v)F (u)([α]) = [vu(α)] ∼ = [α], we have F (v)F (u) ∼ =
Id F (D) . Similarly we get F (u)F (v) ∼ = Id F (C) . Thus F (D) ≃
F (C).
⊓⊔
Proposition 2.3.10. Consider the canonical functors C←F t (C) →C s o
Then
1. F (C) is cofibred over both C and C op . It implies, for any
x ∈ Ob C, the natural functor i x : t −1 (x) → t/x has a
left adjoint, and, for any y ∈ Ob C op , the natural functor
i y : t −1 (y) → s/y has a left adjoint;
2. for any x ∈ Ob C, the category t −1 (x) ∼ = (Id C /x) op has an
initial object;

72 2 Simplicial methods
3. for any y ∈ Ob C op , the category s −1 (y) ∼ = y\Id C has an
initial object.
Consequently both t and s induce homotopy equivalences.
Proof. The category F (C) is precofibred over C because we
can define h x ([α], α ′ ) = [α ′ α] for any x ∈ Ob C and ([α], α ′ ) ∈
Ob t/x such that
Hom t −1 (x)(h x ([α], α ′ ), [β]) ∼ = Hom t/x (([α], α ′ ), i x (β)).
We can compute t −1 (x) of t : F (C) → C for every x ∈ Ob C
and find t −1 (x) ∼ = (Id C /x) op . It has an initial object [1 x ]
and thus is contractible. This means that t/x is always contractible
and thus t induces an equivalence by Quillen’s Theorem
A. Moreover from the description of t −1 (x) we can see
that F (C) is cofibred over C.
We can similarly prove that s : F (C) → C op induces an
equivalence as well, with s −1 (y) ∼ = y\Id C .
⊓⊔
A hidden fact in the proof is that whenever there is a category
C, we can produce many over- and undercategories Id C /x
and x\Id C , all of which are contractible. This will be used for
an important construction later on.
Remark 2.3.11. Here the homotopy equivalence Bt : BF (C) →
BC usually is not a simplicial homotopy equivalence as we do
not expect to have a functor from C to F (C) giving a homotopy
equivalence BC → BF (C). If such a functor were to
exist, then we would obtain a functor inducing the homotopy
equivalence BC → BC op , and hence inducing isomorphism on
homology H∗(C, R) → H∗(C op , R), which rarely happens. See
Example 2.2.14.
Definition 2.3.12. The functor ∇ = (t, s) : F (C) → C e =
C × C op is called the skew diagonal functor. The category C e
is called the enveloping category of C.

2.3 Quillen’s work on classifying spaces 73
The above construction is a special case of a more general
definition F (u) for any u : D → C, naturally equipped with a
functor F (u) → D × C op . Indeed F (Id C ) = F (C), see Section
2.4.3.
Example 2.3.13. When C is a poset, both F (C) and C e are
posets and ∇ : F (C) → C e is an embedding. For the poset
C = 1, we have F (1) ↩→ 1 e as follows
[1 ← 0] (1, 0)
[0 ← 0]
[1 ← 1]
↩→ (0, 0)
(1, 1)
(0, 1)
Proposition 2.3.14. The category F (C) is cofibred over C e .
It implies that, for any (y, x) ∈ Ob C e , the functor i (y,x) :
∇ −1 (y, x) → ∇/(y, x) has a left adjoint. Thus N∇ −1 (y, x) ≃
N∇/(y, x).
Furthermore ∇ −1 (y, x) ∼ = Hom C (x, y).
Proof. We first prove that F (C) is precofibred over C e . To
this end, we need to show that for every (y, x) ∈ Ob C e , the
functor i (y,x) : ∇ −1 (y, x) → ∇/(y, x) has a left adjoint. It is
easy to identify ∇ −1 (y, x) with the discrete set Hom C (x, y).
Then we define h (y,x) ([α], (α ′ , α ′′ )) = [α ′ αα ′′ ] ∈ Hom C (x, y)
which leads to an isomorphism
Hom ∇ −1 (y,x)([α ′ αα ′′ ], [β]) ∼ = Hom ∇/(y,x) (([α], (α ′ , α ′′ )), ([β], (1 y , 1 x ))).
Note that in any case either morphism set has at most one
element.
If (γ ′ , γ ′′ ) : (y, x) → (y ′ , x ′ ) is a morphism, then (γ ′ , γ ′′ ) ⋆ is
given by
[α] ↦→ ([α], (1 y , 1 x )) ↦→ ([α], (γ ′ , γ ′′ )) ↦→ [γ ′ αγ ′′ ],

74 2 Simplicial methods
for any α ∈ Hom C (x, y). Hence one can verify that F (C) is
cofibred over C e .
⊓⊔
We note that some of the overcategories or undercategories
or fibres can be empty. As a reminder, the homology of ∅ is
constantly zero.
Remark 2.3.15. In the previous two propositions F (C) are
cofibred over C, C op as well as C e . It is not fibred over these
categories. For instance in Example 2.3.13, i x : ∇ −1 (1, 0) =
[1 ← 0] → ∇/(1, 0) ∼ = F (1) has only a left adjoint and no
right adjoint because [1 ← 0] is a terminal object in F (1). In
these notes, it is very important to compute limits over various
over and undercategories. We often do it by establishing
adjoint functors between these categories and some simpler
categories. Then by Proposition 1.2.4 we will know whether
we can simplify the computations of either direct or inverse
limits (usually not both!). We shall see many explicitly calculations
from Chapter 3.
2.4 Further categorical and simplicial constructions
In this section we will introduce the definition of a Grothendieck
construction which is important when we deal with finite
categories constructed from groups. Grothendieck constructions
are a certain kind of homotopy colimits. By Thomason’s
Homotopy Colimit Theorem, every homotopy colimit
is a Grothendieck construction. In order to provide the interested
reader a necessary understanding of these constructions,
we will define bisimplicial sets, and since bisimplicial sets are
introduced, there is no reason not to give proofs of Quillen’s
and Thomason’s theorems. Although this chapter is the most
suitable place to present these materials, it is possible to postpone
reading them until they are needed. Indeed, only un-

2.4 Further categorical and simplicial constructions 75
derstanding the Grothendieck constructions and knowing the
statements are sufficient for our purposes in these notes.
2.4.1 Grothendieck constructions
Let C be a small category and F : C → Cat a functor. Then
we can construct a small category, called the Grothendieck
construction Gr C F, as follows.
The objects are pairs of the form (x, a) with x ∈ Ob C and
a ∈ Ob F(x). Let (x, a) and (y, b) be two objects. A morphism
(x, a) → (y, b) is a pair (α, f) such that α ∈ Hom C (x, y) and
f ∈ Hom F(b) (F(α)(a), b). It can be pictured as
x
α
α
y A ′
B ′
C ′
D ′
y A ′
B ′
x
Note that there always exists a functor π : Gr C F → C.
Example 2.4.1. Let u : D → C be a functor between small
categories. Then the over- and undercategories can be realized
as Grothendieck constructions. Given an object x ∈ Ob C, we
can define two functors u = Hom C (u(−), x) : D → Set and
u = Hom C (x, u(−)) : D → Set. One can verify that Gr D u
and Gr D u are isomorphic to u/x and x\u, respectively.
We record one more important Grothendieck construction,
although it is not needed in this chapter.
Definition 2.4.2. A small category C is an EI category if all
endomorphisms are isomorphisms.
C ′
D ′

76 2 Simplicial methods
Typical examples of EI categories are groups and posets.
For each EI category C, one can introduce a partial order on
the set of isomorphism classes of objects by [x] ≤ [y] if and
only if Hom C (x, y) ≠ ∅. Let us denote by [C] the poset of
isomorphism classes of objects in C. Obviously there exists a
natural functor p : C → [C]. Since [C] is a poset, we know the
barycentric subdivision of [C], written as S([C]), is another
poset whose objects are chains of non-isomorphism objects
[x] n = [x 0 ] → [x 1 ] → · · · → [x n ] and there is a morphism
[x] n → [y] m if and only if [y] m is a subchain of [x] n . We define
a functor
˜p : S([C]) → Cat
by asking ˜p([x] n ) to be a category whose objects are the y n =
y 0 → y 1 → · · · → y n in NC n such that p(y i ) = [x i ] for
all 0 ≤ i ≤ n, and whose morphisms are (n + 1)-tuples of
morphisms in C making the following diagram commutative
α
y 1 0 y 1
α 2
· · · αn
y n
γ 0
γ 1
γ n
y ′ 0 β 1
y ′ 1 β 2
· · · βn
y ′ n
Since C is EI, all γ i are isomorphism. This means that ˜p([x] n )
is always a groupoid. One can readily verify that for any
x n → y m in Mor(sd[C]), there is a well defined functor
˜p(x n ) → ˜p(y m ) given by first removing objects in x n which
are not isomorphic to any object in y m , and then composing
appropriate morphisms in what is left of x n .
Definition 2.4.3 (Slominska). Let C be an EI category.
Then its subdivision, written as S(C), is defined to be Gr S([C])˜p.
Alternatively S(C) is a category whose objects are chains of
non-isomorphisms x n = x 0 → x 1 → · · · → x n in C, and a
morphism from x n to y m = y 0 → y 1 → · · · → y m is an (m +

2.4 Further categorical and simplicial constructions 77
1)-tuple of isomorphisms (γ 0 , · · · , γ m ), making the following
diagram commutative
α 1
x n1
α 2
· · · αn x nm
y 0
γ 1
β1
y 1
γ n
β2
· · · βn
y m
x n0
γ 0
in which every x ni ∈ {x i } n i=0 and x ni
∼ = yi in C. From this
description there exists a functor s : S(C) → C by x n ↦→ x 0 .
Thus there are maps s ∗ : H∗(S(C), R) → H∗(C, R) and s ∗ :
H ∗ (C, R) → H ∗ (S(C), R).
Proposition 2.4.4 (Slominska). The map s induces a homotopy
equivalence BS(C) ≃ BC. Hence we have isomorphisms
s ∗ : H∗(S(C), R) → H∗(C, R) and s ∗ : H ∗ (C, R) →
H ∗ (S(C), R).
Proof. We shall prove all undercategories are contractible. In
order to do so, we show S(C) is prefibred over C. Then one
can easily verify that for each x ∈ Ob C, s −1 (x) has x as a
terminal object and hence is contractible.
To show S(C) is prefibred over C we only have to find a
right adjoint to the inclusion i x : s −1 (x) → x\s. Suppose
(α, x 0 → x 1 → · · · → x n ) is an object in x\s. Then we define
a functor x\s → s −1 (x) by
(x 0
α 1
→x1 → · · · → x n , α) ↦→ (x α 1α
→x 1 → · · · → x n ).
Moreover if (γ 0 , γ 1 , · · · , γ m ) is a morphism from (α, x 0 →
x 1 → · · · → x n ) to (α ′ , x ′ 0 → x ′ 1 → · · · → x ′ m), then its image
is defined as (1 x , γ 1 , · · · , γ n ). Now one can check this functor
is right adjoint to i x . Hence we are done.
⊓⊔
These isomorphisms will be generalized in Chapter 4.

78 2 Simplicial methods
There are two or three occasions in this book that we need
to understand the classifying spaces of some categories. Thus
we provide two examples on subdivisions of categories.
Example 2.4.5.1. Let C be the following EI category
G
x
α
y H .
Then its underlying poset [C] is x → y, and S([C]) is the
poset [x] ← [x → y] → [y]. One can easily find that ˜p[x]
and ˜p[y] are
G x , y H
respectively. The groupoid ˜p[x → y] is
G×H {x α
y} ,
in which we denote by α the only object in ˜p[x → y]. The two
functors ˜p[x → y] → ˜p[x] ˜p[x → y] → ˜p[y] are the obvious
projections. Thus the category S(C) has objects of the forms
([x], x), ([y], y) and ([x → y], x α →y). All the morphisms in
S(C) are of the following forms
•([x → y] → x, g), ∀g ∈ Mor(˜p[x]) = Aut C (x) = G;
•([x → y] → y, h), ∀h ∈ Mor(˜p[y]) = Aut C (y) = H;
•(1 [x→y] , (g ′ , h ′ )), ∀(g ′ , h ′ ) ∈ Mor(˜p[x → y]) = Aut C (x) ×
Aut C (y) = G × H.
As one can see the subdivision S(C) is more complicated
than C itself so it seems not helpful in terms of understanding
BC through BS(C). However, in the next section, we shall
see it does help if we combine with Thomason’s Homotopy
Colimit Theorem.
For future reference, we record two more finite EI categories,
which share the same underlying poset with C. The reader
may try to figure out their subdivisions. We shall give only
partial descriptions of these constructions.

2.4 Further categorical and simplicial constructions 79
2. Let D be the following category
h
x
1 x
α
β
y {1 y }
,
where h 2 = 1 x and βh = α. Then we also have poset [D] =
x → y, and S([D]) = [x] ← [x → y] → [y]. We can find ˜p[x]
and ˜p[y] are
h
1 x
x , y {1 y }
respectively. The third groupoid ˜p[x → y] has two isomorphic
objects x α →y and x β →y. Its skeleton is the trivial category.
3. Let E be the following EI category
h
g
x
1 x
α
β
gh
y {1 y }
,
where g 2 = h 2 = 1 x , gh = hg, αh = α, αg = β, βh =
β, βg = β. Then we again have poset [E] = x → y, and
S([E]) = [x] ← [x → y] → [y]. By direct calculations, ˜p[x]
and ˜p[y] are
h
g
x
1 x
, y {1 y }
gh
respectively. The groupoid ˜p[x → y] has two isomorphic
objects x→y α and x→y. β However in this case, its skeleton
is isomorphic to C 2 = {1 x , h}, which is the subgroup of
Aut C (x) × Aut C (y) that stabilizes the set {α, β}.
We will come back to these examples at the end of the next
section.

80 2 Simplicial methods
2.4.2 Bisimplicial sets and homotopy colimits
Definition 2.4.6. A bisimplicial set is a contravariant functor
△ × △ to Set.
Because
Hom Cat (△ op ×△ op , SimpSet) ∼ = Hom Cat (△ op , Hom Cat (△ op , SimpSet
clearly we may regard a bisimplicial set as a simplicial object
in the category of simplicial sets. Thus a bisimplicial set is
indeed a simplicial simplicial set.
Alternatively, a bisimplicial set is a bigraded sequence of sets
X p,q for p, q ≥ 0, together with horizontal face and degeneracy
maps d h : X p,q → X p−1,q and s h : X p,q → X p+1,q as well as
vertical face and degeneracy maps d v : X p,q → X p,q−1 and s v :
X p,q → X p,q+1 . These maps must satisfy simplicial identities
in either direction, and in addition every horizontal map must
commutes with every vertical map. The total complex of a
bisimplicial set, Tot(X), is the simplicial set
Tot(X) n := ⊕ n≥0
X n,∗ × △ n / ∼,
in which the equivalence is given by (d h i x, y) ∼ (x, di y) and
(s h i x, y) ∼ (x, si y).
We can also consider bisimplicial objects in an abelian category
A-Mod for some associative ring. Then by Dold-Kan
Correspondence, the category of bisimplicial A-modules is
equivalent to the category of first quadrant double complexes
of A-modules.
Fix a commutative ring R. Each bisimplicial set gives rise
to a first quadrant double complex R[X] p,q with differentials
∂ h p,q = ∑ i (−1)i d h i and ∂ v p,q = (−1) p ∑ i (−1)i d v i . Consequently

2.4 Further categorical and simplicial constructions 81
the category of bisimplicial R-modules is equivalent to the category
of first quadrant double chain complexes of R-modules.
Definition 2.4.7. The diagonal, diagX, of a bisimplical set
X is the simplicial set obtained via restriction along the diagonal
functor △ → △ × △.
Theorem 2.4.8 (Bousfield-Kan ). Let X, Y be two bisimplicial
sets.
1. There is a natural simplicial isomorphism Tot(X) ∼ = diagX.
2. Given a bisimplicial map Φ p,q : X p,q → Y p,q , if for all q,
X ∗,q → Y ∗,q is a weak homotopy equivalence, then so is
diagΦ : diagX → diagY .
Proof. For the first part, we define a map Ψ, induced by the
maps X p,∗ × △ p → diagX p given by
(x, d p y) ↦→ d h px ∈ X p,p ,
for any (x, d p y) ∈ X p,q × △ p n. One may readily check that Ψ
is a simplicial isomorphism.
As for Part (2), we will not provide a proof here but it is
not surprising from Part (1). See [9], [Tornehave], [Reedy] and
[8, 16].
Definition 2.4.9. Suppose G : C → SimpSet is a functor.
⊕Then the simplicial replacement of G is the bisimplicial set
∗
G which in dimension n consists of the coproduct
( ⊕ G) ⊕
n :=
G(x 0 ).
∗
x 0 →···→x n ∈NC n
The horizontal face map d i maps G(s(σ)) to G(s(d i σ)) by the
identity map if i > 0 and by α 0 if i = 0. The horizontal degeneracy
map s i maps the simplicial set G(s(σ)) to G(s(s i σ))
by the identity map.

82 2 Simplicial methods
Definition 2.4.10. Let G : C → SimpSet be a functor.
The homotopy colimit of G, hocolim C G, is diag( ⊕ ∗ G).
For example one can easily see that hocolim C Npt = NC.
Theorem 2.4.11 (Thomason’s Homotopy Colimit Theorem).
Suppose C is a small category and F : C → Cat is
a functor. Let Gr C F be the Grothendieck construction on F.
Then there is a natural weak equivalence
NGr C F ≃ hocolim C NF.
The proof is in the next section. It is clear from the proof of
Thomason’s theorem, and Definitions 2.2.2 and 2.4.9 that
BGr C F ≃ hocolim C BF.
Remark 2.4.12. The homotopy colimit is a replacement in
T op of the colimit we introduced earlier. In the category T op,
the colimit of a diagram of spaces does exit. However it does
not behave very well and thus is not good to work with in algebraic
topology. One can find a well-known example in [DH].
There is also a dual notion of homotopy limit but we shall
not touch it in this book.
The following examples provide the only three homotopy
colimits that the reader needs to know for our purposes. In
fact, they are only used in Section 4.2.2, for computing certain
ordinary (co)homology.
Example 2.4.13. When C is the poset P : a ← b → c, for any
F : P → Cat the homotopy colimit hocolim P BF is called a
homotopy pushout because it fits into the right lower corner
of the following diagram

2.4 Further categorical and simplicial constructions 83
BF(b→a)
BF(b) BF(b→c)
BF(a)
BF(c)
hocolim P BF
such that it is commutative up to homotopy of continuous
maps, and hocolim C BF enjoys a certain universal property,
analogous to that of a colimit.
Example 2.4.14. In Example 2.4.5, the classifying spaces of
the subdivisions of all three categories can be realized as homotopy
pushouts, according to Example 2.4.13. For instance,
in Example 2.4.5 (3) we have BE ≃ B(C 2 ×C 2 )/BC 2 because
we have the following homotopy pushout diagram
inclusion
B˜p([x → y]) = BC
2 B(C 2 × C 2 ) = B˜p([x])
Bpt
B˜p([y]) = B•
B(C 2 × C 2 )/BC 2 ≃ hocolim P B˜p ≃ BS(E
Similarly, in Example 2.4.5 (1), we get BC ≃ BG ∗ BH (a
join of spaces), and in Example 2.4.5 (2) BD ≃ BC 2 . At
least in these small examples, subdivisions are useful to obtain
the homotopy type of classifying spaces of categories. This is
one of the the reasons why we introduced the Grothendieck
construction earlier.
2.4.3 Proofs of Quillen’s Theorem A and Thomason’s theorem
The proofs are taken directly from the original papers of
Quillen [61] and Thomason [77].
In order to prove Quillen’s Theorem A, we shall first generalize
the construction of category of factorizations.
Definition 2.4.15. Suppose u : D → C is a functor between
two small categories. We define F (u) to be a category

84 2 Simplicial methods
whose objects are (a, α, x) with a ∈ Ob D, x ∈ Ob C and
α ∈ Hom C (x, u(a)). A morphism from (a, α, x) to (b, β, y)
is a pair of morphisms (f, γ) with f ∈ Hom D (a, b) and
γ ∈ HomC(y, x) such that β = u(f)αγ as in the diagram
u(a)
u(f)
u(b)
Note that there exists two natural functors P t : F (u) →
D op and P s : F (u) → C, as well as P t × P s : F (u) →
D × C op . For any category C, F (Id C ) is exactly the category
of factorizations in C, F (C), such that P s = s and P t = t.
Let us notice that any element (a 0 , α 0 , x 0 ) → (a 1 , α 1 , x 1 ) →
· · · → (a n , α n , x n ) ∈ NF (u) n can be pictured as a commutative
diagram
x 0
α 0
x 1
α 1
α
β
x
y
γ
· · ·
x n
α n
u(a 0 )
u(a 1 )
· · ·
u(a n )
Proof (of Quillen’s Theorem A). We will only prove the case
where all undercategories associated wtih u are contractible.
The other case can be shown similarly.
Let F (u) be as above. We define a functor N[Id C /u(−)] ∗ :
⊕D → SimpSet and consider its simplicial replacement X =
∗ N[Id C/u(−)] ∗ the bisimplicial set with X p,q consists of
pairs of sequences of morphisms in C and D, respectively,
(x p → · · · → x 0 → u(a 0 ), a 0 → a 1 → · · · → a q ).
The horizontal and vertical face and degeneracy maps are
given in the obvious way. We have a simplicial map X ∗q →
ND q as well as a simplicial map X p∗ → NCp op . Since the

2.4 Further categorical and simplicial constructions 85
diagonal of X is the nerve of F (u), these two functors provide
two more simplicial maps
⊕
N(Id C /u(a 0 )) op
q → ⊕
N• q = ND q
a 0 →···→a q a 0 →···→a q
and ⊕
N(x 0 \u) p →
⊕
N• p = NCp op .
x p →···→x 0 x p →···→x 0
The geometric realizations of these two maps are homotopy
equivalences because both Id D /u(a 0 ) and x 0 \u are contractible
(the latter by assumption). Hence by Theorem 2.4.7
(2) we obtain two homotopy equivalences
BD
BP t
BP
BF (u)
s
BC op .
By examining the following diagram
C op
P s
C op
P s =s
F (u)
u ′
F (C)
P t
D
u
P t =t C
where u ′ is defined by u ′ (a, α, x) = (u(a), α, x). It forces Bu ′
and thus Bu to be homotopy equivalences.
⊓⊔
The proof of Thomason’s homotopy colimit theorem is in
spirit similar to that of Quillen’s Theorem A. We remind the
reader that NF (u) ≃ diagX = hocolim D N[Id C /u(−)] in the
above proof, and one verifies that Gr D Id C /u(−) ∼ = F (u).
Proof (of Thomason’s theorem). We shall establish a natural
map
η : hocolim C NF → NGr C F
and construct a new functor ˜F : C → Cat so that we have
natural equivalences

86 2 Simplicial methods
λ 1
λ
hocolim C NF
hocolim C N ˜F 2
NGr C F
satisfying ηλ 1 ≃ λ 2 .
Step 1: Similar to the proof of Quillen’s Theorem A, hocolim C NF
is the diagonal of ⊕ ∗
NF such that for any integer n ≥ 0,
( ⊕ ∗ NF) n consists of n-simplicies
α
(x
1 α
0→ · · ·
n f 1 f n
→xn , a 0→ · · · →an )
where x 0 → · · · → x n ∈ NC n and a 0 → · · · → a n ∈ NF(x 0 ) n .
We define η by
α
η n (x
1 α
0→ · · ·
n f 1 f n
→xn , a 0→ · · · →an )
= (x 0 , a 0 ) (α 1,F(α 1 )(f 1 ))
−→ (x 1 , F(α 1 )(a 1 )) (α 2,F(α 2 α 1 )(f 2 ))
−→ · · ·
· · · (α n,F(α n···α 1 )(f n ))
−→ (x n , F(α n · · · α 1 )(a n ))
Step 2: Let us construct λ 1 . We define ˜F : C → Cat to be
the functor
˜F(x) = π/x
for π : Gr C F → C. Recall that the objects in ˜F(x) are of the
form ((y, a), α) such that y ∈ Ob C, a ∈ Ob F(y) and α : y →
x, and moreover any α : y → x induces ˜F(α) : ˜F(y) → ˜F(x).
There exists a canonical functor
˜F(x) → F(x)
for any x ∈ Ob C, given by ((y, a), α) ↦→ F(α)(a). This functor
has a right adjoint F(x) → ˜F(x) by a ↦→ ((x, a), 1x ). Consequently
N ˜F(x) → NF(x) is a simplicial homotopy equivalence.
Since all functors constructed previously, ˜F(x) → F(x),
assemble to a natural transformation

2.4 Further categorical and simplicial constructions 87
˜F → F
of functors C → Cat, we obtain a simplicial homotopy equivalence
N ˜F → NF which gives rise to a homotopy equivalence,
by Theorem 2.4.7 (2),
λ 1 : hocolim C N ˜F
≃
→hocolimC NF.
Step 3: Now we show there is a simplicial homotopy equivalence
λ 2 : hocolim C N ˜F → GrC F.
Similar to Step 1, hocolim C N ˜F is the diagonal of
⊕∗ N ˜F such
that for any integer n ≥ 0, ( ⊕ ∗ N ˜F)n consists of n-simplicies
α
(x
1 α
0→ · · ·
n (h 1 ,l 1 )
→xn , c 0 → · · · (h n,l n )
→ c n )
where x 0 → · · · → x n ∈ NC n and c 0 → · · · → c n ∈
N ˜F(x0 ) n = [N(π/x 0 )] n . Note that each c i is of the form
((y i , b i ), β i ) such that y i ∈ Ob C, b i ∈ Ob F(y i ) and β i : y i →
x 0 . Moreover, h i+1 : y i → y i+1 satisfies β i+1 h i+1 = β i , and
l i+1 : F(h i+1 )(b i ) → b i+1 , for every i ≥ 0. Note that there are
morphisms α i · · · α 1 β i : y i → x i . We define λ 2 by
α
λ 2 (x
1 α
0→ · · ·
n
→xn , ((y 0 , b 0 ), β 0 ) (h 1,l 1 )
→ · · · (h n,l n )
→ ((y n , b n ), β n ))
= (y 0 , b 0 ) (h 1,l 1 )
→ · · · (h n,l n )
→ (y n , b n ).
Let us consider NGr C F as a bisimplicial set with (p, q)-
simplicies NGr C F p,q = NGr C F q . This means it is constant
in horizontal direction. Clearly NGr C F ∗ = diagNGr C F ∗,∗ .
The simplicial map λ 2 is the diagonalization of an obvious
bisimlicial map λ : ⊕ ∗ Gr C ˜F → NGr C F ∗,∗ .
Thus, by Theorem 2.4.7 (1), we only have to show for each q,
λ ∗,q : ⊕ ∗ N ˜F∗,q → NGr C F ∗,q is a homotopy equivalence. But
λ ∗,q is the coproduct over all q-simplicies (x 0 , c 0 ) → · · · →

88 2 Simplicial methods
(x q , c q ) of NGr C F of the map ⊕ p x q\Id C → ⊕ p
•. Then by
Theorem 2.4.7 (2) λ ∗,q , and so λ 2 , are homotopy equivalences.
Step 4: Finally we establish a simplicial hopmotopy
: hocolim C N ˜F × N1 → NGrC F
from ηλ 1 to λ 2 . We define it by
(i+1) zeros
α
(x
1 α
0→ · · ·
n
{ }} {
→xn , ((y 0 , b 0 ), β 0 ) → · · · → ((y n , b n ), β n ))×( 0, · · · , 0, 1, · · ·
= (y 0 , b 0 ) (h 1,l 1 )
→ · · · (h i,l i )
→ (y i , b i ) (α i+1···α 1 β i ,F(α i+1···α 1 β i+1 )(l i+1 ))
−→
(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 ))
−→
(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 )).
One may verify that all simplicial identities are met. Hence
we are done.
⊓⊔

Chapter 3
Category algebras and their representations
Abstract The concept of a category algebra is introduced
here. The relation ship between the module category of a category
algebra and an appropriate functor category is demonstrated
at the very beginning. We shall characterize the intrinsic
properties of category algebras. Especially we show
how one may compare a category algebra with a cocommutative
bialgebra so such algebras enjoy similar homological
properties. We are particularly interested in finite EI category
algebras because we may classify their indecomposable
projective and simple modules. When dealing with category
algebras, the usual homological tools, such as induction and
coinduction, are replaced by Kan extensions. Examples are
supplied to show how one may compute Kan extensions of
various modules.
Throughout this chapter, the base ring R is always a commutative
ring with identity. A module will normally be a left
module, unless otherwise specified. If S is a set, then RS
stands for the free R-module generated by S.
3.1 Basic concepts and examples
3.1.1 Category algebras
The category algebra is a natural way to linearize a category.

90 3 Category algebras and their representations
Definition 3.1.1. Let C be a small category and R a commutative
ring. The category algebra RC is a free R-module
whose basis is the set of morphisms of C. We define a product
on the basis elements of RC by
{
α ◦ β, if α and β can be composed in C;
α ∗ β =
0 , otherwise
and then extend this product linearly to all elements of RC.
With this product, RC becomes an associative R-algebra.
If Ob C is finite, it is easy to see that ∑ x∈Ob C 1 x is the identity
of RC where 1 x is the identity of Aut C (x). If a category
C is finite, then Ob C is finite and RC is of finite R-rank.
Another way to linearize a category C over a ring R is to
construct a new category C R whose objects are the same as C
while the morphisms between any two objects x, y ∈ C R are
Hom CR (x, y) := RHom C (x, y). This category C R is additive
and one can obtain the category algebra RC by forgetting the
category structure in an obvious way.
Example 3.1.2. Suppose C is the following finite category
1 x
x
with g 2 = 1 y and α = gα. Then the category algebra RC is
of rank 4 with identity 1 RC = 1 x + 1 y . It contains two group
algebras R{1 x } and R{1 y , g}.
We say C is connected if C as a (directed) graph is connected.
Every category C can be written as the disjoint union
of connected components C = ⊎ i∈J C i , where each C i is a connected
full subcategory and J is an index set. As a consequence
the category algebra RC becomes a direct product of
α
1 y
y
g

3.1 Basic concepts and examples 91
ideals RC i , i ∈ J. Thus in order to study the properties of RC
it suffices to study the properties of each RC i . For simplicity
and some technical reasons we often make the connectedness
assumption.
3.1.2 Representations of categories and Mitchell’s Theorem
We shall show that a fundamental property of the category
algebra RC is that it provides a mechanism for investigating
R-representations of C, which we define now.
Definition 3.1.3. An (R-)representation of a category C is
a (covariant) functor M : C → R-mod.
When the base ring is understood, we often abbreviate an
R-representation as a representation of C. All representations
of C form a functor category (R-mod) C , which is an abelian
category with enough projectives and injectives so we can
talk about subfunctors and quotient functors. The following
fundamental theorem of category representations tells us an
alternative description of the functor category.
Theorem 3.1.4 (Mitchell). For any small category C with
finitely many objects, there exist functors ι : (R-mod) C →
RC-mod and σ : RC-mod → (R-mod) C such that
1. σ ◦ ι ∼ = Id (R-mod)
C, and
2. ι is fully faithful, and moreover if Ob C is finite then ι◦σ ∼ =
Id RC-mod .
Thus if Ob C is finite, the R-representations of C can be
identified with the unital RC-modules.
Proof. Assume F : C → R-mod is a representation of C.
We construct a free R-module M F = ⊕ x∈Ob C
F (x). For any
m ∈ F (x) and morphism α ∈ Mor C, we ask α·m = F (α)(m),
if t(α) = x, or α · m = 0 if t(α) ≠ x. By extending this

92 3 Category algebras and their representations
operation linearly we obtain an RC-module structure on M F .
This construction can be easily verified to define a functor
ι : (R-mod) C → RC-mod.
Conversely, if M is an RC-module, we may define a functor
F M by F M (x) = 1 x · M. Since, if α ∈ Hom C (x, y) and m ∈
1 x · M, α · m = (1 y ◦ α) · m = 1 y · (α · m) ∈ 1 y · M, we see
that F M : C → R-mod is well defined. It induces a functor
σ : RC-mod → (R-mod) C .
We may readily check these two functors satisfy 1 and 2. ⊓⊔
Similar statements can be made between right RC-modules
and contravariant functors from C to R-mod.
Remark 3.1.5. Throughout these notes we will be particularly
interested in RC-modules lying in ι{(R-mod) C }. One reason
is that many important modules are indeed of this form. Another
reason is that, on top of module-theoretic methods, we
may apply simplicial or homotopy-theoretic tools. Hence we
want to focus on categories with finitely many objects, although
many results make sense for arbitrary small categories.
Moreover since there is a well developed representation theory
of finite-dimensional algebras, from now on we will restrict
our attention to the finite categories. Consequently for practical
reasons we will not distinguish between RC-mod and
(R-mod) C , and normally refer to an object in these categories
as an RC-module.
Among all RC-modules, there is a distinguished one that we
introduce below. It plays a key role throughout these notes.
Definition 3.1.6. For any category C, the constant functor
or trivial module R : C → R-mod, is defined by R(x) = R
for all x ∈ Ob C and R(α) = Id R for all α ∈ Mor C.

3.1 Basic concepts and examples 93
We provide several examples of simplicially constructed
modules where both the module and functor aspects are described.
Example 3.1.7.1. The constant functor R corresponds to the
module R Ob C, on which RC acts via α · x = y, if α ∈
Hom C (x, y), or zero otherwise.
2. The regular module RC corresponds to a functor such that
RC(x) = 1 x · RC = RHom C (−, x).
3. The opposite algebra (RC) op is isomorphic to the algebra of
the opposite category C op , and thus a set of base elements
are Mor C op . Consider the enveloping category of C, namely
C e = C × C op . Then we have
RC e ∼ = (RC) e = (RC) ⊗ R (RC) op ,
the enveloping algebra of RC. Consequently RC is an RC e -
module via (α, β op ) · γ = αγβ, for any α, β, γ ∈ Mor C. As
a functor RC : C e → R-mod, we have
RC(y, x) = RHom C (x, y)
if (x, y) ∈ Ob C e .
Occasionally we will need to discuss right RC-modules. A
dual version of Mitchell’s theorem says that the right RCmodules
corresponds to the contra-variant functors/representations
of C. There exists a natural functor (−) ∧ = Hom R (−, R) :
RC-mod → mod-RC. We give an explicit description of the
dual module M ∧ = Hom R (M, R) (a right RC-module) of
M ∈ RC-mod. In case R is a field, ∧ induces an antiisomorphism
between RC-mod and mod-RC.
Lemma 3.1.8. Suppose M ∈mod-RC. Then its dual M ∧ ∈
RC-mod has values M ∧ (x) = M(x) ∧ = Hom R (M(x), R),
and each α ∈ Mor C acts via M(α) ∧ = Hom R (M(α)(−), R).

94 3 Category algebras and their representations
Proof. As a free R-module, M ∧ = ⊕ x∈Ob C M(x)∧ . Each α ∈
Hom C (x, y) acts as M(α) : M(y) → M(x) and thus induces
a map M(α) ∧ : M(x) ∧ → M(y) ∧ . One can readily check that
M ∧ (x) is exactly M(x) ∧ so we know the structure of M ∧ as
a right RC-module.
⊓⊔
The next statement follows from direct calculation.
Corollary 3.1.9. We have R ∧ = R. Here the first R is a
left module and the second is a right module.
3.1.3 Three examples
Let us consider three motivating examples.
Example 3.1.10. Let G be a (discrete) group. Then the group
can be regarded as a category with only one object •, whose
morphisms are the elements of G. The group algebra RG is
the same as the category algebra RG. A left RG-module M
can be regarded as the representation of G given by a certain
functor Φ : G → R-mod, sending • to M and Aut G (•) = G
into Aut R (M). The trivial RG-module R is exactly the trivial
module of RG.
Note that in this case G ∼ = G op and thus RG ∼ = RG op and
RG e ∼ = RG ⊗ RG ∼ = R(G × G).
A quiver q = (Γ 0 , Γ 1 ) is a directed graph having Γ 0 and
Γ 1 as the set of vertices and the set of arrows, respectively.
The path algebra of q can be thought as a category algebra
as follows. Any directed graph G = (Γ 0 , Γ 1 ) may be used to
generate a category C G on the same set Γ 0 of objects, where
the morphisms of this category are the strings of composable
arrows of G. It is called the free category generated by G.
Example 3.1.11. The path algebra of a quiver q is the category
algebra RC q of the free category C q .

3.1 Basic concepts and examples 95
It is necessary to point out that, given a category, its category
algebra is usually different from its path algebra, if we
consider the category as a quiver at the same time. Nevertheless,
there is a relationship between these two algebras, as we
now explain.
Proposition 3.1.12. Let q be a category. We may regard q
as a quiver and form the free category C q over q. There is a
natural functor u : C q → q, which extends to a surjective homomorphism,
still denoted by u : RC q → Rq, from the path
algebra of q to the category algebra of q, such that its kernel
I is generated by { α 1
← α 2
← − α ←
1α 2
}, where α1 and α 2 are arrows
of q. This epimorphism induces a natural isomorphism of
R-algebras RC q /I and Rq.
Proof. The functor u is defined as follows. For each x ∈ Ob C q ,
φ(x) = x. For each α ∈ Mor C q , u(α) = the composite of
the maps in the string α. It can be extended linearly to an
epimorphism u : RC q → Rq, having the kernel I. ⊓⊔
The last example of a category algebra is the incidence algebra
of a locally finite poset (partially ordered set). A (closed)
interval of P is a subposet [x, y] ⊂ P which consists of all
objects z such that x ≤ z ≤ y for a given pair of objects
x, y in P. The incidence algebra I(P, R) is an R-algebra of all
functions f : Int(P) → R, where Int(P) is the set of intervals
of P. The multiplication (also called convolution) if defined
by
(fg)([x, y]) = ∑
f([x, z])g([z, y]).
x≤z≤y
It has an identity δ such that
{
1, if x = y;
δ([x, y]) =
0 , otherwise.

96 3 Category algebras and their representations
On the other hand since whenever x ≤ y in a poset we can
replace ≤ with an arrow x → y, a poset is always a category.
Example 3.1.13. Let P be a finite poset. Then the incidence
algebra I(P, R) of P is isomorphic to the category algebra
RP.
3.2 A closed symmetric monoidal category
Suppose C is finite. We describe the intrinsic structure of RC
which make it comparable with a cocommutative bialgebra.
Later on we shall see RC-mod possesses similar homological
properties as the module category of a cocommutative bialgebra.
3.2.1 Tensor structure and an intrinsic characterization of category algebras
Suppose G is a group. The category RG-mod is a symmetric
monoidal category equipped with a tensor product − ⊗ R −
and a tensor identity R. The reason why RG-mod enjoys such
good properties is that RG is a cocommutative Hopf algebra.
For general properties of Hopf algebras, we refer the reader
to []. For an arbitrary unital associative R-algebra there are
maps
1. There is a map µ : A ⊗ R A → A, called the multiplication.
2. The following diagram is commutative (associativity)
A ⊗ A ⊗ A µ⊗Id
Id⊗µ
A ⊗ A µ
A ⊗ A
µ
3. There is a map ι : R → A, called the unit, and the following
diagram is commutative
A

3.2 A closed symmetric monoidal category 97
R ⊗ A ι⊗Id
A ⊗ A
∼ =
µ
Id⊗ι
∼ =
A ⊗ R
A
If A = RG, it is also a counital coalgebra, which means, on
top of the previous maps, we have the following extra maps,
reversing maps in 1-3.
4. There is a comultiplication, RG → RG ⊗ R RG.
6. The following diagram is commutative (coassociativity)
∆
RG
RG ⊗ RG
∆
RG ⊗ RG
∆⊗Id
Id⊗∆
RG ⊗ RG ⊗ RG,
5. The group algebra has an augmentation map ɛ : RG → R,
also called the counit.
R ⊗ RG
∼ =
ɛ⊗Id
RG
∆
∼ =
RG ⊗ RGId⊗ɛ RG ⊗ R
Any R-module satisfying conditions 4-6 is called a counital
coassociative coalgebra. Any R-algebra satisfying 4-6 is
named a bialgebra.
A group algebra is usually not commutative, but
7. there is a twist map τ : RG ⊗ R RG → RG ⊗ R RG, given
by τ(a ⊗ b) = b ⊗ a, such that the following diagram is
commutative
∆
RG ⊗ RG
RG
τ
∆
RG ⊗ RG.

98 3 Category algebras and their representations
An R-coalgebra or bialgebra satisfying 7 is called cocommutative.
8. Moreover there is an antipode η : RG → RG in the sense
that if δ(a) = ∑ i b i ⊗ c i then ∑ i b iη(c i ) = ∑ i η(b i)c i =
ɛ(a)e.
If a bialgebra has an antipode, then it is called a Hopf algebra.
Remark 3.2.1. For a group algebra, η is explicitly given by
g → g −1 . The comultiplication is given by g ↦→ g ⊗ g.
A cocommutative bialgebra A has the significant property
that A-mod is a symmetric monoidal category with tensor
identity R. More precisely it means that for any two A-
modules M and N, M ⊗ R N is still an A-module, M ⊗ R N ∼ =
N ⊗ R M, and M ⊗ R R ∼ = M. In other words, A-mod inherits
various constructions on R-mod. Note that the tensor identity
R plays the role of both the unit and co-unit of A. If A
is Hopf, then even Hom R (M, N) becomes an A-module such
that
Hom A (L ⊗ M, N) ∼ = Hom A (L, Hom R (M, N)).
In the literature, Hom R (M, N) is called a function object or
the internal hom. Furthermore when R = k is a field, A
is cocommutative Hopf and M, N are finite-dimensional, we
have an isomorphism of A-modules
M ∧ ⊗ N ∼ = Hom R (M, N).
Since R ∈ A-mod, M ∧ is a left A-module under the circumstance,
satisfying (M ∧ ) ∧ ∼ = M.
In what follows, we begin with a description about how R-
mod gives rise to a symmetric monoidal category structure
on RC-mod. Then we characterize the structure of RC using

3.2 A closed symmetric monoidal category 99
some structure maps comparable to those of a cocommutative
bialgebra. In this way we demonstrate why a category algebra
and a cocommutative bialgebra, as well as their module categories,
are similar yet different. It is the intrinsic structure of a
category algebra that makes it a natural and interesting subject
of investigation. We note that R-mod itself is the module
category of the R-category algebra of the trivial category •,
based on Mitchell’s theorem.
Fix a finite category C. The so-called internal product on
(R-mod) C ≃ RC-mod, in which the tensor product is denoted
by ˆ⊗ R , is defined by (M ˆ⊗ R N)(x) = M(x) ⊗ R N(x)
for any M, N ∈ RC-mod ≃ (R-mod) C and x ∈ Ob C. The
module structure of M ˆ⊗N can be viewed as given by the comultiplication
∆ : RC → RC ⊗ R RC, induced by the canonical
diagonal functor ∆ : C → C × C whose action on each
α ∈ Mor C is ∆(α) = α ⊗ α. One can easily verify that the
trivial module R is the tensor identity with respect to ˆ⊗ R . For
the sake of simplicity, we shall write ⊗ for ⊗ R , and ˆ⊗ for ˆ⊗ R ,
throughout this book. We note that in the literature (see for
example [25]), the symbol ⊗ is often used instead of ˆ⊗ for the
internal tensor product of functors. The new notation ˆ⊗ is introduced
because we need to distinguish between M ⊗ N and
M ˆ⊗N. In fact, as R-modules, the inclusion M ˆ⊗N ⊂ M ⊗N,
for any RC-modules M and N, is often strict.
As we mentioned above, the diagonal functor ∆ : C → C ×C
induces a co-multiplication on the category algebra RC. The
co-multiplication ∆ : RC → RC ⊗ RC almost gives us a coalgebra
structure on RC except that there is not a suitable
choice of a map RC → R which would serve as the co-unitary
map. Recall from Example 3.1.7 (1) that the trivial module
R can be realized by R Ob C. The next result states a natural
augmentation map from RC to R Ob C.

100 3 Category algebras and their representations
Lemma 3.2.2. There exists a surjective linear map
defined on base elements by
where t(α) is the target of α.
RC ɛ ↠R Ob C = R,
ɛ(α) = t(α),
Proof. The surjective map induces a RC-module structure on
R Ob C as follows. If x ∈ Ob C and β ∈ Hom C (x ′ , y), then
βx = y when x = x ′ , and βx = 0 otherwise.
⊓⊔
If we return to ∆, we realize that the image of it really
lies in RC ˆ⊗RC, a subspace of RC ⊗ RC, and moreover
∆ : RC → RC ˆ⊗RC becomes a RC-map, since RC ˆ⊗RC, unlike
RC ⊗ RC, is a well-defined RC-module. The above observations
hint that, in order to get a sound “co-algebra” structure,
one needs to use ˆ⊗ other than ⊗ to define the structure
maps. This motivates us to write down the following maps
which resemble almost all of the structure maps for a cocommutative
bialgebra. Abusing terminology, we adopt the same
names for the structure maps of a category algebra, such as
co-multiplication and co-unit, as their counterparts for a bialgebra.
In a coalgebra, the augmentation map and co-unit are
the same, so by analogy the map ɛ in the preceding lemma
will occasionally be called the co-unit. We emphasize that the
unit, given by the natural inclusion map R ∼ ι
= R · 1 RC →RC,
is different from the co-unit.
Proposition 3.2.3. Let RC be the category algebra of a finite
category C. Then we have the following R-linear maps:
a co-multiplication ∆ : RC → RC⊗RC, defined by ∆( ∑ ∑
α λ αα) =
α λ αα ⊗ α, a co-unit ɛ : RC → R defined as above, and a
twist map τ : RC ⊗RC → RC ⊗RC defined on base elements

3.2 A closed symmetric monoidal category 101
by τ(α ⊗ α ′ ) = α ′ ⊗ α, such that the following diagrams are
commutative: 1. co-associativity
∆
RC
RC ⊗ RC
∆
RC ⊗ RC
2. co-unitary property
R ˆ⊗RC
∼ =
ɛ⊗Id
and 3. co-commutativity
∆
RC ⊗ RC
∆⊗Id
Id⊗∆ RC ⊗ RC ⊗ RC,
RC
∆
∼ =
RC ˆ⊗RC
Id⊗ɛ RC ˆ⊗R
RC
τ
∆
RC ⊗ RC.
If we denote by µ the multiplication, then we have furthermore
three commutative diagrams: 4. Multiplication and comultiplication
RC ⊗ RC
∆⊗∆
RC ⊗ RC ⊗ RC ⊗ RC
µ
RC
∆
RC ⊗ RC
Id⊗τ⊗Id
5. Unit and co-multiplication:
RC
and 6. Unit and co-unit
µ
R ⊗ R
∆
ι⊗ι
µ⊗µ
RC ⊗ RC ⊗ RC ⊗ RC,
RC ˆ⊗RC,

102 3 Category algebras and their representations
RC
ι
R
ɛ
where the R-linear map η : R → R is defined by 1 ↦→
∑
x∈Ob C x,
We note that the only missing diagram is the compatibility
of multiplication and co-unit. The reason is that one usually
cannot give R a meaningful algebra structure so that
ɛ : RC → R becomes an algebra homomorphism. The existence
of the above tensor structure ˆ⊗ is well known. Notably
it has been used to define the internal product in functor
homology theory, see for example [25].
Remark 3.2.4. If we remove the finiteness condition on C, then
there is no identity in the algebra kC. Nevertheless many of
the constructions are still valid, although in this case, we are
forced to use the full subcategory V ect C k
because kC-mod does
not have a tensor structure.
Remark 3.2.5. If C = P happens to be a poset, then there
is another co-multiplication that one can find in [72]. Let
α ∈ Mor P, one has ¯∆ : kP → kP ⊗ kP such that
¯∆(α) = ∑ {β,γ|βγ=α}
β ⊗ γ. Nevertheless it is possible to give
an incidence algebra a Hopf algebra structure. The augmentation
map ¯ɛ : kP → k is given by ¯ɛ(α) = 1 if α is an identity,
or ¯ɛ(α) = 0 otherwise. The antipode kP → kP can also be
explicitly constructed. Suppose α ∈ Mor P is a morphism in
the poset. Then a factorization of α is a way of writing α as
a product α 1 α 2 · · · α n for some non-identity α i ∈ Mor P. The
length l(α) of α is the maximal n for such a factorization to
exist. Then the antipode is given by α ↦→ ∑ β (−1)l(β) β, in
which β runs over all possible factors of α.
η
R,

3.2 A closed symmetric monoidal category 103
Indeed one can do this for the category algebras of the socalled
(finite) Möbius categories. A main feature of a Möbius
category is that each object only has one automorphism.
It seems that RC can only be a (cocommutative) Hopf algebra
if C is of the two extreme structures: either a group
(all morphisms are invertible), or a Möbius category (identity
morphisms are the only invertible morphisms) .
Remark 3.2.6. Let A be an R-algebra. Although A-mod is not
monoidal in general, the module category A e -mod is always
equipped with a tensor product ⊗ A such that A is the tensor
identity.
When A = RC is a category algebra, (RC) e ∼ = RC e becomes
the category algebra of the category C e := C × C op , and hence
there are two distinct monoidal structures on RC e -mod. The
monoidal structure given by ⊗ RC is more interesting to us, and
we shall deal with it in Chapter 5 on Hochschild cohomology.
3.2.2 The internal hom
Usually one can not define an antipode for a category algebra.
This causes a problem when one attempts to define
Hom k (M, N) as a sensible kC-module (generalizing the group
case). We shall give a remedy below. Another relevant fact is
that the product of two projective kC-modules is in general
not projective, see Example 3.. These make many homological
properties, such as the cohomology theory, of a finitedimensional
category algebra different from those of a finitedimensional
cocommutative Hopf algebra.
Proposition 3.2.7 (Swenson). Suppose C is a finite category
(or at least Ob C is finite). Let M, N ∈ RC-mod. Then
we can define an internal hom Hom(M, N) ∈ RC-mod.

104 3 Category algebras and their representations
Proof. We want to define Hom(M, N) such that there is an
isomorphism
Hom RC (L ˆ⊗M, N) ∼ = Hom RC (L, Hom(M, N)).
Let RC = RC · 1 x . Assume the above isomorphism. Then the
right hand side is Hom RC (RC·1 x , Hom(M, N)) ∼ = Hom(M, N)(x),
and the left hand side is Hom RC (RC · 1 x ˆ⊗M, N). Hence we
may define Hom(M, N) by
Hom(M, N)(x) = Hom RC (RC · 1 x ˆ⊗M, N).
When C = G is a group, it is straightforward from Swenson’s
definition that Hom(M, N) ∼ = Hom R (M, N) as RG-modules.
Alternatively consider the diagonal functor ∆ : C → C × C.
Because R(C ×C) ∼ = RC ⊗RC, for L, M, N ∈ RC-mod, L⊗M
is a R(C × C)-module, and
Hom RC (L ˆ⊗M, N) = Hom RC (Res ∆ (L ⊗ M), N)
∼ = HomR(C×C) (L ⊗ M, RK ∆ N)
∼ = HomRC⊗RC (L ⊗ M, RK ∆ N)
∼ = HomRC (L, Hom RC (M, RK ∆ N)).
Here in Hom RC (M, RK ∆ N), RK ∆ N ∈ RC ⊗ RC-mod is
considered as a (R · 1 RC ) ⊗ RC-module and hence a RCmodule.
The RC ⊗ RC-, or rather the RC ⊗ (R · 1 RC )-, module
structure on RK ∆ N provides a RC-module structure on
Hom RC (M, RK ∆ N). Then we may verify that Hom(M, N) =
Hom RC (M, RK ∆ N).
With the internal hom, one might attempt to define a dual
module of M by Hom(M, R) in order to generalize the group
case. The following example explain why this is not a good
idea.
⊓⊔

3.3 Functors between module categories 105
Example 3.2.8. Let k be a field of characteristic two and C
the following category
{1 x }
x
α
β
y
{1 y ,g}
with g 2 = 1 y , gα = α and gβ = β.
We consider two modules S x,k and S y,k such that S x,k (x) =
k, S x,k (y) = 0, S y,k (x) = 0 and S y,k (y) = k. Then Hom(S x,k , k)
can be easily shown to be isomorphic to S x,k . However Hom(S y,k , k) ∼ =
S 2 x,k ⊕ S y,k because kC1 x ˆ⊗S y,k
∼ = S
2
y,k
and kC1 y ˆ⊗S y,k
∼ = kC1y .
Thus Hom(Hom(S y,k , k), k) ≁ = Sy,k .
3.3 Functors between module categories
We investigate functors for comparing two module categories
over category algebras.
3.3.1 Restriction on algebras and modules
We prove some basic properties, many of which follow directly
from simple reasoning.
Definition 3.3.1. Suppose u : D → C is a (covariant) functor.
We define Res u : RC-mod → RD-mod to be the restriction
along u. Given a module M ∈ RC-mod, we have
Res u M = M ◦ u ∈ RD-mod.
Lemma 3.3.2. Let u : D → C be a functor. Then Res u R =
R.
Since a functor u : D → C also extends linearly to a natural
map of R-modules ū : RD → RC, it is reasonable to ask if
ū is always an algebraic homomorphism because if it is, the
so-called change-of-base-ring or the representation-theoretic
restriction, ↓ RC
RD : RC-mod → RD-mod, should coincide with

106 3 Category algebras and their representations
Res u . The answer to the question is no, and here is a simple
example. Let D = 1 and C = •. There is a unique functor pt :
D → C. The induced map ¯pt : RD → RC is not an algebra
homomorphism since the product of the two isomorphisms in
D is zero while the product of their images is not.
Proposition 3.3.3. A functor u : D → C extends linearly
to an algebra homomorphism ū : RD → RC if and only
if u is injective on Ob D. When this happens, the induced
functor followed by 1 RD , 1 RD ◦ ↓ RC
RD : RC-mod → RD-mod, is
exactly Res u .
Proof. We know u(βα) = u(β)u(α) for any pair of composable
morphisms α, β in D. The injectivity of u implies two
morphisms α, β ∈ Mor(D) are composable if and only if
u(α), u(β) ∈ Mor(C) are composable.
If u is injective on Ob D, then we define a map ū : RD →
RC as the linear extension of functor u, i.e., ū( ∑ ∑
i r iα i ) =
i r iū(α i ) for any r i ∈ R, α i ∈ Mor(D). This ū is indeed
an algebra homomorphism because our previous observation
of u implies ū(( ∑ j r jβ j )( ∑ i r iα i )) = ū( ∑ j r jβ j )ū( ∑ i r iα i ) is
always true.
On the other hand if the linear extension ū : RD → RC is
an algebra homomorphism then we must have ū(0) = 0 and
then ū(1 x )ū(1 y ) = ū(1 x · 1 y ) = 0 unless x = y. This suggests
that u is injective on Ob C ′ .
When ū : RD → RC is an algebra homomorphism, we
show 1 RD ◦ ↓ RC
RD = Res u : RC-mod → RD-mod. Let M be an
RC-module and γ ∈ Mor(D). Then it corresponds to some
F M ∈ (R-mod) C . The RD-modules Res u M and 1 RD·(M ↓ RC
RD )
are isomorphic because Res u M = M Resu (F M ), γ · M = u(γ)M
for any γ ∈ Mor D and 1 RD kills the elements of M ↓ RC
RD that
are not supported on any objects of D.
⊓⊔

3.3 Functors between module categories 107
Proposition 3.3.4. Let D and C be equivalent small categories.
Then (R-mod) C ≃ (R-mod) D , an equivalence which
sends the constant functor to the constant functor. If both
Ob C and Ob D are finite then RC and RD are Morita equivalent.
Proof. We show that the two functor categories (R-mod) C and
(R-mod) D are equivalent. Then it implies the module categories
RC-mod and RD-mod are equivalent, hence RC and
RD are Morita equivalent. In fact if u : D → C and v : C → D
are equivalences, we have Res u Res v
∼ = IdRD := Id RD-mod : (Rmod)
D → (R-mod) D because of the following diagram
M(vu(x)) = (Res u Res v M)(x) ∼ =
(Id RD M)(x) = M(x)
M(vu(α))=(Res u Res v M)(α)
M(vu(y)) = (Res u Res v M)(y) ∼=
(Id RD M)(α)=M(α)
(Id RD M)(y) = M(y)
where M ∈ (R-mod) D , α : x → y ∈ Mor D and Id RD is the
identity functor. Similarly we can show Res v Res u
∼ = IdRC : (Rmod)
C → (R-mod) C . Clearly the constant functor restricts to
the constant functor always.
⊓⊔
Remark 3.3.5. Recall from Chapter 2 that if D ≃ C then
ND ≃ NC (and BD ≃ BC). The above result is an algebraic
consequence of a category equivalence. On one hand the existence
of a Morita equivalence implies that the two (category)
algebras are the same, in terms of homological properties.
On the other hand there are non-equivalent finite categories
whose algebras are Morita equivalent, see Example 3.1.16 (5).
By contrary a simplicial/topological equivalence does not
provide a Morita equivalence. For instance the posets 0 and
1 are simplicial homotopy equivalent, because there are natural
adjoint functors between them. However if k is a field
then k0 and k1 are not equivalent as algebras. The reason

108 3 Category algebras and their representations
is that k0 ∼ = k has only one simple module k while k1 has
two simple modules because its identity can be written as a
sum of orthogonal primitive idempotents 1 0 + 1 1 . Since their
category algebras have different numbers of non-isomorphic
simple modules, k0-mod and k1-mod cannot be equivalent
as categories.
3.3.2 Kan extensions of modules
Since not every functor u : D → C has an adjoint, we have to
look for other functors for comparing RD-mod and RC-mod.
Clearly the restriction Res u possesses two adjoint functors,
the left and right Kan extensions LK u and RK u . Here will
be our first attempt to study Kan extensions of modules. We
note that, because Res u is exact, LK u preserves projectives
while RK u preserves injectives.
In light of Proposition 3.2.3, if one is careful enough, the
restriction can be written using the usual module-theoretic
notation ↓ C D :=↓RC RD , and then in the literature one may see that
its left adjoint and right adjoint are denoted by ↑ C D := RC ⊗ RD
−, the induction, and ⇑ C D := Hom RD(RC, −), the coinduction.
They are the Kan extensions in different forms. Since there
is plenty of description of the induction and coinduction for
algebra representations, in these notes we shall mainly discuss
the functor-theoretic aspect of them through Kan extensions.
A special situation of Kan extensions, the group case, can
be found in Example 1.2.11. One can see that using moduletheoretic
methods it is not easy to obtain most of the following
results for general category algebras.
Suppose M ∈ RD-mod and u : D → C is a functor. Then
we obtain an RC-module LK u M. As a functor, for each x ∈
Ob C, (LK u M)(x) is given by a direct limit, and if α : x → y

3.3 Functors between module categories 109
is a morphism, it induces a map (LK u M)(x) → (LK u M)(y)
by the universal property of −→
lim. This specifies the RC-action
on LK u M.
Suppose P is a poset and Q is a subposet. We call Q an
ideal of P if for a pair of object x ∈ Ob Q and y ∈ Ob P
such that x ≤ y in P then y belongs to Q. Before we start
the first calculation we mention that if D is a full subcategory
of a small category C then any functor over D can be naively
considered as a functor over C by asking its value to be zero
on any objects that do not belong to D. In other words, any
RD-module is naturally an RC-module under the assumption.
Proposition 3.3.6. Suppose P is a poset and Q is an ideal
together with the inclusion i : Q ↩→ P. Then for any M ∈
RQ-mod, the RP-module LK i M is exactly M, regarded as
an RP-module.
Proof. One can find that (LK i M)(x) ∼ = lim
−→Q≤x
M, where Q ≤x
is a subposet of Q consisting of objects smaller or equal to x.
By assumption this poset is either empty or has a terminal
object x. Consequently (LK i M)(x) is either zero if x ∉ Ob Q
or M(x) otherwise. Hence we are done.
Alternatively we can compute RP ⊗ RQ M. For any x ∈
Ob P, we have
1 x·(RP⊗ RQ M) = ∑ y≤x
RHom P (y, x)⊗M = ∑ y≤x
RHom P (y, x)⊗1 y·M.
This is not zero if and only if 1 y · M ≠ 0 which means y ∈
Ob Q. By assumption, it forces x belongs to Q. Thus 1 x ·
(RP⊗ RQ M) ≠ 0 if and only if x ∈ Ob Q. When this happens,
1 x · (RP ⊗ RQ M) = 1 x · M. Thus we obtain the same result.
⊓⊔

110 3 Category algebras and their representations
The concept of an idea can be defined for a certain class of
categories, which are called EI-categories and will be introduced
shortly. The above result stays true in this generality.
We will also give a dual concept of coideal later on.
Proposition 3.3.7. Suppose u : D → C satisfies the condition
that for every x ∈ Ob C u/x is connected. Then
LK u R = R.
Especially for t : F (C) → C, we have LK t R ∼ = R as an
RC-module.
Proof. When u/x is connected for an x ∈ Ob C, we have
lim R = R. Then by the universal property of direct limit
−→u/x
we see any α : x → y must induce the identity morphism on
R. Thus LK u R = R. ⊓⊔
For any M ∈ RF (C)-mod we can compute LK t M explicitly.
If x ∈ Ob C, then −→t/x
lim M ∼ = lim −→t
M by Propositions
2.3.10 and 1.2.4. The category t −1 (x) has an initial ob-
−1 (x)
ject [1 x ] and thus in general we cannot simplify the direct
limit. We want to construct an example based on this description
showing that LK t (M ˆ⊗N) ≁ = LKt (M) ˆ⊗LK t (N) for
M, N ∈ RF (C)-mod.
Example 3.3.8. Consider the following poset P and its factorization
category F (P) (a poset too)
2
α
1 [α] [β]
β
3
, [1 2 ]
We define an RF (P)-module M by
0
R
0
R
0
[1 1 ]
[1 3 ]

3.3 Functors between module categories 111
Since we know explicitly t −1 (1), t −1 (2) and t −1 (3), we can
compute directly that LK t M(1) = R 2 , LK t M(2) = LK t M(3) =
0. Thus the RP-module LK t M ˆ⊗LK t M is pictured by
0
R 4
On the other hand, since M ˆ⊗M = M, LK t (M ˆ⊗M) ≁ =
LK t M ˆ⊗LK t M.
Proposition 3.3.9. Let C be a small category and ∇ :
F (C) → C e the skew diagonal functor. Then LK ∇ R ∼ = RC
as an RC e -module.
Proof. We compute (LK ∇ R)(y, x) = lim −→∇/(y,x)
R for any
(y, x) ∈ Ob C e . By Proposition 2.3.13 the functor i (y,x) :
∇ −1 (y, x) → ∇/(y, x) has a left adjoint. Hence Proposition
1.2.4 implies that −→∇/(y,x)
lim R ∼ = lim −→∇
R. However
−1 (y,x)
∇ −1 (y, x) ∼ = Hom C (x, y) so we have −→∇
lim R = RC(y, x),
−1 (y,x)
where RC is considered as a functor C e → R-mod. We are
done.
⊓⊔
0
For a finite group algebra RG and M, N ∈ RG-mod, we
have a canonical isomorphism Hom RG (M, N) ∼ = Hom RG (R, Hom R (M,
where Hom R (−, −) is the internal hom for RG-mod. The
above proposition allows us to generalizes this isomorphism,
but without using the previously defined internal hom for RCmod.
Corollary 3.3.10. Let M, N ∈ RC-mod. Then Hom R (M, N) ∈
RC e -mod and
Hom RC (M, N) ∼ = Hom RF (C) (R, Res ∇ Hom R (M, N))
∼ = HomRC (R, RK t Res ∇ Hom R (M, N)).

112 3 Category algebras and their representations
Proof. We can define an RC e -module structure on Hom R (M, N)
by
[(α, β op ) · f](m) = αf(βm),
for any f ∈ Hom R (M, N) and m ∈ M. Then
Hom RC (M, N) ∼ = Hom RC e(RC, Hom R (M, N))
∼ = HomRC e(LK ∇ R, Hom R (M, N))
∼ = HomRF (C) (R, Res ∇ Hom R (M, N))
∼ = HomRF (C) (Res t R, Res ∇ Hom R (M, N))
∼ = HomRC (R, RK t Res ∇ Hom R (M, N)).
Since F (G) is equivalent to G as categories, under Morita
equivalence between RF (G) and RG, RK t Res ∇ Hom R (M, N) ∈
RG-mod is exactly the usual RG-module Hom R (M, N). In
Chapter 5, we shall give a higher version of this corollary
in terms of Ext, based on Hom RC (M, N) = Ext 0 RC(M, N) =
Ext 0 RC e(RC, Hom R(M, N)). The reader should compare the
above isomorphism with Hom RC (M, N) ∼ = Hom RC (R, Hom(M, N)).
We will explain later on that the latter usually does not admit
an higher version using Ext.
Proposition 3.3.11. Let p : C × C op → C be the projection
to the first component. Then LK p RC = R as an RC-module.
Proof. For each x ∈ Ob C one can easily verify that p/x ∼ =
Id C /x × C op . Since Id C /x has an terminal object (x, 1 x ),
the inclusion {(x, 1 x )} → Id C /x has a left adjoint. Thus
the natural functor C op ∼ = • × C op → Id C /x × C op ∼ = p/x
also has a left adjoint. By Proposition 1.2.4 (LK p RC)(x) =
lim RC ∼ = lim
−→p/x −→C opRC. As a functor from C op to R-mod
(RC)(x) = RHom C (x, −) and thus we can define a morphism
(RC)(x) → R by ∑ i r iα i ↦→ ∑ i r i so that R fits into the
defining diagram for direct limit. In fact if M fits into the
⊓⊔

3.3 Functors between module categories 113
limit defining diagram
RHom C (y, −)
RHom C (x, −)
θ y
θ x
M ,
then θ y (α) = θ x (β) for any β ∈ Hom C (y, −) and α ∈
Hom C (x, −). Thus (LK p RC)(x) = R and we are done. ⊓⊔
The last proposition can be generalized to a functor isomorphism
LK p
∼ = − ⊗RC R, and we shall come back to it in
Chapter 4.
Note that if we have two functors u : D → C and v :
E → D then Res uv = Res v Res u and as a consequence
LK uv = LK v LK u . It is enlightening if we summarize our
three calculations in one diagram for t = p∇.
R
RF (C)-mod
LK ∇
RC e -mod
RC
LK t
LK p
∼ =−⊗RC R
RC-mod .
R
Based on explicit formulas for computing LK ∇ and LK p ,
one should be able to find examples, similar to Example 3.3.8,
that they do not commute with corresponding tensor products.
Definition 3.3.12. Let C be a small category and x an object.
Then we define a full subcategory C x which consists of
one object x and all of its endomorphisms. We also define C [x]
to be the full subcategory of C containing all objects that are
isomorphic to x.

114 3 Category algebras and their representations
Proposition 3.3.13. Let C be a small category and x ∈
Ob C. Consider the inclusion ι : C x ↩→ C. Then LK ι [REnd C (x)] ∼ =
RHom C (x, −).
Proof. For every y ∈ Ob C, the overcategory ι/y has objects
{(x, α)} where α runs over the set Hom C (x, y). A morphism
(x, α) → (x, β) is an endomorphism g ∈ End C (x) such that
α = βg. Hence we can fit RHom C (x, y) into the commutative
diagram
[REnd C (x)](x, α)
α◦−
β◦−
[REnd C (x)](x, β)
RHom C (x, y) ,
in which [REnd C (x)](x, α) = [REnd C (x)](x, β) = REnd C (x),
and α ◦ −, β ◦ − are compositions. If M is another R-module
that can replace RHom C (x, y) and make a new commutative
diagram, then, by examining the images of End C (x), one can
easily establish a canonical map RHom C (x, y) → M such
that the whole diagram is commutative. This means that
RHom C (x, y) ∼ = lim −→ι/y
[REnd C (x)] and thus LK ι [REnd C (x)] ∼ =
RHom C (x, −).
⊓⊔
3.3.3 Dual modules and Kan extensions
In group representations, the two Kan extensions are isomorphic.
Many important properties of group representations rely
on this fact. Unfortunately it is not the case for category representations.
To obtain a balanced understanding of the homological
properties of category algebras, we describe some
compatibility between the two Kan extensions. Here we assume
the base ring is a field R = k.
Let M ∈ mod-kC be a right module. Its dual M ∧ =
Hom k (M, k) becomes a left kC-module. The functor (−) ∧ =

3.3 Functors between module categories 115
Hom k (−, k) is an anti-isomorphism between the two module
categories. Suppose G is a group and H is a subgroup. In
group representation theory, there are two well-known isomorphisms
(M ↓ G H )∧ ∼ = M ∧ ↓ G H , if M ∈ kG-mod, and
(N ↑ G H )∧ ∼ = N ∧ ↑ G H , if N ∈ kH-mod. We shall extend these
to category representations.
Lemma 3.3.14. Let τ : D → C be a functor between finite
categories. Then we have Res τ M ∧ ∼ = (Res τ M) ∧ , for any
M ∈mod-kC.
Proof. The isomorphism follows from Lemma 3.1.8.
Lemma 3.3.15. With the same assumption as above, we
have RK τ N ∧ ∼ = (LK τ N) ∧ , for any N ∈ mod-kD.
Proof.
RK τ N ∧ ∼ = Hom kC -mod(kC, RK τ N ∧ )
∼ = HomkD-mod(Res τ kC, N ∧ )
∼ = Hommod-kD(N, (Res τ kC) ∧ )
∼ = Hommod-kD(N, Res τ kC ∧ )
∼ = Hommod-kC(LK τ N, kC ∧ )
∼ = HomkC -mod(kC, (LK τ N) ∧ )
∼ = (LKτ N) ∧ .
The last isomorphism can be immediately combined with
computations from the preceding section to get the right Kan
extensions of certain modules.
Since this duality exchanges projective and injective modules,
whenever the base ring is a field or a complete local ring
if we can characterize the projective modules then the structures
of injective modules are understood through duality.
⊓⊔
⊓⊔

116 3 Category algebras and their representations
3.4 EI categories, projectives and simples
In this section, we investigate the representation theory of EIcategories.
It will help us in computing various (co)homology
groups. For technical reasons we always assume in this section
the base ring R is a field or a complete discrete valuation ring,
in order to have the unique decomposition property for every
RC-module.
Some of the general theory of EI-categories is given by tom
Dieck in [15] (I.11), much of which was due to Lück, see also
[51].
3.4.1 EI condition and its implications
Recall from Definition 2.4.2 that EI-category is a small category
C in which all endomorphisms are isomorphisms.
One of the important features of EI-categories is described
as follows. Given an EI-category C, there is a preorder defined
on Ob C, that is, y ≤ x if and only if Hom C (y, x) ≠ ∅. Let [y]
be the isomorphism class of an object y ∈ Ob C. This preorder
induces a partial order on the set Is C of isomorphism classes
of Ob C (specified by [y] ≤ [x] if and only if Hom C (y, x) ≠
∅), which plays an important role in studying representations
and cohomology of EI-categories. Because of the existence of
an order for the isomorphism classes of objects in any EIcategory,
EI-categories are sometimes referred to as ordered
categories by some authors, see [57], [42].
Definition 3.4.1. For any EI category C and any object
x ∈ Ob C, we can define a full subcategory D ≤x ⊂ D consisting
of all y ∈ Ob D such that [y] ≤ [x], or equivalently
Hom C (y, x) ≠ ∅. Similarly we can define other full subcategories
of D: D x .

3.4 EI categories, projectives and simples 117
An object in an EI category C is called maximal if C >x = ∅,
and is minimal if C

118 3 Category algebras and their representations
and otherwise Mˆx (y) = 0. For each minimal x we have a short
exact sequence
0 → Mˆx → M → M/Mˆx → 0
such that M/Mˆx is atomic. In either case, we can repeat the
process for Mˆx . Since C is finite, we will eventually obtain a
filtration of M with every factor atomic. We can further refine
this filtration until every factor is simple.
For each RC-module, even if C is not finite, the above analysis
can visualize its structure, which will help us in future
investigations. For example, every simple RC-module has to
be atomic.
3.4.2 Some representation theory
Here we recall some standard facts in representation theory.
Suppose A is a finite-dimensional algebra over a field k. An A-
module M is indecomposable if M is not a direct sum of two
non-trivial submodules. An element e ∈ A is an idempotent if
e ≠ 0 and e 2 = e. For example 1 is always an idempotent and
if e ≠ 1 is an idempotent then so is 1 − e. Two idempotents
e 1 and e 2 are orthogonal if e 1 e 2 = e 2 e 1 = 0. An idempotent
e is primitive if one cannot write e = e 1 + e 2 such that both
e 1 and e 2 are orthogonal idempotents.
Suppose e ∈ A is a non-identity idempotent. Then e and
1−e are a pair of orthogonal idempotents with 1 = e+(1−e).
It results in a decomposition
A = Ae ⊕ A(1 − e).
The module Ae is projective and one can check that it is
indecomposable (not a direct sum of two non-trivial modules)
if and only if e is primitive.

3.4 EI categories, projectives and simples 119
Since A is finite-dimensional, 1 can be decomposed as a sum
of finitely many pairwise orthogonal primitive idempotents
(called a primitive decomposition)
1 = e 1 + e 2 + · · · + e n .
Consequently the regular module A becomes a direct sum
A = Ae 1 ⊕ Ae 2 ⊕ · · · ⊕ Ae n .
and we obtain a list of indecomposable projective A-modules,
Ae i . Up to isomorphism this is a complete list of indecomposable
projective A-modules.
In fact if 1 = e ′ 1 + e ′ 2 + · · · + e ′ m is another sum of primitive
idempotents in A, then m = n and there exists an invertible
element a ∈ A such that 1 = a1a −1 = ae ′ 1a −1 +ae ′ 2a −1 +· · ·+
ae ′ na −1 such that up to an reordering one has e i = ae ′ i a−1
for all possible i. Following from this fact, the automorphism
of A, induced by a, maps each direct summand Ae ′ i of A =
Ae ′ 1⊕Ae ′ 2⊕· · ·⊕Ae ′ n isomorphically to Ae i . Thus the number
of idempotents in a primitive decomposition is equal to the
number of indecomposable direct summand of A.
Example 3.4.4. Suppose C is a finite category and R is a commutative
ring with identity. Then 1 RC is a sum of orthogonal
idempotents ∑ x∈Ob C 1 x. Hence we have a decomposition of
the regular module
RC = ⊕
RC · 1 x = ⊕
RHom C (x, −),
x∈Ob C
x∈Ob C
and each summand RHom C (x, −) is a projective RC-module.
However since an idempotent 1 x may be decomposable in RC,
the above decomposition can be refined, in a way depending
on the choice of R. If this happens, the corresponding projec-

120 3 Category algebras and their representations
tive module RHom C (x, −) is a direct sum of other projective
modules. See Example 3.4.7.
In order to describe the simple A-modules, we have to introduce
a few more terminologies. Let J(A) be the Jacobson
radical of A. It is the intersection of all left maximal ideals of
A. In particular it is nilpotent in the sense that there exists an
integer n > 0 such that J(A) n = 0. By contrast an element
a ∈ A is nilpotent if a n = 0 for some positive integer n. If an
ideal satisfies the condition that every element is nilpotent,
then this ideal is contained in the radical.
Example 3.4.5. If C is an EI-category, the any non-isomorphism,
as an element in the category algebra RC is nilpotent. Since
they form an ideal of RC, it implies that all non-isomorphisms
are contained in J(RC). A important fact is that every element
in J(A) is nilpotent, but the converse if not true (see
Example 3.4.7 (4)).
For each A-module, the radical of M, RadM, is the intersection
of all maximal submodules of M. For example the
regular module has its radical RadA = J(A) because left
ideal of A is exactly the same as a left submodule of A. In
general RadM = J(A)M. It is easy to see that M/RadM has
a trivial radical. Any A-module with trivial radical is called
semi-simple. A semi-simple module is called simple if it is indecomposable.
Equivalently an A-module is simple if it does
not contain any non-trivial submodule. For example A/RadA
is semi-simple. Every simple A-module occurs as a direct summand
in this semi-simple module up to isomorphism.
The quotient A/RadA is itself an algebra with identity ¯1,
the image of 1 ∈ A. A pairwise-orthogonal primitive decomposition
1 = e 1 + e 2 + · · · + e n gives rise to ¯1 ∈ A/RadA,
¯1 = ē 1 + ē 2 + · · · + ē n , which is again a sum of pairwiseorthogonal
primitive idempotents in A/RadA. This simple

3.4 EI categories, projectives and simples 121
observation actually establishes a one-to-one correspondence
between the sets of isomorphism classes of indecomposable
projective A-modules and of simple A-modules.
Proposition 3.4.6. Every indecomposable projective A-module,
up to isomorphism, is of the form Ae, for some primitive
idempotent e ∈ A. Moreover, Ae/Rad(Ae) is a simple A-
module and every simple A-module arises in this way.
Moreover the number of idempotents in a primitive decomposition
of 1 ∈ A, the number of indecomposable summands
of A and the number of indecomposable summands
of A/Rad(A) equal to each other.
There exists a large collection of good references on representation
theory of associative algebras. However for those who
do not plan to go over the whole theory, just bear in mind the
basic constructions and important facts that we record here.
Then through upcoming examples one can see how they work.
We shall use them to classify projective and simple modules
of certain finite category algebras. It will be sufficient for us
to develop (co)homology theory of categories and modules.
Example 3.4.7.1. Let G = {g ∣ g 2 = 1 • } be the cyclic group
of order 2, regarded as a category with one object. If k = C
is the field of complex numbers, the identity 1 • can be written
as 1 •+g
, a decomposition into a sum of orthogonal
primitive idempotents. The regular module is a direct sum
of two one dimensional modules CG ∼ = C(1 • +g)⊕C(1 • −g).
Thus both C(1 • + g) and C(1 • − g) are projective. They are
simple as well because they cannot have non-trivial submodules.
It means CG is semi-simple with trivial radical. The
module C(1 • +g) is the trivial module and C(1 • −g) is called
the sign representation.
However when k is a field of characteristic 2, 1 • is primitive.
Hence kG is indecomposable. The regular module has ex-
2
+ 1 •−g
2

122 3 Category algebras and their representations
actly one non-trivial submodule k(1 • + g), which has to be
the radical Rad(kG). Then kG/Rad(kG) is one-dimensional
and is simple. It is the only simple kG-module, the trivial
module.
2. The poset 1 = 0 → 1 is a category with two objects 0 and 1.
For any field k, the identity 1 k1 = 1 0 +1 1 in the category algebra
k1. The two identity morphisms 1 0 and 1 1 are primitive
orthogonal idempotents so k1 = k{1 0 , α} ⊕ k1 1 . The first
indecomposable summand has exactly one non-trivial submodule
k{α}, the radical Rad(k{1 0 , α}) of k{1 0 , α}. Then
it gives rise to a one-dimensional simple module S 0 . As a
functor, S 0 (0) = k and S 0 (1) = 0. The second summand is
of dimension one so it is already simple. If we denote it by
S 1 . As a functor S 1 (0) = 0 and S 1 (1) = k.
3. Now we examine the category C of Example 3.1.2 that is
neither a group nor a poset.
1 x
x
α
1 y
y
g
with g 2 = 1 y and α = gα. We always have 1 kC = 1 x + 1 y so
kC = k{1 x , α} ⊕ k{1 y , g}. Similar to 2, the first summand,
named P x , is indecomposable and has radical k{α}. The
quotient of P x by its radical is a one-dimensional simple
module S x (analogues to S 0 as above). According to 1, the
second direct summand is decomposable if k = C. Whence
we have CG ∼ = C(1 • + g) ⊕ C(1 • − g). It means when CC
has three indecomposable projective modules and the same
number of simple modules.

3.4 EI categories, projectives and simples 123
The category algebra CC
Indecomposable projective modules Simples module
P x,1 = C{1 x , α} S x,1 = C{1 x }
P y,1 = C{1 y + g} S y,1 = C{1 y + g}
P y,−1 = C{1 y − g} S y,−1 = C{1 y − g}
When k is of characteristic 2, k{1 y , g} is indecomposable.
Whence kC only has two indecomposable projective and
simple modules.
The category algebra kC, chark = 2
Indecomposable projective modules Simple modules
P x,1 = k{1 x , α} S x,1 = k{1 x }
P y,1 = k{1 y , g} S y,1 = P y,1 /k{1 y + g}
4. A useful example to bear in mind is the following category
(a groupoid) D that is equivalent to •
x
α
α −1
One can write RD = RD · 1 x ⊕ RD · 1 y = R{1 x , α} ⊕
R{1 y , α −1 }. Observe that we have an isomorphism (−) ◦
α −1 : R{1 x , α} → R{1 y , α −1 }. When R = k is a field,
every non-zero element of a1 x + bα ∈ kD · 1 x generates the
whole module. Hence kD · 1 x has no non-trivial submodule
and thus is simple. Note that it corresponds to the functor
S x,1 which takes values S x,1 (x) = k1 x and S x,1 (y) = kα.
Similarly kD · 1 y is also a simple module, corresponding to
S y,1 given by S y,1 (x) = kα −1 and S y,1 (y) = k1 y , which is
isomorphic to S x,1 . The algebra RD is semi-simple but it
contains two nilpotent elements α and α −1 .
5. Finally let us consider the following category E
y.

124 3 Category algebras and their representations
x
α
β
such that αβ = e, βe = β, eα = α and e 2 = e. Note that α
and β are not invertible in E. Hence E is not a groupoid and
the two objects are not isomorphic in E. However the category
algebra RE is isomorphic to R{1 x , α, β, e} × R{1 y } ∼ =
RD × R•.
This category is a simple example of inverse categories, see
[51]. The category algebra of an inverse category is canonically
isomorphic to a direct product of groupoid algebras.
Sometimes we may want to use injective modules, so we finish
this section with several remarks concerning the injectives.
In general injective modules behave better than the projectives
in the sense that for any ring A and any A-module M,
there exists a minimal injective A-module I M such that M
admits an injection into I M . This module is called the injective
hull of M. There are module categories with enough
injective but not with enough projectives. However if A is a
finite-dimensional algebra, then for any M ∈ A-mod there
exists a minimal projective module P M which admits a surjection
onto M. It is called the projective cover of M. For
instance, in the tables of Example 3.4.7 (3), each row consists
of a simple module as well as its projective cover. When we do
representation theory we often prefer working with projective
modules because simple modules come from their quotients.
As we pointed out earlier one has an anti-isomorphism
(−) ∧ = Hom k (−, k) from A-mod to mod-A. Suppose P is a
projective A-module, then P ∧ is an injective right A-module.
The anti-isomorphism provides a bijection between projective
left (resp. right) A-modules and injective right (resp. left) A-
1 y
y
e

3.4 EI categories, projectives and simples 125
modules. Thus knowing all projectives leads to getting all
injectives. An important case is the group algebra of a finite
group G, or more generally a finite-dimensional cocommutative
Hopf algebra. They are self-injectives, which means the
regular module is an injective module. In this case, the sets
of projective and injective modules coincide.
In the end we mention an important concept in algebra.
Definition 3.4.8. Let A and B be two rings. Then A is
Morita equivalent to B if A-mod is equivalent to B-mod.
There are many interesting invariants under a Morita equivalence.
For instance, a Morita equivalence preserves the number
of isomorphism classes of simple modules. Hochschild
(co)homology is invariant under Morita equivalence. Here we
shall focus on category algebras only. In fact we will prove that
a category equivalence induces a Morita equivalence between
category algebras. Thus in Example 3.4.7 (4) RC is Morita
equivalent to R = R•.
3.4.3 Classifications of projectives and simples
Now we start describing the projective and simple modules of
an EI category algebra. This part of the work is due to Lück
[51], as is described by tom Dieck in [15]. The base ring R is
assumed to be a field or a complete discrete valuation ring.
Let C be a small category and x ∈ Ob C an object. Suppose
P x is a projective RC x -module (or in other words a projective
REnd C (x)-module). Then its left Kan extension LK ι P x ,
along ι : C x ↩→ C, is a projective kC-module. Especially
LK ι [REnd C (x)] ∼ = RHom C (x, −) by Proposition 3.2.11.
Let us assume furthermore C is a finite EI category. Then
End C (x) = Aut C (x) for every x ∈ Ob C. From Example 3.4.4
we already learned that RC decomposes into a direct sum

126 3 Category algebras and their representations
⊕ x∈Ob C RC · 1 x . Now we try to analyze the indecomposable
direct summands.
Suppose Is C is the full subcategory of C, consisting of all
objects and all isomorphisms. Then R Is C is a subalgebra of
RC.
Lemma 3.4.9. If 1 RC = ∑ n
i=1 e i is a primitive decomposition
of 1 RC in R Is C, then it is also a primitive decomposition
of 1 RC in RC.
Proof. Given any primitive decomposition of 1 RC in RC, the
number of idempotents in this decomposition is equal to the
number of indecomposable direct summands of the regular
module of RC, which is equal to the number of indecomposable
direct summands of RC/Rad(RC) by Proposition 3.4.5.
Let us take the decomposition 1 RC = ∑ n
i=1 e i. We need to
show it is primitive. To this end, we prove n equals the number
of indecomposable direct summands in RC/Rad(RC).
Since all non-isomorphisms generate an ideal I of RC, which
is contained in Rad(RC) and which induces an algebra isomorphism
RC/I ∼ = R Is C, from the isomorphism RC/Rad(RC) ∼ =
(RC/I)/(Rad(RC)/I), we know the two sides have the same
numbers of indecomposable direct summands. From definition
one can check that Rad(RC)/I is the radical of RC/I. Then
by Proposition 3.4.5, applied to both RC and RC/I ∼ = R Is C,
we see the primitive decompositions of 1 RC in both RC and
R Is C must have the same number of idempotents. Hence we
are done.
⊓⊔
The category Is C is a disjoint union of groupoids, each of
which comes from an isomorphism class of some object. Recall
that, for each object x ∈ Ob C, we denote by [x] the set of objects
isomorphic to x, and C [x] the full subcategory consisting
of these objects.

3.4 EI categories, projectives and simples 127
Lemma 3.4.10.1. If x ∼ = y are two isomorphic objects, and
f y ∈ Hom C (x, y) is an isomorphism, then the assignment
α ↦→ α · f y for each α ∈ RC · e defines an isomorphism of
RC-modules RC · 1 y → RC · 1 x .
2. If 1 x = ∑ n
i=1 e i is a primitive decomposition in RAut C (x),
then 1 y = ∑ n
i=1 f ye i fy
−1 is a primitive decomposition in
RAut C (y). Furthermore if we fix for each y ∼ = x an isomorphism
f y ∈ Hom C (x, y), then
∑ n∑
f y e i fy
−1
y∈Ob C [x]
i=1
is a primitive decomposition of the identity 1 RC[x] in the
groupoid algebra RC [x] .
Proof. The isomorphism is straightforward to prove. Now if
1 x = ∑ n
i=1 e i is a primitive decomposition in RAut C (x), certainly
1 y = ∑ n
i=1 f ye i fy
−1 is a decomposition in RAut C (y). It
has to be primitive, because if it were not, then a primitive
decomposition would be a sum of more than n idempotents
in RAut C (y). However fy
−1 (−)f y maps such a primitive decomposition
of 1 y to a decomposition of 1 x , which contradicts
with the assumption that 1 x = ∑ n
i=1 e i is a primitive decomposition.
⊓⊔
The reader can compare the above statements with Example
3.4.7 (4).
Corollary 3.4.11. Let C be a finite EI category. One can
write
1 RC = ∑ ∑n x
e xj ,
x∈Ob C
∑where n x is a positive integer for each x ∈ Ob C and 1 x =
nx
j=1 e xj. As a consequence, RC = ⊕ x∈Ob C ⊕ n x
j=1 RC · e xj
j=1

128 3 Category algebras and their representations
for some primitive pairwise orthogonal idempotents e xj ∈
RAut C (x), x ∈ Ob C.
Any projective RC-module is isomorphic to a direct sum of
indecomposable projective modules of the form RC · e, where
e ∈ RAut C (x) is a primitive idempotent, for some x ∈ Ob C.
Given a primitive orthogonal decomposition 1 RC = ∑ i e i
such that each e i belongs to some group algebra RAut C (x),
each summand of RC ∼ = ⊕ e i
RC · e i is indeed a left Kan
extension
LK ι [RAut C (x)e i ] = {LK ι [RAut C (x)]}e i
∼ = RHomC (x, −)e i = RCe i
because LK ι commutes with direct sums. Here ι : C x ↩→ C is
the inclusion. In each RCe i , its radical contains ∑ z ≁ =x
RHom x,z e i
which are linear combinations of non-isomorphisms in Hom C (x, −).
We continue to characterize the simple RC-modules. Directly
from the EI condition we have seen that a simple module
S has to be atomic. It matches with our description of
indecomposable projective modules. The quotient of RCe i ,
for a primitive idempotent e i ∈ RAut C (x), by its radical is an
atomic module supported on C [x] . Moreover for each y ∼ = x,
S(y) must be a simple RAut C (y)-module. In fact all simple
RC-modules are exactly those simple modules of R Is C, which
are obtained from simple modules of automorphism group algebras
RAut C (x).
Theorem 3.4.12 (Lück). Let C be a finite EI-category.
The isomorphism classes of the simple RC-modules biject
with the pairs ([x], V ), where x ∈ Ob C and V is a simple
RAut C (x)-module, taken up to isomorphism.
Proof. First of all, we already know that all simple RCmodules
are atomic. Thus simple RC-modules are exactly
those simple RC [x] -modules, with x running over Ob C.

3.4 EI categories, projectives and simples 129
Secondly for a fixed x, RC [x] -mod is equivalent to RAut C (y),
for any y ∼ = x, through restrictions induced by the equivalences
C y → C [x] and C [x] → C y . Hence simple RC [x] -modules
biject with simple RC y -modules, for any y ∼ = x. Since C y is
the group Aut C (y), we have proved the assertion. ⊓⊔
Because of the above theorem, it is natural to denote a simple
RC-module by S x,V , if it comes from a simple RAut C (x)-
module V , for some x ∈ Ob C. For consistency, we use P x,V for
the projective cover of S x,V , whose structure is determined by
its value at the object x. If RAut C (x)·e is the projective cover
of the simple RAut C (x)-module V , then RC · e is the projective
cover of S x,V . The reader may revisit Example 3.4.7 to
get better understanding of our results and notations in this
section.
Example 3.4.13. Let k be a field of characteristic two and C
the following category
{1 x }
x
α
β
y
{1 y ,g}
with g 2 = 1 y , gα = α and gβ = β. Indeed the algebra kC has
two (one-dimensional) simples S x,k , S y,k and their projective
covers are P x,k = k{1 x , α, β}, and P y,k = k{1 y , g}, respectively.
The product P x,k ˆ⊗P x,k
∼ = Px,k ⊕ Sy,k 2 is not projective
because S y,k ≠ P y,k .
Remark 3.4.14. Using the tensor product, one can introduce
a “representation ring” of RC, namely a(RC), which consists
of Z-linear combinations of symbols like [M], representing
an isomorphism class of a simple RC-module M. For any
two elements [M] and [N], the multiplication is defined by
[M] · [N] = [M ˆ⊗N]. However this product does not exist in
K 0 (RC), which is spanned over the set of isomorphism classes
of indecomposable projectives.

130 3 Category algebras and their representations
With the description of indecomposable projectives, we can
show when the trivial module R is projective.
Proposition 3.4.15. Let C be a finite EI-category. Then R
is projective if and only if each connected component of C
has a unique isomorphism class of minimal objects [x], with
the properties that for all y in the same connected component
as x, Aut C (x) acts transitively on Hom(x, y), and
|Aut C (x)| is invertible in R.
Proof. Without loss of generality, we may assume C is connected.
If R is projective then R ∼ = ⊕ P y,W for certain indecomposable
projective modules P y,W . Since R is indecomposable
and takes constant value at all objects, we must have that
R ∼ = P x,V for some x ∈ Ob C, x is minimal, and all minimal
objects are isomorphic. Moreover because P x,V (x) = R,
the projective cover of the simple kAut C (x)-module V , we get
V = R and R must be projective as an RAut C (x)-module.
Thus R is projective if and only if R ∼ = P x,R , all minimal
objects are isomorphic to x and |Aut C (x)| −1 ∈ R.
Now, assume all minimal objects are isomorphic to x and
|Aut C (x)| −1 ∈ R. By Proposition 3.2.11 RHom C (x, −) ∼ =
LK ι [RAut C (x)]. Then R ∼ = P x,R can
∑
be explicitly constructed,
1
using the idempotent e =
|Aut C (x)| g∈Aut C (x)
g, as
R ∼ = P x,R
∼ = LKι [RAut C (x)e] = {LK ι [RAut C (x)]}e = RHom C (x, −)e
because LK ι commutes with direct sum. It is equivalent to
saying that at each y ∈ Ob C, R ∼ = RHom C (x, y)e. This happens
if and only if Aut C (x) to act transitively on Hom C (x, y).
⊓⊔

3.4 EI categories, projectives and simples 131
3.4.4 Projective covers, injective hulls and their restrictions
Let C be a finite category and R = k a field. Then any finitely
generated kC-module M admits a minimal projective resolution
P ∗ → M → 0
in the sense that if P ′ ∗ → M → 0 is another projective resolution
of M, then Id M induces a split injection of complexes
from the minimal resolution to the latter. Similarly we can
define a minimal injective resolution of M, 0 → M → I ∗ .
Proposition 3.4.16. Suppose C is an EI category.
1. If D ⊂ C is a coideal and P ∈ kC-mod is an indecomposable
projective module, then Res ι P ∈ kD-mod is either an
indecomposable projective or zero.
2. If D ⊂ C is an ideal and I ∈ kC-mod is an indecomposable
injective module, then Res ι I ∈ kD-mod is either an
indecomposable injective or zero.
Proof. Let P = P x,V be an indecomposable projective kCmodule.
If x ∈ Ob C, then Res i P x,V is brutally truncated from
P x,V and is indecomposable projective as an kD-module. On
the other hand, if x ∉ Ob D, then Res i P x,V = 0.
For the case of injective modules we recall (Res i P ) ∧ ∼ =
Res i P ∧ for any right projective module, by Lemma 3.2.12.
Note that Statement 1 stays true for right projective modules
if we replace the term “coideal” by “ideal”. Now we combine
this with the duality between (indecomposable) right projectives
and left injectives.
⊓⊔
For example if x is an object in C, then C ≤x is a coideal. Take
any indecomposable projective module P x,V , then Res i P x,V
∼ =
P V , the projective cover of the simple kAut C (x)-module V .
This module P V is an indecomposable projective kC ≤x -module.

132 3 Category algebras and their representations
Definition 3.4.17. Let C be an EI category. Suppose M is a
kC-module. Then we define C M ⊂ C to be the smallest ideal
satisfying the condition that if x ∉ Ob C M then M(x) = 0.
Suppose M is a kC-module. Then we define C M ⊂ C to be
the smallest coideal satisfying the condition that if x ∉ Ob C M
then M(x) = 0.
Suppose (M, N) is an ordered pair of kC-modules. We define
a full subcategory C N M to be C M ∩ C N .
Obviously if N ⊂ M then C N ⊂ C M and C N ⊂ C M .
Lemma 3.4.18. Let C be a finite EI category and M a kCmodule.
Then the projective cover P M of M satisfies the
condition that C PM ⊂ C M , and the injective hull I M of M
satisfies the condition that C I M
⊂ C M .
This result allows us to give a characterization of the minimal
projective and injective resolutions of a module.
Corollary 3.4.19. Let C be a finite EI category and M ∈
kC-mod. Suppose P ∗ and I ∗ are minimal projective and injective
resolutions of M. Then for every n ≥ 0, C Pn ⊂ C M
and C I n
⊂ C M .
Proof. The kernel K 0 of P 0 ↠ M satisfies C K0 ⊂ C P0 ⊂ C M .
We use the preceding lemma repeatedly. Similar argument
can be made on I ∗ .
⊓⊔
The last corollary will be useful when we compute cohomology
of modules.

Chapter 4
Cohomology of categories and modules
Abstract We begin through investigation of (co)homology
theories in this chapter. Extensions of modules over a category
algebra is a concept of paramount importance here. We
shall discuss various ways to examine Ext groups, multiplicative
structure and their relationship with previously defined
simplicial and singular (co)homology. A particular important
situation is when the first module is trivial. In this case, on
top of the module theoretic tools, simplicial methods are applicable.
We shall provide a discussion of the bar resolution
and its Kan extensions. Examples are used to illustrate various
computational methods. In the end, the Grothendieck
spectral sequences are introduced and we will study them in
a couple of special cases.
4.1 General theory
4.1.1 Cohomology of modules
Since RC-mod is an abelian category with enough projectives
and injectives, for any two modules M, N ∈ RC-mod
we can consider the Ext groups Ext i RC(M, N), with i ≥ 0.
It is the i-th right derived functor of the left exact functor
Hom RC (M, −) (or Hom RC (−, N)). In general for any
M ∈ RC-mod Ext ∗ RC(M, M) has a ring structure with product
given by the Yoneda splice. Usually it is not graded com-

134 4 Cohomology of categories and modules
mutative. However it is in the case of M = R. In this chapter
we shall compare the Yoneda splice with the cup product
introduced earlier.
Similarly for a right RC-module M ′ and a left RC-module
N ′ , we can study Tor RC
i (M ′ , N ′ ) as the i-th right derived functors
of M ′ ⊗ RC − (or − ⊗ RC N ′ ). In these notes we shall focus
on cohomology. However we shall remark on homology whenever
it is appropriate.
We recall some basics about extensions of modules and their
relationship with cohomology classes.
Definition 4.1.1. Let M, N be A-modules. An n-fold extension
, n ≥ 1, of M by N is an exact sequence of A-modules
0 → N → L n−1 → · · · → L 0 → M → 0.
Two n-fold extensions of M by N are equivalent if there is
a commutative diagram
0
N L
n−1 · · ·
L
0 M 0
0
N L ′ n−1
· · ·
L ′ 0
M 0
Then we can extend this by symmetry and transitivity to
an equivalence relation among n-fold extensions of M by N.
Proposition 4.1.2. There is an one-to-one correspondence
between elements of Ext n A(M, N) and equivalent classes of
n-fold extensions of M by N.
Proof. Let P ∗ → M → 0 be a projective resolution. Then
an extension determines an element in Ext n RC(M, N) by the
following lifting

4.1 General theory 135
· · ·
P n+1
P n
∂ n
P n−1
· · ·
P
0 M 0
f
0
N L n−1
· · ·
L 0
M 0
We see from here that two equivalent extensions give rise to
the same element in Ext n A(M, N).
Conversely if two n-fold extensions determine the same elements
in the group Ext n A(M, N), then we can construct a
commutative diagram (by enlarging P ∗ we may assume f is
surjective)
0 N
L n−1
L n−2
· · ·
L 0
M 0
0 N P n−1 /∂ n (Kerf)
0 N
L ′ n−1
P n−2
L ′ n−2
· · ·
P
0 M 0
· · ·
L ′ 0
M 0
Definition 4.1.3. Let M ∈ A-mod. Then we can define the
Yoneda splice on Ext ∗ A(M ′ , M) and Ext ∗ A(M, M ′′ ) so that
for any η ∈ Ext i A(M ′ , M) and η ′ ∈ Ext j A (M, M ′′ ), η ′ ∗ η ∈
Ext i+j
A
(M ′ , M ′′ ) is given by
0 → M ′′ → N j−1 · · · → N 0 → N ′ i−1 → · · · → N ′ 0 → M ′ → 0,
if η is represented by 0 → M → N ′ i−1 · · · → N ′ 0 → M ′ → 0
and η ′ is represented by 0 → M ′′ → N j−1 · · · → N 0 → M →
0.
The Yoneda splice gives Ext ∗ A(M, M) a ring structure which
is not graded commutative in general. Moreover Ext ∗ A(M, M)
acts on Ext ∗ A(M, N) and Ext ∗ A(N, M) for another N ∈ A-
mod.
Definition 4.1.4. We call Ext ∗ RC(R, R) = ⊕ i≥0 Exti RC(R, R)
the ordinary cohomology ring of the category algebra A. The
⊓⊔

136 4 Cohomology of categories and modules
product in this ring is defined by the Yoneda splice of Ext
classes.
We shall show later on that this ring is isomorphic in a
natural way to H ∗ (C, R) ∼ = H ∗ (BC, R) so it deserves the name.
Now based on the tensor structure on RC-mod, we provide
a module theoretic description to the ring Ext ∗ RC(R, R).
In the meantime we pave the road to studying the ring action
of Ext ∗ RC(R, R) on various Ext groups. We comment that
since (RC-mod, ˆ⊗, R) is a monoidal category with an exact
tensor product, it gives rise to a suspended monoidal category
(D − (RC), ˆ⊗, R) and then following a general statement
[71] on the endomorphisms of the identity in a suspended
monoidal category, End D − (RC)(R) is a graded commutative
ring. It will be clear in this section that this endomorphism
ring is isomorphic to what we call the ordinary cohomology
ring Ext ∗ RC(R, R).
Let M, M ′ , N, N ′ ∈ RC-mod which are projective as R-
modules. We will define the cup product to be
∪ : Ext i RC(M, N)⊗Ext j RC (M ′ , N ′ ) → Ext i+j
RC (M ˆ⊗M ′ , N ˆ⊗N ′ ).
Since R is the identity with respect to ˆ⊗, this will give us
a ring structure on Ext ∗ RC(R, R), as well as an action of
Ext ∗ RC(R, R) on Ext ∗ RC(M, N) for arbitrary M, N ∈ RC-mod.
One shall compare our construction with [, Section 3.2] for
cocommutative Hopf algebras.
Now we are ready to give a precise definition to the cup
product. Suppose ζ ∈ Ext m RC(M, N) is represented by an exact
sequence
0 → N → L m−1 → P m−2 → · · · → P 0 → M → 0,
where P i ’s are projective RC-modules, and ζ ′ ∈ Ext n RC(M ′ , N ′ )
is represented by an exact sequence

4.1 General theory 137
0 → N ′ → L ′ n−1 → Q n−2 → · · · → Q 0 → M ′ → 0,
where Q j ’s are projective RC-modules. Since projective RCmodules
are projective R-modules, all modules in these two
exact sequences are projective R-modules. Furthermore an
RC-module L being a projective R-module is equivalent to
L(x) being projective for all x ∈ Ob C. Then applying the
Künnethe formula to the following two complexes of projective
R-modules
and
0 → N(x) → L m−1 (x) → P m−2 (x) → · · · → P 0 (x)
0 → N ′ (x) → L ′ n−1(x) → Q n−2 (x) → · · · → Q 0 (x),
for all x ∈ Ob C, we get exact sequences
0 → (N ˆ⊗N ′ )(x) → (L m−1 ˆ⊗N)(x) ⊕ (N ˆ⊗L ′ n−1)(x) → · · ·
→ (P 0 ˆ⊗Q 0 )(x) → (M ˆ⊗M ′ )(x) → 0,
with x running over Ob C. Thus we get an exact sequence of
RC-modules
0 → N ˆ⊗N ′ → (L m−1 ˆ⊗N ′ ) ⊕ (N ˆ⊗L ′ n−1) → · · · → P 0 ˆ⊗Q 0 → M ˆ⊗M
which is defined to be the cup product of ζ and ζ ′ ,
ζ ∪ ζ ′ ∈ Ext m+n
RC (M ˆ⊗M ′ , N ˆ⊗N ′ ).
Lemma 4.1.5. Let ζ, ζ ′ be as above. The cup product ζ ∪ ζ ′
is the Yoneda splice of
with
ζ ˆ⊗N ′ ∈ Ext i RC(M ˆ⊗N ′ , N ˆ⊗N ′ )
M ˆ⊗ζ ′ ∈ Ext j RC (M ˆ⊗M ′ , M ˆ⊗N ′ ).

138 4 Cohomology of categories and modules
There exists a natural map φ M = − ˆ⊗M : Ext ∗ RC(R, R) →
Ext ∗ RC(M, M) lies in the graded center, for any M ∈ RCmod.
Particularly, Ext ∗ RC(R, R) is graded commutative.
Proof. By using exact sequences representing ζ and ζ ′ , one
can easily establish a map between (n + m)-fold extensions
(ζ ˆ⊗Id N ′) ∗ (Id M ˆ⊗ζ ′ ) → ζ ∪ ζ ′ . Let
and
D : 0 → N → P m−1 = L m−1 → P m−2 → · · · → P 0
C : 0 → N ′ → Q n−1 = L ′ n−1 → Q n−2 → · · · → Q 0
come from the given two extensions. Then we have a map of
(m + n)-fold extensions
0 N ˆ⊗N ′
(P m−1 ˆ⊗N ′ ) ⊕ (N ˆ⊗Q n−1 )
· · ·
P 0 ˆ⊗Q
0 M ˆ⊗M ′
0
f m+n−1
0 N ˆ⊗N ′
P m−1 ˆ⊗N ′
· · ·
M ˆ⊗Q 0
M ˆ⊗M ′
0
given by
{
(D ˆ⊗C)
f i :
i → P 0 ˆ⊗Q i → M ˆ⊗Q i , 0 ≤ i ≤ n − 1;
(D ˆ⊗C) i → P i−n ˆ⊗N ′ , n ≤ i ≤ n + m − 1.
Thus cup product is a Yondea splice.
Next for an m-fold extension (all modules chosen to be R-
projective)
ζ : 0 → R → L m−1 → P m−2 → · · · → P 0 → R → 0,
the tensor product with M
0 → M → L m−1 ˆ⊗M → P m−2 ˆ⊗M → · · · → P 0 ˆ⊗M → M → 0,
stays exact because
0 → R = R(x) → L m−1 (x) → P m−2 (x) → · · · → P 0 (x) → R(x) = R
f 0

4.1 General theory 139
is split exact and thus
0 → M(x) → L m−1 (x)⊗M(x) → P m−2 (x)⊗M(x) → · · · → P 0 (x)⊗M
is exact for all x ∈ Ob C. Thus − ˆ⊗M gives us a map, commuting
with Yoneda splice,
φ M = − ˆ⊗M : Ext ∗ RC(R, R) → Ext ∗ RC(M, M),
and hence is a ring homomorphism.
If ζ ′ ∈ Ext n RC(M, M), then φ M (ζ) ∗ ζ ′ = ζ ∪ ζ ′ and ζ ′ ∗
φ M (ζ) = ζ ′ ∪ ζ. We want to show ζ ∪ ζ ′ = (−1) mn ζ ′ ∪ ζ.
This comes from the fact that, given the cocommutativity
τ∆ = ∆, we can establish an isomorphism of complexes of
RC-modules
C ˆ⊗D → D ˆ⊗C,
by a ⊗ b ↦→ (−1) deg a deg b b ⊗ a for any two homogeneous elements.
⊓⊔
In terms of projective resolutions, we can describe the cup
product as follows. Let M, M ′ , N, N ′ be RC-modules which
are projective as R-modules. Take two projective resolutions
P ∗ → M → 0 and Q ∗ → M ′ → 0. Then by our previous observation,
P ∗ ˆ⊗Q ∗ → M ˆ⊗M ′ → 0 is an exact sequence. This
is usually not a projective resolution as the tensor product of
two projective is not projective, in contrast to the case of a cocommutative
Hopf algebra. However we can build a projective
resolution, unique up to chain homotopy, R ∗ → M ˆ⊗M ′ → 0
such that there exists a chain map δ : R ∗ → P ∗ ˆ⊗Q ∗ and a
commutative diagram
R ∗
δ
M ˆ⊗M ′
0
P ∗ ˆ⊗Q ∗
M ˆ⊗M ′
0.

140 4 Cohomology of categories and modules
If for two integers m, n, ζ ∈ Ext m RC(M, N) and ζ ′ ∈ Ext n RC(M ′ , N ′ )
are represented by two cocycles f : P m → N and g : Q n →
N ′ , then the product ζ ∪ ζ ′ is represented by (f ˆ⊗g) ◦ δ :
R m+n → N ˆ⊗N ′ . In Theorem 4.1.13 we shall show how to establish
the algebra isomorphism Ext ∗ RC(R, R) ∼ = H ∗ (BC, R).
4.1.2 Cohomology of a small category with coefficients in a functor
In this section we introduce a particular important case, the
Baues-Wirsching cohomology theory of small categories, and
go over some basic properties. The cohomology theory of small
categories has been discussed in various places in literature,
see Baues-Wirsching [3], Generalov [29] and Oliver []. One
can also find in Gabriel-Zisman [28] and Hilton-Stammbach
[35] the homology theory of small categories.
Definition 4.1.6. Let C be a small category and R a commutative
ring. We define H n (C; M), the n-th cohomology of
C with coefficients in a covariant functor M : C → R-
mod, as the homology of the following cochain complex
{C i (C; M), δ i } i≥0 , where
∏
C i (C; M) = {f : NC i →
M(x i ) ∣ f([x 0 → · · · → x i ]) ∈ M
[x 0 →x 1 →···→x i ]
for all i ≥ 0; and for f ∈ C i (C; M),
δ n φ
i∑
(f)([x 0 → · · · → x i→xi+1 ]) = (−1) j f([x 0 → · · · → ˆx j → · · · →
+(−1) i+1 M(φ)(f([x 0 → · · · → x i ]))
We define Hn(C; M), the n-th homology of C with coefficients
in a covariant functor M : C → R-mod, as the homology of
the following chain complex {C i (C; M), δ i } i≥0 , where
j=0

4.1 General theory 141
C i (C; M) =
⊕
[x 0 →x 1 →···→x i ]
M(x 0 ),
for all i ≥ 0; and for any m [x0 →x 1 →···→x i ] ∈ C i (C; M) which
is an element in some m ∈ M(x 0 ) indexed by x 0 → · · · →
ˆx j → · · · → x i ,
δ i ( ⊕ m [x0
φ
→x1 →···→x i−1 →x i ] ) = [M(φ)(m)] [x 1 →···→x i−1 ]
+
i∑
(−1) j m [x0 →···→ ˆx j →···→x i ],
j=0
in which m [x0 →···→ ˆx j →···→x i ] means m ∈ M(x 0 ) is considered
as an element of C i−1 (C; M) indexed by the (i − 1)-sequence
x 0 → · · · → ˆx j → · · · → x i .
We now give different interpretations to the above homology
and cohomology theory. They will give us better understanding
of the preceding definitions and then lead us to a more general
cohomology theory. Recall that for any functor u : D → C
between two small categories, we can naturally define two
functors u/− : C → Cat, the category of small categories,
and B(u/−) : C → T op, the category of topological spaces.
Thus for any u : D → C, we may write C ∗ (u/x, R), x ∈ Ob C
as the chain complex coming from the simplicial set associated
to the small category u/x, which can be regarded as the
cellular chain complex on the space B(u/x).
Definition 4.1.7. Let C be a small category. We define the
bar resolution of R ∈ RC-mod as B∗
C = C ∗ (Id C /−, R), a
complex of RC-modules.
The bar resolution is a complex reconstructed from C ∗ (C, R)
and its name is justified as follows.

142 4 Cohomology of categories and modules
Proposition 4.1.8. For any small category C, B C ∗ is a complex
of projective RC-modules such that
1. B C 0 ∼ = RC;
2. B C ∗ → R → 0 is an exact sequence of RC-modules;
3. There is a degree 1 isomorphism of complexes of R-modules
from
C ∗ (Id C /−, R) → R → 0
to R[NC] ∗ = C ∗ (C, R) → 0.
4. Hom RC (B C ∗, M) ∼ = ∏ x 0 →···→x i
M(x i ).
Remark 4.1.9. We emphasize that there is no conflict between
2 and 3. A complex of RC-modules is exact if and only if it is
pointwise exact.
Proof. For each i ≥ 0, C i (Id C /−, R) : C → R-mod is a well
defined functor and thus an RC-module. There is an RCmap
∂ i : C i+1 (Id C /−, R) → C i (Id C /−, R) as follows. For any
x ∈ Ob C,
∂ i ((x 0 , α 0 ) → · · · → (x j , α j ) → · · · → (x i+1 , α i+1 ))
= ∑ i+1
j=0 (−1)j [(x 0 , α 0 ) → · · · → (x ̂ j , α j ) → · · · → (x i+1 , α i+1 )],
where α j ∈ Hom C (x j , x). Thus for each x ∈ Ob C, C ∗ (Id C /x, R) →
R → 0 is the augmented chain complex of Id C /x. When we assemble
these augmented chain complexes together, C ∗ (Id C /−, R) →
R → 0 becomes a complex of RC-modules. Since every Id C /x
has a terminal object (x, 1 x ) and thus is contractible, the complex
C ∗ (Id C /−, R) → R → 0 is acyclic because its evaluation
at any x, i.e. C ∗ (Id C /x, R) → R → 0 computes
the reduced homology ˜H∗ (Id C /x, R) and is acyclic. Moreover,
we can show every C i (Id C /−, R) is projective. Indeed
for each i > 0, C i (Id C /x, R) has base elements of the form
(x 0 , α 0 ) → · · · → (x i , α i ). The following bijection provides a
different presentation of these base elements

4.1 General theory 143
{(x 0 , α 0 ) β 1
→ · · · →(x βi
α i β i···β 1 α i β i α
i , α i )} {x 0 −→ · · ·
i
−→xi →x}.
This bijection in fact induces an isomorphism of complexes of
R-modules (see Definitions 1.2.5 and 2.2.4)
C i (Id C /−, R) ∼ =
−→R[NC] i+1 = C i+1 (C, R), ∀i ≥ 0.
We get immediately C 0 (Id C /−, R) ∼ = C 1 (C, R) ∼ = RC. In general
let us take the new R-basis of C i (Id C /−, R), {x 0 → · · · →
x i → x}, and regroup them by putting two such sequences
into one subset of the basis if, after removing the rightmost
object, they become identical. Then each subset determines
an RC-module
R{x 0 → · · · → x i
α
→−
∣ ∣ s(α) = x i },
isomorphic to the projective module RHom C (x i , −). It means
as RC-modules
C i (Id C /−, R) ∼ ⊕
=
x 0 →x 1 →···→x i
RHom C (x i , −).
Hence B∗
C → R → 0 is a projective resolution. In the end
for any M ∈ RC-mod, there is an isomorphism by Yoneda
lemma
Hom RC (C i (Id C /−, R), M) ∼ =
∏
x 0 →···→x i
M(x i ).
We remind the reader that R can be defined as R Ob C ∼ =
R[NC] 0
∼ = C0 (C, R). Thus we have a degree one isomorphism
of chain complexes (of R-modules) from C ∗ (Id C /−, R) →
R → 0 to R[NC] ∗ = C ∗ (C, R) → 0 if in the first complex
we insert R in degree -1.
⊓⊔
Example 4.1.10. In Example 2.2.6 (1) we considered the following
category with two objects, two identity and non-

144 4 Cohomology of categories and modules
identity morphisms
x
α
β
Its nerve gives rise to a non-normalized chain complex C ∗ (C, R)
which is of rank 2 in degree zero, rank 4 in degree 1 and rank
2n + 2 at degree n ≥ 2. It is certainly not exact.
The bar resolution B C ∗ can be written down by computing
the two overcategories Id C /x and Id C /y. In fact Id C /x is a
trivial category with one object (x, 1 x ), and Id C /y is a poset
with three objects (x, α), (x, β) and (y, 1 y ).
(x, α)
α
(y, 1 y )
y
β
(x, β)
Hence the bar resolution is given by B C ∗(y) and B C ∗(x). The
former has rank 3 in degree zero, rank 5 in degree 1, and
rank 2n + 3 in degree n ≥ 2; while the latter has rank 1 at
every degree. Thus B C ∗ has rank 4 in degree zero, rank 6 in
degree 1 and rank 2n + 4 in degree n ≥ 2. One can see that
Rank(B C i ) =Rank(C i+1 (C, R)) for all i ≥ 0. This equality is
actually induced by the R-isomorphism we explained in the
proof of Proposition 3.1.3. Note that both B C ∗(y) and B C ∗(x)
are exact because they come from the nerves of the two overcategories
Id C /y and Id C /x which are contractible. Furthermore
we can explicitly verify that B C 0 ∼ = RC, B C 1 ∼ = P x,1 ⊕P 3 y,1,
etc.
If one examines the normalized complexes of C ∗ (C, R),
C ∗ (Id C /y, R) and C ∗ (Id C /x, R) then the above equality between
ranks are not true any more.
Remark 4.1.11. Proposition 4.1.8 summarizes to a characterization
of the bar resolution in terms of the nerve, as well as

4.1 General theory 145
an isomorphism
Hom RC (B C i , M) ∼ =
∏
x 0 →···→x i
M(x i ).
By contrast, when we consider R as a right RC-module. The
bar resolution of it is C ∗ (−\Id C , R). We shall also denote it
by B∗ C when it does not cause any confusion. For each i ≥ 0,
C i (−\Id C , R) ∼ ⊕
=
x 0 →x 1 →···→x i
RHom C (−, x 0 ).
Since RHom C (−, x 0 ) = 1 x0 · RC, we can verify that
Bi C ⊗ RC M ∼ ⊕
=
x 0 →x 1 →···→x i
M(x 0 ).
When we examine the special case for M = R, we get isomorphisms
of complexes
Hom RC (Bi C , R) ∼ =
∏
and
B C i ⊗ RC R ∼ =
x 0 →···→x i
R ∼ = Hom R (R[NC] i , R).
⊕
x 0 →···→x i
R ∼ = R[NC] i .
Proposition 4.1.12. Let C be a small category and R a
commutative ring. If M : C → R-mod is a functor, then
lim
−→ n M ∼ = Tor RC
C n (R, M) ∼ = Hn(C; M),
and
lim
←− n M ∼ = Ext n C RC(R, M) ∼ = H n (C; M).
Proof. Let U be an arbitrary R-module. Then we have
Hom R (U, Hom RC (R, M)) ∼ = Hom RC (R⊗ R U, M) ∼ = Hom RC (U, M),

146 4 Cohomology of categories and modules
where U ∈ RC-mod is a constant functor. Hence by Proposition
1.1.23, ←−C
lim M ∼ = Hom RC (R, M).
Similarly
Hom R (R⊗ RC M, U) ∼ = Hom RC (M, Hom R (R, U)) ∼ = Hom RC (M, U),
and consequently R ⊗ RC M ∼ = lim −→C
M.
We will only prove the isomorphisms for cohomology because
the statements for homology can be obtained similarly.
Since the kernel of the differential δ 0 : C 0 (C; M) → C 1 (C; M)
can be identified with Hom RC (R, M), we get H 0 (C; M) ∼ =
Hom RC (R, M).
n
Since lim ←−C
is left exact and ←−
lim is its n-th right derived
C
n
functor, we have ←−
lim M ∼ = Ext n C RC(R, M). Hence we only have
to show Ext n RC(R, M) ∼ = H n (C; M). Because Ext n RC(R, M) can
be computed by using any projective resolution of R in RCmod,
we can use the bar resolution to do it. But we have an
isomorphism for each i ≥ 0
Hom RC (Bi C , M) ∼ =
∏
x 0 →···→x i
M(x i ) ∼ = C i (C; M).
Suppose C = G is a group and M, N are two left RGmodules,
since a left RG-module can be naturally regarded as
a right RG-module (and vice versa), it makes sense to consider
both Tor RG
∗ (M, N) and Tor RG
∗ (N, M). Indeed we always have
Tor RG
∗ (M, N) ∼ = Tor RG
∗ (N, M), see [12, Chapter 3]. It is not
the case in general for category homology.
We deduce now that the preceding homology and cohomology
theories are the same as the simplicial and singular homology
and cohomology theories we introduced in Chapter
2.
⊓⊔

4.1 General theory 147
Theorem 4.1.13. We have isomorphisms
and
Tor RC
∗ (R, R) ∼ = H∗(C; R) ∼ = H∗(C, R) ∼ = H∗(BC, R)
Ext n RC(R, R) ∼ = H ∗ (C; R) ∼ = H ∗ (C, R) ∼ = H ∗ (BC, R).
The latter are algebra isomorphisms.
Proof. In order to prove the isomorphisms between graded R-
modules, we only need to show that H∗(C; R) ∼ = H∗(C, R) and
H ∗ (C; R) ∼ = H ∗ (C, R). But these follow from Remark 4.1.11
which compare Definitions 2.2.4 with 4.1.6.
In order to prove the isomorphism between cohomology rings
is an algebra isomorphism we just have to compare the cup
products. Here we only need to explain the first algebra isomorphism
because the second is clear.
Using the bar resolution B∗
C → R → 0, we can describe
the cup product on Ext ∗ RC(R, R). In fact, we can explicitly
write out a diagonal approximation map (unique up to chain
homotopy)
R 0
B C ∗
D
B∗ C ˆ⊗B∗
C
R 0
as a natural transformation, given by
α
D x (x
1
n∑
0→x1 → · · · →x αn α α
n→x) = (x
1
α···α i+1
0→ · · · → xi
i=0
→ x)⊗(xi
α i+1
for any x ∈ Ob C and integer n. Since for each n ≥ 0,
(B C ∗ ˆ⊗B C ∗) n = ⊕ i+j=n BC i ˆ⊗B C j and there is a natural map
Hom RC (B C i , R) ⊗ Hom RC (B C j , R) → Hom RC (B C i ˆ⊗B C j , R),
→ · · ·
α
→

148 4 Cohomology of categories and modules
one can easily see, under the isomorphism Hom RC (B C ∗, R) ∼ =
Hom R (R[NC] ∗ , R), that the diagonal approximation map D
induces a map
Hom RC (B C ∗, R) ⊗ Hom RC (B C ∗, R) → Hom RC (B C ∗, R)
identical to the following
Hom R (R[NC] ∗ , R)⊗Hom R (R[NC] ∗ , R) → Hom R (R[NC] ∗ , R)
induced by the Alexander-Whitney map. Since these two
maps are used to calculate cup products in the two cohomology
rings, our observations imply that Ext ∗ RC(R, R) ∼ =
H ∗ (BC, R) as algebras.
⊓⊔
Recall that the bar resolution can also be constructed via the
nerve of overcategories associated with the identity functor
Id C : C → C, as B∗ C ∼ = C ∗ (Id C /−, R) := RN ∗ (Id C /−). In this
form, Bn(x) C ∼ = C n (Id C /x), for each x ∈ Ob C and integer
n ≥ 0, consists of the following chains as base elements
(x 0 , β 0 ) γ 1
→(x 1 , β 1 ) → · · · → (x n−1 , β n−1 ) γ n
→(x n , β n ),
in which β i is a morphism in Hom C (x i , x), and γ i ∈ Hom C (x i−1 , x i )
such that β i−1 = β i γ i−1 . The previously defined diagonal approximation
map D is given by
D x ((x 0 , β 0 ) γ 1
→(x 1 , β 1 ) → · · · → (x n−1 , β n−1 ) γ n
→(x n , β n ))
= ∑ n
i=0 [(x 0, β 0 ) γ 1
→ · · · →(x γi
i , β i )] ⊗ [(x i , β i ) γ i+1
→ · · · →(x γn
n , β n )]
This is exactly the Alexander-Whitney map for the chain
complex coming from the nerve of Id C /x.
All homology and cohomology theories of small categories
that we have introduced so far coincide whenever they are
comparable. Hence we shall only deal with the most general
form Ext ∗ RC(M, N) and Tor RC
∗ (M ′ , N) from now on, where
M, N ∈ RC-mod and M ′ ∈ mod-RC.

4.1 General theory 149
Proposition 4.1.14. Suppose u : D → C is a functor. Then
we have a restriction Res u : RC-mod → RD-mod. Given two
RC-modules M, N, Res u induces a natural map
res u : Ext ∗ RC(M, N) → Ext ∗ RD(Res u M, Res u N),
called the restriction.
Proof. Take a projective resolution P ∗ of M. Then Res u P ∗ →
Res u M → 0 is an exact sequence of RD-modules. Hence
for any projective resolution Q ∗ → Res u M → 0, the usual
lifting, a chain map, Q ∗ → Res u P ∗ induces a cochain map
Hom RC (P ∗ , N) → Hom RD (Q ∗ , Res u N) because we have a
cochain map Hom RC (P ∗ , N) → Hom RD (Res u P ∗ , Res u N). This
is a well defined map since it does not depend on the choice
of P ∗ and Q ∗ .
⊓⊔
When M = N = R, by using bar resolutions, one can easily
see that the above restriction is the same as the one we defined
for simplicial cohomology in Chapter 2.
Remark 4.1.15. Suppose u : D → C is a functor. Let us consider
res u : Ext ∗ RC(R, N) → Ext ∗ RD(R, Res u N).
One way to obtain some information about the restriction is to
examine its action in degree zero. In fact, the restriction is the
same as Hom RC (R, N) → Hom RC (LK u R, N) ∼= →Hom RD (R, Res u N)
induced by the counit LK u R → R. When LK u R ∼ = R, res u
becomes an isomorphism in degree zero. Giving LK u R ∼ = R
is equivalent to saying that every overcategory of u is R-
acyclic, that is, ˜H∗ (u/x, R) vanishes for every x ∈ Ob C. In
other words, every overcategory u/x has to be connected, see
Proposition 3.2.7.

150 4 Cohomology of categories and modules
4.1.3 Extensions of categories and low dimension cohomology
We describe low dimension cohomology groups Ext n RC(M, N)
for n = 1, 2. The results and their proofs are taken from
Webb’s notes [80]. Let us recall that there is an augmentation
map ɛ : RC → R by sending α to t(α), its target. The kernel
of ɛ is a left ideal of RC, which we call the (left) augmentation
ideal, and write as IC := Kerɛ.
Lemma 4.1.16. The augmentation ideal IC is a free R-
module with basis the elements α − 1 t(α) , where α runs over
all the non-identity morphisms in C.
Definition 4.1.17. Let M ∈ RC-mod. We define a derivation
d : C → M to be a mapping from Mor C to M so that
d(α) ∈ M(t(α)) and so that d(αβ) = M(α)d(β) + d(α). The
set of derivations forms an R-module Der(C, M).
Given any set of elements {u x ∈ M(x) ∣ ∣x ∈ Ob C} we obtain
a derivation specified by d(α) = M(α)(u s(α) ) − u t(α) . Any
derivation obtained in this way is called an inner derivation.
The inner derivations form an R-module IDer(C, M).
Proposition 4.1.18.1. Der(C, M) ∼ = Hom RC (IC, M) as R-
modules.
2. H 1 (C; M) ∼ = Der(C, M)/IDer(C, M).
Proof. Given a homomorphism δ : IC → M, we can define
a derivation d : C → M by d(α) = δ(α − 1 t(α) ). It makes
sense because M(α)d(β) + d(α) = δ[(α(β − 1 t(β) )] + d(α) =
δ(αβ − α) + d(α) = δ(αβ − 1 t(αβ) )) = d(αβ) since t(αβ) =
t(α). Conversely given a derivation d we can define a RChomomorphism
δ : IC → M by δ(α − 1 t(α) ) = d(α), and we
can verify that δ is an RC-map. Hence we have proved (1).
As for (2), consider the short exact sequence 0 → IC →
RC → R → 0 and apply Ext ∗ RC(−, M). We obtain an exact

4.1 General theory 151
sequence
0 → Hom RC (R, M) → Hom RC (RC, M) → Hom RC (IC, M) → Ext 1 RC(R
The image of Hom RC (RC, M) ∼ = M consists of f m | IC , where
f m is the RC-homomorphism determined by some m ∈ M via
the canonical map m ↦→ f m such that f m (1 RC ) = m. Because
f m (α − 1 t(α) ) = [M(α)](m) − [M(1 t(α) )](m) = [M(α)](1 s(α) ·
m) − [M(1 t(α) )](1 t(α) · m), f m | IC is an inner derivation determined
by the set of elements {1 x · m ∈ M(x) ∣ ∣ x ∈ Ob C}.
Moreover all inner derivations are of this form. We are done.
⊓⊔
In order to characterize H 2 (C; K), we have to introduce category
extensions in the sense of G. Hoff.
Definition 4.1.19 (Hoff). An extension E of a category C
via a category K is a sequence of functors
K ι
−→E π
−→C,
which has the following properties:
1. Ob K = Ob E = Ob C, ι is injective and π is surjective on
morphisms; and
2. if π(α) = π(β), for two morphisms α, β ∈ Mor(E), if and
only if there is a unique g ∈ Mor(K) such that β = ι(g)α.
We may readily deduce some useful information from the
above definition.
Proposition 4.1.20.3. If αι(h) exists for α ∈ Mor(E) and
h ∈ Mor(K), then there exists a unique h α ∈ Mor(K) such
that ι(h α )α = αι(h). Moreover any α ∈ Hom E (x, y) induces
a group homomorphism K(x) → K(y); and
4. for any α ∈ Hom C (x, y), K(y) acts regularly (i.e. freely
and transitively) on π −1 (α).

152 4 Cohomology of categories and modules
Proof. Suppose αι(h) exists for a given α ∈ Mor E. Then
since π(αι(h)) = π(α), by Definition 4.1.19 (2), there exists
a unique h α ∈ Mor K such that αι(h) = ι(h α )α. If h 1 , h 2 ∈
K(x), then we have αι(h 1 h 2 ) = ι[(h 1 h 2 ) α ]α. On the other
hand, we also get αι(h 1 h 2 ) = αι(h 1 )ι(h 2 ) = ι(h 1 ) α [αι(h 2 )] =
ι(h α 1)ι(h α 2)α. Hence h α 1h α 2 = (h 1 h 2 ) α and α induces a group
homomorphism.
Now assume β ∈ Hom E (x, y) and π(β) = α. Let h ∈ K(y).
If ι(h)β = β, then we have π(ι(h)β) = π(β) = α. Definition
4.1.19 (2) forces h = 1 y because of the uniqueness property.
⊓⊔
One can see that K is indeed a disjoint union of the groups
π −1 (1 x ) for all 1 x ∈ Mor(C) (regarded as categories), and can
be identified with a functor K : E → Groups. Usually from
the context, one knows when we take K to be a category and
when it is regarded as a functor.
Since in practice one often crosses a concept dual to the
category extension, for future reference, we first state its definition.
Definition 4.1.21. An opposite extension E of C via K is
a sequence of functors K ι →E π →C such that the following sequence
is an extension of C op
K op ι op
−→E op π op
−→C op ,
Sometimes we just say K → E → C is an opposite extension.
Dually there are characterizations as follows.
1 op . Ob K = Ob E = Ob C, ι is injective and π is surjective on
morphisms; and
2 op . if π(α) = π(β), for two morphisms α, β ∈ Mor(E), if and
only if there is a unique g ∈ Mor(K) such that β = αι(g).

4.1 General theory 153
3 op . If ι(h)α exists for α ∈ Mor(E) and h ∈ Mor(K), then there
exists a unique h ′ ∈ Mor(K) such that αι(h ′ ) = ι(h)α; and
4 op . for any α ∈ Hom C (x, y), K(x) acts regularly on π −1 (α).
One of the main difference between extensions and opposite
extensions is that, for an extension, K can be considered as
a covariant functor from C to Groups while, for an opposite
extension, K gives a contra-variant functor. When we study
H 1 and H 2 , extensions are used because we deal with left modules.
If dually we want to obtain corresponding statements for
right modules, then we have to replace extensions by opposite
extensions in wherever it applies.
Example 4.1.22.1. Any category C is an extension and opposite
extension of itself by ⊎ Ob C
•, a disjoint union of some
trivial groups.
2. Any extension of a group is both an extension and an opposite
extension if we consider the group and its extension as
categories.
3. The following sequence is an extension of the rightmost category
by a group which is the disjoint union of a trivial group
and a cyclic group of order 2:
x
C 2
{1 y }
y
C 2
y
ι π
α
x
C 2
x
{1 x }
α
C 2
y
.
4. The following sequence is an opposite extension of the rightmost
category by a group which is the disjoint union of a
trivial group and a cyclic group of order 2:

154 4 Cohomology of categories and modules
C 2
y
C 2
y
{1 y }
y
x
{1 x }
ι π
α
x
C 2
α
x .
C 2
One shall easily see the difference and relationship between
the first two examples. We note that in both examples, the
categories at the two ends have finitely generated cohomology
rings, while the ones in the middle do not, see Example
2.2.4 (4). and Proposition 2.2.38.
5. Let G be a finite group and p a prime dividing the order of G.
A collection P of p-subgroups of G is a set of p-subgroups
which is closed under conjugations in G. The transporter
category G ∝ P = Gr G P has various quotient categories,
e.g. the orbit category O P (G), many of which form extension
sequences, e.g.
K s → G ∝ P → O P (G),
where K s (H) = H for any H ∈ Ob P = Ob(G ∝ P). We
shall come back to this in Chapter 6.
Definition 4.1.23. Two extensions are equivalent if we have
the following commutative diagram
K E
C
K E ′
C
Definition 4.1.24. An extension is split if it admits a functor
s : C → E such that π ◦ s = 1 C .
Proposition 4.1.25. An extension is split if and only if it
is equivalent to a Grothendieck construction.

4.1 General theory 155
Proof. Assume an extension K → E → C is split. Then
K can be restricted to a functor C → E → Groups. The
Grothendieck construction Gr C K has the same objects of the
form (x, • x ) where x ∈ Ob C and • x is the only object of
K(x). A morphism from (x, • x ) to (y, • y ) is a pair (α, h) with
α ∈ Hom C (x, y) and h ∈ K(y). Hence we can define a funtor
Gr C K → E by (x, • x ) ↦→ x and (α, h) ↦→ hα.
On the other hand for an extension K → Gr C K → C, we can
define a functor C → Gr C K by x ↦→ (x, • x ) and α ↦→ (α, 1 y )
if α ∈ Hom C (x, y). Thus this extension is split.
⊓⊔
Proposition 4.1.26. The equivalence classes of extensions
of C by K are in bijection with H 2 (C; K) in such a way that
the zero element corresponds to the Grothendieck construction
Gr C K.
Proof. We will provide correspondence between equivalence
classes of extensions and elements of H 2 (C; K).
Let K → E → C be an extension. We can choose a section for
E → C, that is, for each morphism α : x → y in C a morphism
q(α) : x → y in E with pq(α) = α. If α : x → y and β : y → z
are two morphisms in C, then q(βα) = ι[τ(β, α)]q(β)q(α) for
a unique τ(β, α) ∈ K(z). Thus we obtain a mapping τ :
C × C → C. The associativity of morphisms implies the 2-
cocycle condition by direct calculation
τ(γβ, α) + τ(γ, β) = τ(γ, βα) + K(γ)τ(β, α).
If f ∈ Hom RC (RC, M), then mappings of the form
˜f(β, α) := f(βα) − f(β) − K(β)f(α)
automatically satisfies the 2-cocycle condition. If we examine
Hom RC (B C ∗, M), used to define H ∗ (C; M) in Definition 3.1.1,
we can verify that all these mappings τ and ˜f biject with
the 2-cocycles and 2-coboudaries in the the cochain complex

156 4 Cohomology of categories and modules
Hom RC (B∗, C M). In fact we already learned that
Hom RC (B2, C M) ∼ =
∏
K(z)
and
x α →y β →z
Hom RC (B C 1, M) ∼ = ∏ v γ →w
K(w).
With explicitly given differential we can prove the above bijection.
In the end, we may verify that 2-cocycles are homologous
if and only if the extensions which produce them are equivalent,
and also a change in the choice of a section gives rise to
a cohomologous 2-cocycle. Hence we are done.
⊓⊔
To finish this section, we comment on the connection between
H i (C; M) and Ext i RC(R, M), for i = 1, 2. In fact from
the short exact sequence 0 → IC → RC → R → 0, we get
Ext i RC(R, M) ∼ = Ext i−1
RC (IC, M).
Let 0 → M → E → IC → 0 be an RC-module extension
representing an element in Ext 1 RC(IC, M). We can construct
an extension of categories
M → Gr C E → Gr C IC.
There is a splitting functor C → Gr C IC, which is identity on
objects and sends a morphism α to (α − 1 t(α) , α). Now we
form a pullback diagram
M
E
C
M Gr C E Gr C IC

4.2 Classical methods for computation 157
which gives rise to an extension M → E → C, an element in
H 2 (C; M). Note that if 0 → M → E → IC → 0 splits, then
Gr C E ∼ = Gr GrC ICM, in H 1 (C; M).
Conversely if M → E → C is an extension and IM is the
kernel of the algebra homomorphism RE → RC. Then we can
define an extension of RC-modules
0 → M → IE/(IM · IE) → IC → 0,
representing an element in Ext 2 RC(R, M) ∼ = Ext 1 RC(IC, M).
4.2 Classical methods for computation
4.2.1 Minimal resolutions and reduction
Recall in Section 3.3.3 we described the minimal projective
and injective resolutions of a module. With Definition 3.3.13
we establish the following isomorphism.
Proposition 4.2.1. Let C be an EI category, M, N two kCmodules
and M ′ a right kC-module. Then
and
Ext ∗ kC(M, N) ∼ = Ext ∗ kC N M(Res i M, Res i N)
Tor kC
∗ (M ′ , N) ∼ = Tor ∗ (Res
kC M′ i M ′ , Res i N).
Proof. Suppose P ∗ is a minimal projective resolution of M.
Then each direct summand P x,V of any P n must satisfy
the condition that x ∈ Ob C M by Corollary 3.3.15. Hence
Hom kC (M, N) ∼ = Hom kCM (Res i M, Res i N) and Ext ∗ kC(M, N) ∼ =
Ext ∗ kC M
(Res i M, Res i N). Now we apply Corollary 3.3.15 again
to a minimal injective resolution of N when computing
and the isomorphism follows.
Ext ∗ kC M
(Res i M, Res i N)
N

158 4 Cohomology of categories and modules
To prove the isomorphism for Tor the method is the same.
One just need to take the right projective resolution of M ′ .
⊓⊔
This result is useful because one may replace the original
category by a full subcategory before carrying on any computations.
It is particular helpful if both M and N are atomic.
Since any module of an EI category algebra has a filtration by
atomic modules. Repeatedly using the above corollary may
significantly simplify things. In following examples we will
shall how one combines resolutions, module filtrations and
reductions through Proposition 4.2.1 to compute various Ext
groups.
Example 4.2.2.1. Suppose C = n is a poset. Then k ∼ =
kHom n (0, −) is a projective module. Thus Ext ∗ kn(k, M) ∼ =
M(0) for any M ∈ kn-mod.
2. Suppose C is the following category
x
α
β
y.
We will only write down non-vanishing Ext groups in this
example.
The minimal projective resolution of k is 0 → P y,k →
P x,k → k → 0. Thus for any M ∈ kC-mod Ext ∗ kC(k, M)
is given by the homology of 0 → Hom kC (P x,k , M) →
Hom kC (P y,k , M) → 0. If M = k, then Ext 0 kC(k, k) ∼ =
Ext 1 kC(k, k) ∼ = k. If M = S x,k , then Ext 0 kC(k, S x,k ) ∼ = k.
If M = S y,k , then Ext 1 kC(k, S y,k ) ∼ = k.
The minimal projective resolution of S x,k is 0 → Py,k 2 →
P x,k → S x,k → 0. We can use it for direct calculation, but
we want to make use of Proposition 4.2.1. Ext ∗ kC(S x,k , S x,k ) ∼ =
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
k. Proposition 4.2.1 is not useful for computing Ext ∗ kC(S x,k , S y,k )
so we have to compute the homology of 0 → Hom kC (P x,k , S y,k ) →
Hom kC (P 2 y,k , S y,k) → 0, which gives Ext 1 kC(S x,k , S y,k ) ∼ = k 2 .
For Ext ∗ kC(S x,k , k) we can use a resolution but we can also
use the long exact sequence
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
from
0 → S y,k → k → S x,k → 0
after applying Ext ∗ kC(S x,k , −). Since Ext 0 kC(S x,k , k) ∼ = Hom kC (S x,k , k)
k, we know immediately Ext 1 kC(S x,k , k) ∼ = k 2 .
3. Suppose k is a field of characteristic 2. Consider the following
category D
α
y {1 y ,g}
{1 x } x
with gα = α. Then here are two one-dimensional simple
modules S x,k and S y,k , together with their projective
covers P x,k = k{1 x , α} and P y,k = {1 y , g}. Particularly
k ∼ = kHom D (x, −) = P x,k
M ∈ kD-mod, Ext ∗ kD(k, M) ∼ = M(x).
The minimal projective resolution of S x,k is
is projective. Thus for any
· · · → P y,k → P y,k → P x,k → S x,k → 0,
which is infinite and consists of a copy of P y,k in each degree
n > 0. Thus Ext 0 kD(S x,k , S x,k ) ∼ = k and Ext n kD(S x,k , S y,k ) ∼ = k
for all n > 0.
The minimal projective resolution of S y,k is
· · · → P y,k → P y,k → P y,k → S y,k → 0,
which is infinite and consists of a copy of P y,k in each degree.
We get Ext n kD(S y,k , S y,k ) ∼ = k for every n ≥ 0. Moreover
Ext ∗ kD(S y,k , S y,k ) ∼ = Ext ∗ kC 2
(k, k) by Proposition 4.2.1. We

160 4 Cohomology of categories and modules
leave the computation of Ext ∗ kD(−, k) and of all above Ext
groups when k = C to the reader.
One can determine the ring structure of Ext ∗ kC 2
(k, k). In
fact C 2 has a very simple bar resolution so one may quickly
find, using simplicial method, that Ext ∗ kC 2
(k, k) ∼ = k[X] is a
polynomial ring with deg X = 1.
Since Ext ∗ kD(k, k) ∼ = k, both Ext ∗ kD(S x,k , S y,k ) and Ext ∗ kD(S y,k , S y,k )
are not finitely generated modules.
If we look at the opposite category D op , again we have
Ext ∗ kD op(k, k) ∼ = Ext ∗ kD(k, k) ∼ = k, and Ext ∗ kD op(k, S y,k) ∼ =
Ext ∗ kD op(S y,k, S y,k ) is of infinite dimensional. It means we do
not have the finite generation of Ext ∗ kD(k, M) over Ext ∗ kD(k, k)
even if they can be calculated by using the same projective
resolution.
4.2.2 Examples using classifying spaces
Understanding the homotopy type of a classifying space certainly
will be a great help for compute cohomology rings. Here
we list several cases where direct calculation is possible.
Example 4.2.3. In Example 4.2.2, part (1) is a contractible
category n so we know Ext ∗ kn(k, k) ∼ = k. Part (2) is a category
whose classifying space is the circle S 1 . Part (3) is a
contractible category because it has an initial object.
However, there are more things we can take from homotopy
theory. If we know all Ext ∗ kC(k, −), then we may readily write
down Ext ∗ kCop(k, −).
Now we examine some more sophisticated examples. Although
they do not tell much about cohomology with local
coefficients, they do provide substantial information about the
ordinary cohomology ring of a category.
Example 4.2.4.1. Let C be the following category

4.2 Classical methods for computation 161
G
x
α
y H .
Then this is the join of G and H. Thus its classifying space
is a join of spaces BC ≃ BG ∗ BH. Thus its cohomology is
completely determined by G and H. Also the cup product
is trivial.
2. Let D be the following category
h
x
1 x
α
β
y {1 y }
,
where h 2 = 1 x and βh = α. Then the inclusion i :
Aut D (x) → D induces a homotopy equivalence between
their classifying spaces. In particular, we have a ring isomorphism
Ext ∗ RD(R, R) = Ext ∗ RAut D (x)(R, R) for any ring R.
When R = C, the cohomology ring is just C, while when R
is a field of characteristic 2 the ordinary cohomology ring is
a polynomial algebra with indeterminant in degree one.
3. Let E be the following category
h
g
x
1 x
α
β
gh
y {1 y }
,
where g 2 = h 2 = 1 x , gh = hg, αh = α, αg = β, βh =
β, βg = β. Its classifying space is the homotopy pushout of
BC 2
BAut E (y) ∼ = •
BAut E (x) = B(C 2 × C 2 )
where the cyclic group of order 2 is {1 x , h}, the stabilizer of
Hom E (x, y) = {α, β}. Thus BE ≃ BAut E (x)/BC 2 . In fact,
by Definition 2.4.3 and Proposition 2.4.4, we can consider
the subdivision of E, which is a Grothendieck construction

162 4 Cohomology of categories and modules
Gr sd[E] M. Then by Theorem 2.4.10, BGr sd[E] M, and thus
BE, is homotopy equivalent to hocolim sd[E] BM. Since sd[E]
is a poset • ← • → •, this homotopy colimit is just the
homotopy pushout of the above diagram, if one follows Definition
2.4.3 to compute BM.
We can use the exact sequence for computing relative cohomology
(see for instance [34])
H ∗ (BAut E (x), BC 2 ; R)
to find the positive degree cohomology of BE, while in degree
zero we just have R because E is connected. In fact
H ∗ (BE, k), for k a field of characteristic 2, is a subring of
the polynomial ring k[X, Y], with deg X = deg Y = 1 and
with all linear combinations of {X i } i>0 removed. This ring
has no nilpotent elements and is not finitely generated. In
fact, this is the smallest example in commutative algebra
that a subring of a Noetherian ring is not finitely generated.
However when R = C, C is projective (by direct calculation
or by Proposition 3.3.11). That means H 0 (BE, C) = C is
the only non-vanishing cohomology. Note that C as a RE op -
module is not projective, although H ∗ (BE op , C) = C as well.
4.3 Computation via adjoint functors
4.3.1 Adjoint restrictions
It is well-known that if a functor u : D → C has a left adjoint
v : C → D, then Res v is also the left adjoint of Res u . These
are very special Kan extensions and hence can be regarded
as the first applications of Kan extension before we move to
more subtle situations. Because both Res u and Res v are exact,
Res u preserves injectives while Res v preserves projectives.

4.3 Computation via adjoint functors 163
Lemma 4.3.1. If u : D → C has a left adjoint v : C → D,
then
Ext ∗ RC(Res v M, N) ∼ = Ext ∗ RD(M, Res u N)
for any M ∈ RD-mod and N ∈ RC-mod. If M ′ is a right
RD-module, then
Tor RC
∗ (Res v M ′ , N) ∼ = Tor RD
∗ (M ′ , Res u N)
If furthermore u is an equivalence then we have
Ext ∗ RC(M, N) ∼ = Ext ∗ RD(Res u M, Res u N),
for M, N ∈ RC-mod. In particular res u is a ring isomorphism
Ext ∗ RC(R, R) ∼ = Ext ∗ RD(R, R). If M ′′ is a right RCmodule,
then under the circumstance we get
Tor RC
∗ (M ′′ , N) ∼ = Tor RD
∗ (Res u M ′′ , Res u N)
Proof. These follow directly from the definitions of Ext and
Tor, as well as the adjunction.
⊓⊔
Let us briefly go over an important approach of computing
∗
higher limits ←−
lim M. The atomic modules are useful to us
C
because for many interesting category C every module in RCmod
admits a filtration by submodules such that every quotient
is an atomic module. Since the higher limits of a module
can be computed via higher limits of its atomic subquotients,
∗
one can seek ways of computing ←−
lim M for an atomic module
C
M. This method was introduced and studied by Oliver [].
∗
We note that usually ←−
lim M, for an atomic module M concentrated
on x ∈ Ob C, is quite different from the group
C
cohomology H ∗ (Aut C (x), M(x)). Given a projective resolution
P of R, P(x) must be an exact sequence of RAut C (x)-
modules, beginning with R, for any object x ∈ Ob C. But
the modules in P(x) do not have to be projective, and even
if P(x) → R → 0 is a projective resolution the cohomology

164 4 Cohomology of categories and modules
groups H ∗ (C, M) and H ∗ (Aut C (x), M(x)) are not necessarily
isomorphic. We comment here that the modules in P(x) will
be projective if RAut C (x) happens to be semi-simple. Another
possible way to obtain a projective resolution for the
RAut C (x)-module R is formulated as follows. If Aut C (x) acts
freely on Hom C (y, x) for each y ∈ Ob C with Hom(y, x) ≠ ∅,
then every RHom C (y, x) is a free RAut C (x)-module. As an example
the standard resolution evaluated at x, P C (x), becomes
a projective resolution for the RAut C (x)-module R.
4.3.2 Kan extensions of resolutions
For an arbitrary functor u : D → C we can still construct
the left and right adjoints of Res u , i.e. the Kan extensions. In
the situation that u admits a left (or right) adjoint v, then
the left (or right) Kan extension enjoys a particularly simple
form Res v , which is exact. In general, the Kan extensions cannot
have simplified forms, and consequently computing Kan
extensions of an arbitrary module is difficult and the new
modules do not come with explicit descriptions. However if a
module is simplicially constructed, in the sense that it comes
from the nerve of a category, then we do have some satisfactory
results on their Kan extensions. This is a fundamental
step to study cohomology rings and higher limits.
Proposition 4.3.2. Suppose u : D → C is a functor between
small categories. Then
LK u B D ∗
= LK u C ∗ (Id D /−, R) ∼ = C ∗ (u/−, R),
a complex of projective RC-modules.
Particularly for pt : D → •, we have
LK pt B D ∗
= LK pt C ∗ (Id D /•, R) ∼ = C ∗ (pt/•, R) ∼ = C ∗ (D, R),

4.3 Computation via adjoint functors 165
Proof. For the sake of convenience, we suppress the base ring
R in our notations. Suppose x ∈ Ob C. Fix an integer n ≥ 0.
Then
LK u C n (Id D /−) ∼ = lim −→u/x
C n (Id D /−).
If (y, α) ∈ Ob u/x, that is, there is an α : u(y) → x, then by
definition [C n (Id D /−)](y, α) = C n (Id D /y), and we can define
a morphism
θ (y,α) : C n (Id D /y) → C n (u/x)
by [(y 0 , α 0 ) β 1
→ · · · →(y βn
n , α n )] ↦→ [(y 0 , αα 0 ) β 1
→ · · · →(y βn
n , αα n )].
One can easily verify that we have a commutative diagram
for any γ : (y, α) → (z, β)
C n (Id D /y) = [C n (Id D /−)](y, α)
θ (y,α)
γ ∗
C n (u/x)
[C n (Id D /−)](z, β) = C n (Id
θ (z,β)
so that C n (u/x) fits into the limit defining diagram. If L is
another R-module that fits into the limit defining diagram,
equipped with maps θ L (y,α) : [C n(Id D /−)](y, α) → L for all
(y, α) ∈ Ob(u/x). Then we can introduce a morphism Θ :
C n (u/x) → L such that
Θ[(y 0 , αα 0 ) β 1
→ · · · →(y βn
n , αα n )]
:= θ(y L n ,αα n ) [(y 0, u(β n · · · β 1 )) β 1
→(y 0 , u(β n · · · β 2 )) β 2
→ · · · →(y βn
n , 1 u(yn ))].
This is a well defined morphism. In the end, since α n :
(y n , αα n ) → (y, α) in a morphism in u/x, it induces a functor
C n (Id D /y n ) → C n (Id D /y). In particular this functor gives
[(y 0 , u(β n · · · β 1 )) β 1
→(y 0 , u(β n · · · β 2 )) β 2
→ · · · →(y βn
n , 1 u(yn ))]
↦→ [(y 0 , α 0 ) β 1
→ · · · →(y βn
n , α n )],

166 4 Cohomology of categories and modules
by composing with α n . By assumptions on L we get
Θ(y L n ,αα n ) [(y 0, u(β n · · · β 1 )) β 1
→(y 0 , u(β n · · · β 2 )) β 2
→ · · · →(y βn
n , 1 u(yn ))]
= Θ(y,α) L [(y 0, α 0 ) β 1
→ · · · →(y βn
n , α n )].
This equality implies that we can insert Θ into the following
diagram and make it commutative
C n (Id D /y) = [C n (Id D /−)](y, α)
θ (y,α)
θ L (y,α)
γ ∗
C n (u/x)
Θ
L
[C n (Id D /−)](z, β) = C n (Id
θ (z,β)
θ L (z,β)
Consequently C n (u/x) ∼ = lim −→u/x
C n (Id D /−) for all x ∈ Ob C.
Since all these isomorphisms assemble to an isomorphism
C ∗ (u/−) ∼ = LK u C ∗ (Id D /−), by naturality, we are done. ⊓⊔
Note the second statement generalizes the obvious fact
LK pt B∗
D = LK pt RD ∼ = lim −→D
RD ∼ = R⊗ RD RD ∼ = R = R Ob D = C 0 (D
Remark 4.3.3. To understand the complex of projective RCmodules
C ∗ (u/−), we describe the structure of each C n (u/−).
As we observed in the proof, each base element (y 0 , α 0 ) →
· · · → (y n , α n ) ∈ C n (u/−) can be written as α n·[(y 0 , u(β n · · · β 1 )) →
· · · → (y n , 1 u(yn ))]. Thus similar to the special case of bar resolution
where u = Id C , we know
LK u Bn D ∼ = C n (u/−, R) ∼ ⊕
=
RHom C (u(y n ), −).
(y 0 ,u(β n···β 1 )) β 1
→···βn
→(y n ,1 u(y n))
For future reference, the dual version for the right bar resolution
is

4.3 Computation via adjoint functors 167
LK u B D n ∼ = C n (−\u, R) ∼ =
⊕
(1 u(y0 ),y 0 ) β 1
→···βn
→(u(β n···β 1 ),y n )
RHom C (−, u(y 0 )).
Recall that a category is R-acyclic if its reduced (simplicial)
homology with coefficients in R vanishes.
Corollary 4.3.4. Suppose u : D → C is a functor, M ∈
RC-mod and N ∈ mod-RC. Then
1. if u/x is R-acyclic for every x ∈ Ob C,
Ext ∗ RC(R, M) ∼ = Ext ∗ RD(R, Res u M) and Tor RC
∗
2. if x\u is R-acyclic for every x ∈ Ob C,
Ext ∗ RC(R, N) ∼ = Ext ∗ RD(R, Res u N) and Tor RC
∗
(N, R) ∼ = Tor RD
∗ (Res
(R, M) ∼ = Tor RD
∗ (R, R
Proof. We prove part (1). Under the assumption, LK u R ∼ = R
and LK u B∗
D ∼ = C ∗ (u/−, R) → R → 0 becomes a projective
resolution. Hence the isomorphism follows from
Hom RC (C ∗ (u/−, R), M) ∼ = Hom RD (B D ∗ , Res u M).
For the second isomorphism, we also have
N ⊗ RC C ∗ (u/−, R) ∼ = ⊕ N ⊗ (y 0 ,u(β n···β 1 )) β 1
RC RHom C (u(y n )
→···βn
→(y n ,1 u(y n))
∼
⊕ = (Res (y 0 ,u(β n···β 1 )) β 1
uN)(y n )
→···βn
→(y n ,1 u(y n))
∼
⊕ = β
y 1
(Res u N)(y n )
0 →···βn
→y n
∼ = Resu N ⊗ RD Bn D .
⊓⊔
The above isomorphisms are Eckmann-Shapiro type results.
Example 4.3.5. Let D be the category from Example 4.2.4 (2)
h
x
1 x
α
β
y {1 y }
,

168 4 Cohomology of categories and modules
where h 2 = 1 x and βh = α. The inclusion i : Aut D (x) → D
has two overcategories i/x and i/y, both contractible. Thus
we have
Ext ∗ RAut D (x)(R, M) ∼ = Ext ∗ RD(R, Res i M).
This calculation can be generalized to all categories that possess
a unique minimal object x whose automorphism group
acts regularly on every non-empty morphism set Hom D (x, y).
Recall from Section 2.3.1 that we introduced the subdivision
S(C) of an EI category C and showed that their classifying
spaces are homotopy equivalent and hence they have the same
simplicial (co)homology.
Corollary 4.3.6. Let C be an EI category and S(C) its subdivision.
Suppose M ∈ RC-mod and M ′ ∈ RC op -mod. Then
Tor RC
∗ (R, M) ∼ = Tor RS(C)
∗ (R, Res t M)
and
Ext ∗ RC op(R, M ′ ) ∼ = Tor RS(C)op
∗ (R, Res s M ′ )
where s : S(C) → C is the natural functor.
Proof. In the proof of Proposition 2.3., we actually proved
that x\s is contractible for all x ∈ Ob C. This is equivalent
to having s op /x contractible, by Lemma 1.2.8. Hence we can
apply Corollary 4.3.4.
⊓⊔
Remark 4.3.7. Let u : D → C be a functor. We can use LK u
to see how one obtains the maps introduced in Chapter 2
u ∗ : H∗(D, R) → H∗(C, R) and u ∗ : H ∗ (C, R) → H ∗ (D, R). In
fact we have the following diagram
D
u
pt
C
pt
•

4.3 Computation via adjoint functors 169
Thus by LK pt = LK pt LK u we get LK pt B∗ D = LK pt LK u B∗ D .
Since LK u B∗
D ∼ = C ∗ (u/−, R) is a complex of projective modules
and meanwhile there is a canonical map LK u R → R
between RC-modules, there exists a lifting, which is a chain
map unique up to chain homotopy (Comparison Theorem),
· · ·
LK u B∗
D
LK u R
0
· · ·
B C ∗
R
0
Conceptually we see the left Kan extension induces chain
maps
R ⊗ RD B∗ D ∼ = R ⊗ RC LK u B∗ D → R ⊗ RC B∗
C
because R ⊗ RD B∗ D = LK pt B∗ D = LK pt LK u B∗
D ∼ = R ⊗RC
LK u B∗
D and
Hom RC (B C ∗, R) → Hom RC (LK u B D ∗ , R) ∼ = Hom RD (B D ∗ , R).
Hence these give exactly the same map we introduced in
Chapter 2. Alternatively by applying LK pt to the previous
commutative diagram we obtain a chain map
· · · LK pt LK u B∗
D
LK pt LK u R
0
· · ·
LK pt B C ∗
LK pt R
0
The positive degree part LK pt B D ∗ = LK pt LK u B D ∗ → LK pt B C ∗
is exactly the chain map induced by u in Definition 2.2.4, i.e.
C ∗ (D, R) → C ∗ (C, R), in light of Proposition 4.3.2.
Remark 4.3.8. The following descriptions balance the understanding
of left and right Kan extensions. Especially we know
the right Kan extension of a certain injective resolution of k.
Since (−) ∧ establishes a one-to-one correspondence between

170 4 Cohomology of categories and modules
right projective kC-modules and left injective kC-modules, we
can extend the above lemmas to the bar resolution of k ∈modkD
and obtain a complex of left injective kC-modules. Suppose
B∗
D ∼ → k → 0 is the bar resolution of k ∈mod-kD, where
= C ∗ (−\Id D ). Then
B D ∗
0 → k → (B D ∗ ) ∧ ∼ = C∗ (−\Id D ) ∧
is an injective resolution of the left kD-module k. Applying
RK u we get a complex of injective kC-modules, excluding
RK u k, as follows
0 → RK u k → RK u (B D ∗ ) ∧ .
By Lemma 3.2.13 the complex RK u (B D ∗ ) ∧ ∼ = (LK u B D ∗ ) ∧ ∼ =
C ∗ (−\u) ∧ . Thus we have a simplicially constructed injective
resolution of k, and we have control on the complex after
applying the right Kan extension
0 → RK u k → (B D ∗ ) ∧ ∼ = C∗ (−\u) ∧ .
Remark 4.3.9. At this stage, the reader shall be well prepared
to go over Chapter 5 where the main results in Section 5.2
are applications of Proposition 4.3.2.
4.4 Grothendieck spectral sequences
Let u : D → C be a functor. We can construct Grothendieck
spectral sequences. They generalize the Lydon-Hochschil-
Serre spectral sequences for group extensions as well as the
Leray-Serre spectral sequences for fibrations.
4.4.1 Grothendieck spectral sequences for a functor
Let u : D → C be a functor between two small categories.
Consider the composites of following functors

4.4 Grothendieck spectral sequences 171
as well as
RD-mod RK u
RC-mod lim ←−C
R-mod,
RD-mod LK u
RC-mod lim −→C
R-mod.
The second isomorphism of the following is used in Remark
4.3.7.
Lemma 4.4.1. lim ←−C
RK u
∼ = lim ←−D
and lim −→C
LK u
∼ = lim −→D
.
Proof. This is a special case of Corollary 1.2.12 for u : D → C
and pt : C → •.
⊓⊔
Since RK u preserves injectives, there exists a Grothendieck
cohomology spectral sequence for any functor u : D → C
and M ∈ RD-mod, which comes from a double complex
E ∗,∗
0
E ∗,∗
2
(M) that we will describe shortly. The spectral sequence
(M) ⇒ E∗,∗ ∞ (M) is
H ∗ (C; H ∗ (−\u; M)) ⇒ H ∗ (D; M).
i
Remember that for any small category E, one has ←−
lim ∼
E =
H i (E; −) ∼ = Ext i RE(R, −). It means H ∗ (−\u; M) is some sort
of higher right Kan extension of M. Since LK u preserves projectives,
we get a Grothendieck homology spectral sequence
as well. However we will only construct the cohomology spectral
sequence as the construction for the homology spectral
sequence is similar. For future reference, we record the
Grothendieck homology spectral sequence for u : D → C and
M ∈ RD-mod
H∗(C; H∗(u/−; M)) ⇒ H∗(D; M),
in which H∗(u/−; M) should be considered as higher left Kan
extensions of M.

172 4 Cohomology of categories and modules
We shall construct the cohomology spectral sequence and
then assure the reader that there exists a natural pairing of
such double complexes E ∗,∗
0 (M) ⊗ E∗,∗ 0 (N) → E∗,∗ 0 (M ˆ⊗N).
With the pairing, we have a product on each page of the
Grothendieck cohomology spectral sequences
En i,j (M) ⊗ En s,t (N) → En
i+s,j+t (M ˆ⊗N),
and
E∞ i,j (M) ⊗ E∞ s,t (N) → E∞
i+s,j+t (M ˆ⊗N),
Thus we have a ring structure on
H ∗ (C; H ∗ (−\u; R)) ⇒ H ∗ (D; R),
over which the following is a module
H ∗ (C; H ∗ (−\u; M)) ⇒ H ∗ (D; M).
Fix an RD-module M we start with a double complex
(M). First take an injective resolution of M
E ∗,∗
0
0 → M → I 0 → I 1 → I 2 → · · · .
Then we apply RK u to get a complex of injective kC-modules
RK u I 0 → RK u I 1 → RK u I 2 → · · · ,
and consequently a commutative diagram
RK u I 0
RK u I 1
RK u I 2
· · ·
J 0,0
J 1,0
J 2,0
· · ·
J 0,1
J 1,1
J 2,1
· · ·
. . . ,

4.4 Grothendieck spectral sequences 173
in which every column is an injective resolution of the top
module. Now apply ←−C
lim and we obtain a double cochain complex,
denoted by E ∗,∗
0 (M),
lim
←−C J 0,0
lim
←−C J 1,0
lim
←−C J 2,0
· · ·
lim
←−C J 0,1
lim
←−C J 1,1
lim
←−C J 2,1
· · ·
lim
←−C J 0,2
lim
←−C J 1,2
lim
←−C J 2,2
· · ·
. . . ,
and it gives rise to the Grothendieck spectral sequence recorded
above. We omit the details as the construction is standard and
we are more interested in finding a pairing. Suppose we also
have a double complex E ∗,∗
0 (N) for another RD-module N
lim
←−C J ′ 0,0
lim
←−C J ′ 1,0
lim
←−C J ′ 2,0
· · ·
lim
←−C J ′ 0,1
lim
←−C J ′ 1,1
lim
←−C J ′ 2,1
· · ·
lim
←−C J ′ 0,2
lim
←−C J ′ 1,2
lim
←−C J ′ 2,2
· · ·
. . . ,
and furthermore a double complex E ∗,∗
0 (M ˆ⊗N) for the RDmodule
M ˆ⊗N

174 4 Cohomology of categories and modules
lim
←−C J ′′
0,0
lim
←−C J ′′
1,0
lim
←−C J ′′
2,0
· · ·
lim
←−C J ′′
0,1
lim
←−C J ′′
1,1
lim
←−C J ′′
2,1
· · ·
lim
←−C J ′′
0,2
lim
←−C J ′′
1,2
lim
←−C J ′′
2,2
. . . .
We want to establish a natural map
lim J ←−C
i,j ⊗ lim ←−C
J s,t ′ → lim ←−C
J i+s,j+t,
′′
which is compatible with the differentials. In fact, since there
is a unique map, given by the universal property of ←−
lim,
lim J ←−C
i,j ⊗ lim ←−C
J s,t ′ → lim ←−C
J i,j ˆ⊗J s,t,
′
we only need to construct a map ←−C
lim J i,j ˆ⊗J s,t ′ → lim ←−C
J i+s,j+t.
′′
Our definition is again based on the universal property of
lim
←−
, along with the tensor product of complexes of functors
in Section 3.4.1. We emphasize that ←−C
lim J i,j ⊗ lim ←−C
J s,t
′ →
lim J ←−C
i,j ˆ⊗J s,t ′ respects the differentials in E ∗,∗
0 due to its construction
via the universal property. This is the case when we
define lim ←−C
J i,j ˆ⊗J s,t ′ → lim ←−C
J i+s,j+t ′′ and thus we will not verify
the map we are about to construct does respect differentials.
From the two injective resolutions0 → M → I ∗ and 0 →
N → I ∗, ′ we can build a commutative diagram
0
M ˆ⊗N I ∗ ˆ⊗I ′ ∗
0
M ˆ⊗N
I ′′
∗ ,
· · ·

4.4 Grothendieck spectral sequences 175
in which the upper row is an exact sequence and the lower
one is the injective resolution used to define E ∗,∗
0 (M ˆ⊗N). Applying
RK u we obtain a chain map RK u (I ∗ ˆ⊗I ∗) ′ → RK u I ∗ ′′ .
Especially we have for any non-negative integers i and s a
map RK u (I i ˆ⊗I s) ′ → RK u I i+s. ′′ The universal property of ←−
lim
provides a morphism RK u I i ˆ⊗RK u I s ′ → RK u (I i ˆ⊗I s). ′ Thus
we have a natural map
RK u I i ˆ⊗RK u I s ′ → RK u I i+s.
′′
Next we repeat the above tensor construction for the two injective
resolutions 0 → RK u I i → J i,∗ and 0 → RK u I s ′ → J s,∗.
′
It follows from our discussions that there is a commutative diagram
0
RK u I i ˆ⊗RK u I s
′ J i,∗ ˆ⊗J s,∗
′
0
RK u I ′′
i+s
J i+s,∗+∗ ′′ .
In particular there exists J i,j ˆ⊗J s,t ′ → J i+s,j+t, ′′ and consequently
the desired map lim ←−C
J i,j ˆ⊗J s,t ′ → lim ←−C
J i+s,j+t. ′′ Hence
we do obtain a pairing E ∗,∗
0 (M) ⊗ E∗,∗ 0 (N) → E∗,∗ 0 (M ˆ⊗N).
4.4.2 Spectral sequences of category extensions
We introduced category extensions and opposite extensions
in Section 4.1.3. Here we want to show that the Grothendieck
spectral sequences for a functor have simpler forms when the
functor fits into an extension sequence.
Lemma 4.4.2. Let K → E → C be an extension. Then there
exists a natural functor ι : K(y) → π/y such that every
undercategory associated with it is contractible. Hence

176 4 Cohomology of categories and modules
1. H∗(π/−; M) ∼ = H∗(K(−); M(−)) in RC-mod for any M ∈
RE-mod; and
2. H ∗ (−\π op ; N) ∼ = H ∗ (K op (−); N(−)) in RC op -mod for any
N ∈ RE op -mod.
Proof. The category π/y has objects of the form (x, α), where
x ∈ Ob E = Ob C and α ∈ Hom C (x, y). From the definition
of π/y, one can see that the maximal objects are (y, g),
g ∈ Aut C (y), which are isomorphic to each other and have
automorphism groups isomorphic to K(y). Let us take the
full subcategory [(y, 1 y )] of π/y consisting of all maximal objects.
Its skeleton is isomorphic to the group K(y). Using
Quillen’s Theorem A, we show the undercategories associated
with ι : [(y, 1 y )] ↩→ π/y are contractible.
Fix an object (x, α) ∈ Ob(π/y). The undercategory (x, α)\(π/y)
has objects of the form (β, (y, g)), where β : (x, α) → (y, g)
is an morphism in π/y satisfying gπ(β) = α. Since π(β) =
g −1 α, by the definition of a category extension, β = g −1 αh
for a unique h ∈ K(x). From here we can deduce that
(β, (y, g)) ∼ = (β ′ , (y, g ′ )) for any (y, g ′ ) and β ′ : (x, α) →
(y, g ′ ), and that (β, (y, g)) ∈ (x, α)\(π/y) has a trivial automorphism
group. These imply (x, α)\(π/y) is equivalent to a
point, and hence is contractible.
The first isomorphism follows from Corollary 4.3.4 (2)
Tor R(π/y)
∗
(R, M) ∼ = Tor RK(y) (R, Res π M),
and the naturality in y.
The proof of the second is the same. By Lemma 1.2.8 the
overcategories associated with ι op : K(y) op → (π/y) op are
contractible. Then we apply Corollary 4.3.4 (1). ⊓⊔
We can obtain similar statements for opposite extensions.
Keep in mind that since K is a group, K ∼ = K op .
∗

4.4 Grothendieck spectral sequences 177
Lemma 4.4.3. Let K → E → C be an opposite extension.
Then there exists a natural functor ι : K(y) → y\π such
that every undercategory associated with it is contractible.
Hence
1. H∗(π op /−; M) ∼ = H∗(K op (−); M(−)) in RC op -mod for any
M ∈ RE op -mod; and
2. H ∗ (−\π; N) ∼ = H ∗ (K(−); N(−)) in RC-mod for any N ∈
RE-mod.
Proof. Since K op → E op → C op is an extension, we know by
Lemma 4.4.2 that there is a functor ι : K op (y) → π op /y such
that all undercategories are contractible. From Lemma 1.2.8,
this is equivalent to saying that all overcategories associated
to ι op : K(y) → y\π are contractible.
⊓⊔
For brevity we write H ∗ (K; M) etc, instead of H ∗ (K(−); M(−))
etc, for the functors in above lemmas.
Proposition 4.4.4. Given a functor M ∈ RE-mod, there
are two spectral sequences associated with an extension K →
E → C as follows:
1. a homology spectral sequence
E 2 ij = Hi(C; Hj(K; M)) ⇒ Hi+j(E; M);
and
2. a cohomology spectral sequence
E ij
2 = H i (C op ; H j (K op ; M op )) ⇒ H i+j (E op ; M op ).
Dually we have results for opposite extensions. Recall that
RE op -mod = mod-RE. If M ∈ RE-mod, it gives M op ∈ RE op -
mod.
Proposition 4.4.5. Given a functor M ∈ RE-mod, there
are two spectral sequences associated with an opposite extension
K → E → C as follows:

178 4 Cohomology of categories and modules
1. a homology spectral sequence
E 2 ij = Hi(C op ; Hj(K op ; M op )) ⇒ Hi+j(E op ; M op );
and
2. a cohomology spectral sequence
E ij
2 = H i (C; H j (K; M)) ⇒ H i+j (E; M).
Spectral sequences naturally give rise to some long exact
sequences, and we record them below.
Remark 4.4.6. If K → E → C is an extension and M ∈ REmod,
one can obtain two five term exact sequences
H2(E; M) → H2(C; M) → H0(C; H1(K; M)) → H1(E; M) → H1(C; M
and
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
When K → E → C is a group extension then these two exact
sequences are the usual five term sequence in group homology
and cohomology.
Similarly if K → E → C is an opposite extension, we have
H2(E op ; M op ) → H2(C op ; M op ) → H0(C; H1(K op ; M op )) → H1(E op ; M op
and
0 → H 1 (C; M) → H 1 (E; M) → H 0 (C; H 1 (K; M)) → H 2 (C; M) → H 2 (E
In general the finite generation of cohomology rings of both
K and C does not guarantee the cohomology ring of E has the
same property. One of the examples, C 2 , we used to demonstrate
that the cohomology rings of EI-categories are not
finitely generated, is an extension of a contractible category:

4.4 Grothendieck spectral sequences 179
1 y
y
h 2
y
h 2
y
ι π
α
α .
x
h 2
x
h 2
However when K is cohomologically trivial, the cohomology
rings of E and C are isomorphic.
Corollary 4.4.7. Suppose K → E → C is an extension, and
|K(x)| is invertible in R for every object x. Then for any
M ∈ RE-mod
H∗(E; M) ∼ = H∗(C; H0(K; M)) ∼ = H∗(C; −→K
lim M).
H ∗ (E op ; M) ∼ = H ∗ (C op ; H 0 (K op ; M)) ∼ = H ∗ (C op ; ←−K
lim
opM).
Since lim ←−K
opR ∼ = R in RC op -mod, we have H ∗ (C; R) ∼ =
H ∗ (C op ; R) ∼ = H ∗ (E op ; R) ∼ = H ∗ (E; R) as algebras.
Proof. Under the assumption, the E 2 (resp. E 2 ) page of the
cohomology (resp. homology) spectral sequence collapses to
the vertical (resp. horizontal) axis.
⊓⊔
Let K → E → C be an extension. Since there is a natural
correspondence between the subcategories of C and those of
E, one would like to exploit further connections between the
homological properties of C and E. Let D be a subcategory
of C and E D its “preimage” in E. We show the undercategories
(or overcategories) associated with the inclusions are
equivalent, when K → E → C is an extension (or an opposite
extension) of C.
Definition 4.4.8. Let K → E → C be an extension and
D ⊂ C a subcategory. The subextension of D in E via K| D ,
x
1 x

180 4 Cohomology of categories and modules
named E D , is a subcategory of E whose object set is the same
as D and whose morphism set consists of morphisms in E
which are preimages of morphisms in D.
If D is a full subcategory of C then E D is a full subcategory of
E. Given an extension K → E → C, Aut K (x) → Aut E (x) →
Aut C (x) is a subextension for any x ∈ Ob C.
Proposition 4.4.9. Let K → E → C a sequence of functors
and D a full subcategory of C with the inclusion ι D : D → C.
Then
1. if E is an extension of C, K| D → E D → D is the subextension
and ι ED : E D → E is the inclusion, then for any
y ∈ Ob C = Ob E, the undercategory y\ι D is isomorphic to
a subcategory of the undercategory y\ι ED , which is equivalent
to y\ι ED ;
2. if E is an opposite extension of C, K| D → E D → D is
the opposite subextension and ι ED : E D → E is the inclusion,
then for any y ∈ Ob C = Ob E, the overcategory ι D /y
is isomorphic to a subcategory of the overcategory ι ED /y,
which is equivalent to ι ED /y.
Proof. We will only prove (2). In ι ED /y, any two objects (x, α)
and (x, β) are isomorphic if and only if π(α) = π(β). Let
ι ED /y ⊂ ι ED /y be the full subcategory consisting of one object
from each isomorphism class of objects described above.
Then ι ED /y and ι ED /y are equivalent. We prove the former is
isomorphic to ι D /y.
There is a natural bijection between objects sets of these
two categories (x, α) → (x, π(α)) (π is surjective on morphisms).
We show there is a bijection between the morphism
sets and the bijections extend to a functor which gives an
isomorphism between two categories. Any (x, α) →(z, γ β) in
Mor(ι ED /y) gives rise to a morphism (x, π(α)) π(γ)
→ (z, π(β)) in

4.4 Grothendieck spectral sequences 181
ι D /y. On the other hand, a morphism (x, π(α)) π(γ)
→ (z, π(β))
in ι D /y implies π(β)π(γ) = π(α), which means there exists a
unique g ∈ K(x) such that βγ = αg. Thus we have a uniquely
defined morphism (x, α) g−1
→(x, αg) →(z, γ β) = (x, α) γg−1
→ (z, β)
in Mor(ι ED /y). Note that a different γ ′ such that π(γ ′ ) = π(γ)
gives the same morphism (x, α) γg−1
→ (z, β), so the map from
Mor(ι D /y) to Mor(ι ED /y) is well-defined. It’s straightforward
to check these two assignments on morphisms are mutually
inverse to each other.
In order to show the bijections on objects and morphisms
defining an isomorphism between categories, we need to verify
they preserve composition and identity. We will just prove
the former and leave the proof of preserving identity to the
reader. Suppose (x, α) → (z, β) → (w, γ) is a composite
of two morphisms in ι ED /y. Then our map naturally sends
it to a composite of morphisms (x, π(α)) → (z, π(β)) →
(w, π(γ)). Conversely, if (x, π(α)) π(u)
→ (z, π(β)) π(v)
→ (w, π(γ)) =
(x, π(α)) π(vu)
→ (w, π(γ)) is the composite of two morphisms in
ι D /y, then we need to show the two morphisms (x, α) ug−1
→ (z, β) vh−1
→ (w, γ
= (x, α) vh−1 ug
→
−1
(w, γ) and (x, α) vut−1
→ (w, γ) are equal, where
g, h, t are isomorphisms, described in the preceding paragraph.
Since π(vh −1 ug −1 ) = π(vut −1 ), there is a unique isomorphism
s satisfying vh −1 ug −1 = vut −1 s. But then we have
α = γvut −1 = γvh −1 ug −1 = γvut −1 s, and this forces s = 1
because Definition 4.1.21 (4 op ). Hence we get vh −1 ug −1 =
vut −1 .
⊓⊔
The following corollary is a useful outcome of the proposition.

182 4 Cohomology of categories and modules
Corollary 4.4.10. Let K → E → C be a sequence of
functors and D ⊂ C a full subcategory with the inclusion
ι D : D → C. Then
1. if E is an extension of C, then y\ι D is contractible (or
R-acyclic or connected) if and only if y\ι ED is;
2. if E is an opposite extension of C, then ι D /y is contractible
(or R-acyclic or connected) if and only if ι ED /y is.
Example 4.4.11. Let K → E → C be an extension with a
unique maximal object x such that Aut C (x) acts freely and
transitively on Hom C (y, x) for any y ∈ Ob C. Then it’s easy
to check that ι : Aut C (x) ↩→ C induces a homotopy equivalence
since all undercategories associated to it are contractible.
Hence we know Aut E (x) ↩→ E is a homotopy equivalence as
well.
Since any category can be regarded as a trivial extension
of itself, the following result is some sort of generalization of
Corollary 4.3.4.
Corollary 4.4.12. Suppose there is an extension K → E →
C. If ι D : D ↩→ C is an inclusion such that y\ι D is contractible
for every y ∈ Ob C, then H ∗ (E; M) ∼ = H ∗ (E D ; M)
for any M ∈ RE op -mod, and H∗(E; N) ∼ = H∗(E D ; N) for any
N ∈ RE-mod. Here E D is the subextension corresponding to
D.
Suppose there is an opposite extension K → E → C. If
ι D : D ↩→ C is an inclusion such that ι D /y is contractible for
every y ∈ Ob C, then H ∗ (E; M) ∼ = H ∗ (E D ; M) for any M ∈
RE-mod, and H∗(E; N) ∼ = H∗(E D ; N) for any N ∈ RE op -
mod. Here E D is the opposite subextension corresponding to
D.
Proof. We prove the statements for cohomology. Since y\ι D
is contractible for every y ∈ Ob C, y\ι ED is contractible for

4.4 Grothendieck spectral sequences 183
every y ∈ Ob E as well by Corollary 4.4.10. Then we apply
Corollary 4.3.4.
When we have an opposite extension, using Corollary 4.4.10
again we get that ι ED /y is contractible for every y ∈ Ob E.
Hence Corollary 4.3.4 can be used to obtain the isomorphism.
⊓⊔
As an example when K → E → C is an extension (resp.
an opposite extension) and C has a unique maximal (resp.
minimal) object x and Aut C (x) acts regularly on Hom C (y, x)
(or Hom C (x, y)) for any y ∈ Ob C, we have H ∗ (C; M) ∼ =
H ∗ (Aut C (x), M(x)) hence H ∗ (E; M) ∼ = H ∗ (Aut E (x), M(x))
for any contra-variant (resp. covariant) functor M.

Chapter 5
Hochschild cohomology
Abstract We consider Hochschild cohomology of category
algebras here. The upshot is that we can translate Hochschild
cohomology into functor cohomology, which provide a context
for comparing Hochschild and ordinary cohoomology of
a small category. A very general theorem on the relationship
of these two cohomology theories will be stated and proved,
combining module theoretic and simplicial methods. Some
examples are computed to help the reader to understand.
5.1 Hochschild homology and cohomology
Let A be an associative R-algebra with identity. Then we
consider A as an A e -module. We shall assume A is a finitely
presented flat R-module. This implies that A is a projective
R-module.
5.1.1 Definition and general properties
Definition 5.1.1. The acyclic Hochschild complex of A is
{B A n } n≥−1 such that
B A n = A ⊗ R · · · ⊗ R A
} {{ }
(n+2)−copies
and such that the differential ∂ n : A ⊗n → A ⊗(n−1) is given by

186 5 Hochschild cohomology
∂ n (a 0 ⊗ a 1 ⊗ · · · ⊗ a n ) ↦→
n∑
(−1) i a 0 ⊗ · · · ⊗ a i a i+1 ⊗ · · · ⊗ a n .
i=0
By direct calculations, one can see ∂ 2 = 0 and so we do have
a complex of A e -modules. Moreover since we can establish a
chain contraction s n : B A n−1 → B A n by s n (a 0 ⊗ · · · ⊗ a n ) =
a 0 ⊗ · · · ⊗ a n ⊗ 1, this complex is exact.
Suppose M ∈ A e -mod. Then we can regard it as a right A e -
module by m·(a, a ′ ) := a ′·m·a. Thus we shall not distinguish
left and right A e -modules in this chapter.
Definition 5.1.2. For any M ∈ A e -mod, we define
1. the Hochschild homology HH ∗ (A, M) to be the homology of
· · · → M ⊗ A
e B A n → · · · → M ⊗ A
e B A 1 → M ⊗ A
e B A 0 → 0;
and
2. the Hochschild cohomology HH ∗ (A, M) to be the homology
of
0 → Hom A e(B A 0 , M) → Hom A e(B A 1 , M) → · · · → Hom A e(B A n , M) →
Remark 5.1.3. Since for any n ≥ 0 there is an isomorphism
of A e -modules
B A n → A e ⊗ R ˜BA n ,
in which ˜B A 0 = R and ˜B A n for n ≥ 1 is
}
A ⊗ R ·
{{
· · ⊗ R A
}
,
n−copies
and furthermore because ˜B A n is a projective R-module B A n is
a projective A e -module, we actually obtain a complex of A e -
modules with A in degree −1
· · · → B A n → · · · → B A 1 → B A 0 → A → 0.

5.1 Hochschild homology and cohomology 187
We call B A ∗ the bar resolution of the A e -module A. Note that
M ⊗ A eBn A ∼ = M ⊗ R ˜BA n and Hom A e(Bn A , M) ∼ = Hom R (˜B n A , M).
Because of our assumption on A we can also define Hochschild
homology and cohomology as Tor Ae
n (M, A) and Ext n Ae(A, M),
respectively. When M = A, we usually write HH n (A) and
HH n (A) for the Hochschild homology and cohomology.
When we regard the bar resolution B∗
A → A → 0 as a
complex of right A-modules, it splits. Hence for any left A-
module M, Bn A ⊗ A M → A⊗ A M → 0 is a projective resolution
of the left A-module M. Based on this observation, we prove
the following well-known results [14].
Proposition 5.1.4. Let M, N ∈ A-mod and M ′ ∈ mod-A.
Suppose R is a field. Then Hom R (M, N) and N ⊗ R M ′ are
A e -modules such that
Ext ∗ A(M, N) ∼ = H ∗ (A, Hom R (M, N)) ∼ = Ext ∗ A e(A, Hom R(M, N)).
and
Tor A ∗ (M ′ , N) ∼ = H∗(A, N ⊗ R M ′ ) ∼ = Tor Ae
∗ (N ⊗ R M ′ , A).
Proof. We only prove the isomorphism for Ext. The A e -
module structure on Hom R (M, N) is given by (a 1 , a 2 )f(m) =
a 1 f(a 2 m) for any (a 1 , a 2 ) ∈ A e and m ∈ M. Let Bn
A → A
be the previously introduced projective resolution of the A e -
module A. Then
Hom A e(B A n , Hom R (M, N)) = Hom R (˜B A n , Hom R (M, N))
∼ = HomR (˜B A n ⊗ R M, N)
∼ = HomA (B A n ⊗ A M, N).

188 5 Hochschild cohomology
Since R is a field, Bn A ⊗ A M is projective and thus B∗ A ⊗ A M
becomes a projective resolution of M. Consequently the last
term above computes Ext ∗ A(M, N).
⊓⊔
Let B∗
A → A → 0 be the bar resolution of the A e -module
A. There is a chain map D : B∗ A → B∗ A ⊗ A B∗ A given by
n∑
D(a 0 ⊗a 1 ⊗· · ·⊗a n+1 ) = (a 0 ⊗· · · a i ⊗1)⊗(1⊗a i+1 ⊗· · ·⊗a n+1 ).
i=0
Let ζ ∈ H n (A) and ζ ′ ∈ H m (A). Then the cup product ζ ∪ ζ ′
is given by
B∗
A D
→B∗ A ⊗ A B∗
A ζ⊗ζ
−→A ′
⊗ A A = A.
More explicitly ζ ∪ ζ ′ : Bn+m A → A is given by
(ζ∪ζ ′ )(a 0 ⊗a 1 ⊗· · ·⊗a n+m+1 ) = ζ(a 0 ⊗· · · a n ⊗1)ζ ′ (1⊗a n+1 ⊗· · ·⊗a n+m+
Based on previous construction, HH ∗ (A) forms a ring, called
the Hochschild cohomology ring of A. The multiplicative
structure was known to Yoneda when R is a field.
By [67], one can use an arbitrary projective resolution P ∗ →
A → 0 to construct the cup product, and there exists a unique
chain map D : P ∗ → P ∗ ⊗ A P ∗ such that (∂ 0 ⊗ ∂ 0 )D = ∂ 0 . In
fact one just have to show that P ∗ ⊗ A P ∗ → A → 0 is also a
projective resolution.
Theorem 5.1.5 (Gerstenhaber). With the above cup product
the Hochschild cohomology ring
HH ∗ (A) = ⊕ i≥0
HH i (A)
is graded commutative. Moreover the cup product coincides
with the Yondea splice.

5.1 Hochschild homology and cohomology 189
Proof. Let 0 → A → L n−1 → · · · → L 0 → A → 0 and
0 → A → M m−1 → · · · → M 0 → A → 0 be two n-fold and
m-fold extensions. They uniquely determine two cohomology
classes by liftings
· · ·
B A n
B A n−1
· · ·
B A 0
A 0
and
ζ=ζ n
ζ n−1
0
A = L n
L n−1
· · ·
L 0
A 0
· · ·
B A m
B A m−1
ζ 0
· · ·
B A 0
A
0
ξ=ξ m
ξ m−1
0
A = M m
M m−1
· · ·
M 0
A 0.
The Yoneda splice ζ ∗ ξ is given by any lifting of ξ = ξ m
· · ·
Bn+m A Bn+m−1
A · · ·
Bm
A
θ n
θ n−1
θ 0 ξ
0
A
L
n−1 · · ·
L 0
A
ξ 0
B A m−1
· · ·
A 0
M m−1
· · ·
A 0
0 0 .
To finish our proof, we will define two explicit liftings. On one
hand, the composition of the following chain maps
θ ∗ ′ : B∗
A D
−→B∗ A ⊗ A B∗
A Id⊗ξ
−→B∗ A ⊗ A A[m] → B∗ A [m].
provides a lifting of ξ by
θ i ′ : Bi+m
A D
−→(B ∗ A ⊗ A B∗ A Id⊗ξ
) i+m −→Bi
A ⊗ A A[m] → Bi A [m] ξ i
→L i ,
for each 0 ≤ i ≤ n. Here [m] denotes a shift of the chain
complex. From its definition we can check that ζ ∗ ξ = ζ ◦
(Id ⊗ θ n) ′ ◦ D = ζ ∪ ξ.

190 5 Hochschild cohomology
On the other hand we can define ξ ′′
i : B A n+i → BA i
ξ ′′
i (a 0 ⊗· · ·⊗a n+i+1 ) = (−1) ni ξ(a 0 ⊗· · ·⊗a n ⊗1)⊗a n+1 ⊗· · ·⊗a n+i+1 .
They give rise to another lifting of ξ. Then we have
ζ ∗ ξ(a 0 ⊗ · · · ⊗ a n+m+1 ) = ζθ ′′ m(a 0 ⊗ · · · ⊗ a n+m+1 )
= (−1) nm ζ(ξ(a 0 ⊗ · · · ⊗ a n ⊗ 1) ⊗ a n+1 ⊗ · ·
= (−1) nm ξ(a 0 ⊗ · · · ⊗ a n ⊗ 1)ζ(a n+1 ⊗ · · · ⊗
= (−1) nm (ξ ∪ ζ)(a 0 ⊗ · · · ⊗ a n+m+1 ).
The proof is taken from [70].
Theorem 5.1.6 (Dennis,). If A and B are Morita equivalent,
then HH ∗ (A) ∼ = HH ∗ (B) and there is a ring isomorphism
HH ∗ (A) ∼ = HH ∗ (B).
Proof. The proof to the general situation is not difficult, but
one needs to know the functors realizing the given equivalence
between two module categories. See []
⊓⊔
In the situation of category algebras, if D ≃ C are two categories,
then the Hochschild (co)homology of RD and RC are
isomorphic. These isomorphisms can be seen after we express
Hochschild (co)homology as certain ordinary (co)homology.
Hochschild homology and cohomology are important invariants
of rings. However they are very difficult to compute. The
main result in this chapter is that when A = RC is a category
algebra, we can express HH ∗ (RC) by ordinary cohomology,
and this allows effective calculation.
by
⊓⊔
5.1.2 Ring homomorphisms from the Hochschild cohomology ring
In general, if A and B are two associative k-algebras and M is
a A ⊗ k B op -module, or equivalently a A-B-bimodule, we can

5.1 Hochschild homology and cohomology 191
define a ring homomorphism induced by the tensor product
− ⊗ A M
φ M : Ext ∗ A e(Λ, Λ) → Ext∗ A⊗ R Bop(M, M).
Let 0 → A → L n−1 → P n−2 → · · · → P 0 → A → 0 represent
an element in Ext n A e(A, A). We may assume P i are projective
A e -modules. Then considered as a complex of right
A-modules, it is split exact. Thus tensoring with M gives an
exact sequence of A-B-modules
0 → M → L n−1 ⊗ A M → P n−2 ⊗−AM → · · · → P 0 ⊗ A M → M → 0.
This induces a ring homomorphism φ M : Ext ∗ Ae(A, A) →
Ext ∗ A⊗ R B op(M, M). If N is another A ⊗ R B op -module, we
see Ext ∗ A⊗ R B op(M, N) has an Ext∗ Ae(A, A)-module structure
via the ring homomorphisms φ M and φ N together with the
Yoneda splice. We quote the following theorem of Snashall
and Solberg [], which generalizes Gerstenhaber’s theorem.
Theorem 5.1.7. Let A and B be two associative R-algebras.
Let η be an element in Ext n Ae(A, A) and θ an element in
Ext m A⊗ R Bop(M, N) for two A-B-bimodules M and N. Then
φ N (η)θ = (−1) mn θφ M (η).
Proof. Let us fix an element η for n ≥ 1
0 → A → L n−1 → P n−2 → · · · → P 0 → A → 0
with P i projective.
First we consider the case for θ ∈ Hom A⊗B op(M, N) (m = 0).
When n = 0, Ext 0 Ae(A, A) = Z(A). It means each element in
Ext 0 Ae(A, A) is a multiplication by some a ∈ Z(A). Such a
map induces an element in Hom A⊗B op(M, M) by m ↦→ am.
One can easily verify that (η ⊗ A N)θ = θ(η ⊗ A M).
When n = 1, we can construct the following commutative
diagram

192 5 Hochschild cohomology
0
A ⊗ A M L 0 ⊗ A M
A⊗θ
L 0 ⊗θ
A ⊗ A M 0
0 A ⊗ A N X A ⊗ A M
A⊗θ
0
A ⊗ A N
L 0 ⊗ A N
A ⊗ A N 0
where the middle row is (η ⊗ A N)θ = θ(η ⊗ A M).
Next let θ ∈ Ext 1 A⊗ R Bop(M, N) represented by 0 → N →
Y → M → 0. Since all syzygies of A are projective as right
A-modules, we have the following commutative diagram
0
Ω i (A) ⊗ A N
P i−1 ⊗ A N
0
Ω i−1 (A) ⊗ A N
0
0
Ω i (A) ⊗ A Y
P i−1 ⊗ A Y
Ω i−1 (A) ⊗ A Y
0
0
Ω i (A) ⊗ A M P i−1 ⊗ A M Ω i−1 (A) ⊗ A M 0
for all i with Ω 0 (A) = A. Denote the upper row by σ i , the
rightmost column by θ i , the leftmost column by θ i+1 and the
lower row by σ ′ i. Then by the 3 × 3-splice of [52, Lemma 3.3],
we get σ i θ i = −θ i+1 σ ′ i for all i ≥ 1. Since
η⊗ A N = (0 → A⊗ A N → L 0 ⊗ A N → Ω n−1 (A)⊗ A N → 0)σ n−1 · · · σ 1
and
η⊗ A M = (0 → A⊗ A M → L 0 ⊗ A M → Ω n−1 (A)⊗ A M → 0)σ ′ n−1 · · · σ
we obtain an equality (η ⊗ A N)θ = (−1) n θ(η ⊗ A M) by combining
all above.
When m > 1, since every θ ∈ Ext m Ae(M, N) is the Yoneda
splice of m extensions, it follows directly that (η ⊗ A N)θ =
(−1) mn θ(η ⊗ A M). We are done.
⊓⊔

5.2 Hochschild (co)homology of category algebras 193
5.2 Hochschild (co)homology of category algebras
5.2.1 Basic ideas and examples
Let C be a small category. Recall from Section 3.1.1 that we
call C e = C × C op the enveloping category of a small category
C. We also showed in Example 3.1.5 that there is a natural isomorphism
kC e ∼ = (kC) e . As a functor, kC(x, y) = kHom C (y, x)
if Hom C (y, x) ≠ ∅ and kC(x, y) = 0 otherwise. Here (x, y) ∈
Ob C e . This result is just a simple observation. It implies the
enveloping algebra of a category algebra of C is the category
algebra of its enveloping category, so later on we will just use
the terminology kC e when dealing with Hochschild cohomology.
This identification enables us to apply functor cohomology
theory to the investigation of the Hochschild cohomology
theory of category algebras.
A key ingredient is F (C), the category of factorizations in
C. Recall that the category F (C) has the morphisms in C as
its objects. In order to avoid confusion, we write an object in
F (C) as [α], whenever α ∈ Mor C. A morphism from [α] ∈
Ob F (C) to [α ′ ] ∈ Ob F (C) is given by a pair of u, v ∈ Mor C,
making the following diagram commutative
x
u
α
y
x ′ y ′ .
α ′
In other words, there is an morphism from [α] to [α ′ ] if and
only if α ′ = uαv for some u, v ∈ Mor C, or equivalently α is a
factor of α ′ in Mor C. The category F (C) admits two natural
covariant functors to C and C op
v op
C
t s
F (C)
C op ,

194 5 Hochschild cohomology
where t and s send an object [α] to its target and source,
respectively. Using his Theorem A and its corollary, Quillen
showed these two functors induce homotopy equivalences of
the classifying spaces. We will be interested in the functor
∇ = (t, s) : F (C) → C e = C × C op ,
sending an [α] ∈ Ob F (C) to (x, y) ∈ Ob C e if α ∈ Hom C (y, x)
and a morphism (u, v op ) ∈ Mor F (C) to (u, v op ) ∈ Mor(C e ).
The importance of the functor ∇ : F (C) → C e lies in the
fact that its target category gives rise to the Hochschild cohomology
ring of C, while its source category determines the
ordinary cohomology ring of C ≃ F (C). In the situation of
(finite) posets and groups, the functor is well-understood and
in the group case it has been implicitly used to establish the
homomorphism from the Hochschild cohomology ring to the
ordinary cohomology ring.
Example 5.2.1.1. When C is a poset, ∇ : F (C) → C e sends
F (C) isomorphically onto a full category C e ∆ ⊂ Ce , where
Ob C e ∆ = {(x, y) ∈ Ob C e ∣ ∣ Hom C (y, x) ≠ ∅}
(the full subcategory C∆ e is well-defined whenever C is EI).
One can easily see that RC as a functor only takes nonzero
values at objects in Ob C∆ e . Furthermore as a RCe ∆ -
module, RC ∼ = R is the trivial module. Since C∆ e ∼ = F (C)
is a co-ideal in the poset C e , we obtain Ext ∗ RC e(RC, RC) ∼ =
Ext ∗ RC (RC, RC) ∼ = Ext ∗
∆ e RF (C)(k, k) ∼ = Ext ∗ kC(k, k), where the
last isomorphism comes from the fact that BF (C) ≃ BC.
This isomorphism between the two cohomology rings was
first established in [30];
2. When C is a group, the category F (C) is a groupoid and is
equivalent to a subcategory of the one object category C e
with morphism set

5.2 Hochschild (co)homology of category algebras 195
{(g, g −1op ) ∣ ∣ g ∈ Mor C} ⊂ Mor C e .
Based on this description, one can prove the existence of the
surjective homomorphism from the Hochschild cohomology
ring to the ordinary cohomology ring of a group, which is
basically the same as the classical approach. See for example
[].
5.2.2 Hochschild (co)homology as ordinary (co)homology
The two examples of last section hint that we may express
Hochschild (co)homology in terms of ordinary (co)homology.
Moreover they show us the way to establish such an expression.
Let us examine the following commutative diagram of
small categories
F (C)
t
∇
C e = C × C op
p
C ,
where p is the projection onto the first component. Recall from
Section 3.2.2 that the preceding diagram leads to another
R
RF (C)-mod
LK ∇
RC e -mod
RC
LK t
LK p
∼ =−⊗RC R
RC-mod .
R
In the rest of this section, we will establish and describe
the following maps, induced by the three left Kan extensions
LK t , LK p and LK ∇ respectively,

196 5 Hochschild cohomology
t ∗ : Ext ∗ RF (C)(R, R) → Ext ∗ RC(R, R),
p ∗ : Ext ∗ RC e(RC, RC) → Ext∗ RC(R, R)
∇ ∗ : Ext ∗ RF (C)(R, R) → Ext ∗ RCe(RC, RC).
Theorem 5.2.2. Let C be a small category. For any functor
M ∈ RC e -mod, we have
Ext ∗ RC e(RC, M) ∼ = Ext ∗ RF (C)(R, Res ∇ M).
The maps t ∗ is an ring isomorphism, p ∗ ∼ = φ C is a split
surjective ring homomorphism and ∇ ∗ is a split injective
ring homomorphism.
The theorem is proved in a series of lemmas.
Lemma 5.2.3. The following complex of RC e -module
LK ∇ B F (C)
∗ → LK ∇ R → 0
is a projective resolution of the RC e -module RC.
Proof. Let B∗
F (C) → R → 0 be the bar resolution. By Proposition
4.3.2, LK ∇ B∗
F (C) ∼ = C∗ (∇/−, R). In Proposition 2.3.13,
due to Quillen, we asserted that the category F (C) is a cofibred
category over C e . More explicitly for any (x, y) ∈ Ob C e
the overcategory ∇/(x, y) is homotopy equivalent to the fibre
∇ −1 (x, y), which is the discrete category Hom C (y, x). In other
words, for any (x, y) ∈ Ob C e ,
is exact. Thus
C ∗ (∇/(x, y), R) → RHom C (y, x) → 0
C ∗ (∇/−, R) → RC → 0
is a projective resolution of the RC e -module RC. Furthermore
we have LK ∇ R ∼ = RC by direct calculation
(LK ∇ R)(x, y) = lim −→∇/(x,y)
R ∼ = lim −→HomC (y,x) R ∼ = RHom C (y, x).

5.2 Hochschild (co)homology of category algebras 197
⊓⊔
For any functor M ∈ RC e -mod, the isomorphism
Ext ∗ RC e(RC, M) ∼ = Ext ∗ RF (C)(R, Res ∇ M)
is a direct consequence of Lemma 5.2.3.
Before we study the ring homomorphisms, we give a interesting
result on Res ∇ RC. For an arbitrary u : D → C, since
LK u is the left adjoint of Res u , there are natural transformations
Id → Res u LK u and LK u Res u → Id. We pay attention
to the case of ∇ : F (C) → C e . There exists an RF (C)-
homomorphism R → Res ∇ LK ∇ (R) = Res ∇ (RC) as well as
a RC e -homomorphism RC = LK ∇ Res ∇ (R) → R. The latter
gives rise to a RF (C)-homomorphism
Res ∇ (RC) = Res ∇ LK ∇ Res ∇ (R) → R = Res ∇ R.
In case that C is a poset, one has R = Res ∇ (RC). When C is
a group, F (C) is a groupoid, equivalent to the automorphism
group of [1 C ] ∈ Ob F (C), that is, {(g, g −1op ) ∣ g ∈ Mor C}.
If we name the full subcategory of F (C), consisting of one
object [1 C ], by ˜∆C and the inclusion (an equivalence) by i :
˜∆C ↩→ F (C). Then Res ∇i (RC) = Res ∇ (RC)([1 C ]) is a R ˜∆Cmodule
with the action (g, g −1op )·a = gag −1 , a ∈ Res ∇i (RC).
Thus Res ∇i (RC) = ⊕ Rc g , where c g is the conjugacy class of
g ∈ Mor C. In particular R = Rc 1C is a direct summand of
Res ∇i (RC) and it implies R ∣ Res ∇ (RC) as RF (C)-modules
because i is an equivalence of categories.
Lemma 5.2.4. Let C be a small category. Then R ∣ ∣ Res ∇ (RC)
as RF (C)-modules.
Proof. One needs to keep in mind that the restriction of a
module usually has a larger R-rank than the module itself
since ∇ is not injective on objects. We define a RF (C)-

198 5 Hochschild cohomology
homomorphism (a natural transformation) ι : R → Res ∇ (RC)
by the assignments ι [α] (1 R ) = α ∈ Res ∇ (RC)([α]) for each
[α] ∈ Ob F (C). If [β] is another object in Ob F (C) and
(u, v op ) ∈ Hom F (C) ([α], [β]) is an arbitrary morphism, then
by the definition of an F (C)-morphism, (u, v op ) · α = uαv =
β. Hence ι maps R isomorphically onto a submodule of
Res ∇ (RC). On the other hand, we may define a RF (C)-
homomorphism ɛ : Res ∇ (RC) → R such that, for any
[α] ∈ Ob F (C), ɛ [α] : Res ∇ (RC)([α]) → R([α]) = R sends
each base element in Res ∇ (RC)([α]) = RHom C (y, x) to 1 R .
One can readily check the composite of these two maps is the
identity
R ι →Res ∇ (RC) ɛ →R,
and this means R ∣ Res ∇ (RC) or Res ∇ (RC) = R ⊕ N C for
some RF (C)-module N C .
⊓⊔
The module N C as a functor can be described by
N C ([α]) = R{β − γ ∣ ∣ β, γ ∈ Hom C (y, x)},
if [α] ∈ Ob F (C) and α ∈ Hom C (y, x). It will be useful to our
computation since it determines the “difference” between the
Hochschild and ordinary cohomology rings of a category. The
next lemma finishes off our proof of the main theorem.
Using adjunction between LK ∇ and Res ∇ , along with Lemmas
5.2.3 and 5.2.4, we get a commutative diagram
Hom RF (C) (B∗ F (C) , R)
splitting
Hom RF (C) (B F (C)
∗
, Res ∇ RC)
LK ∇
∼ =
Hom RC e(LK ∇ B F (C)
∗ , LK ∇ R)
Hom RC e(LK ∇ B∗
F (C) , RC)
The top row gives rise to ∇ ∗ . From this diagram one can see
it can also obtained as

5.2 Hochschild (co)homology of category algebras 199
Ext ∗ RF (C)(R, R) → Ext ∗ RF (C)(R, Res ∇ RC) ∼ = Ext ∗ RCe(RC, RC).
Now we turn to establish the ring homomorphisms.
Lemma 5.2.5. The map t ∗ : Ext ∗ RF (C)(R, R) → Ext ∗ RC(R, R)
is a ring isomorphism.
Proof. As we explained in Remark 4.3.7, t ∗ is induced by
LK t , and is the same as the map between simplicial/singular
cohomology, induced by the functor t in Chapter 2. Since
F (C) ≃ C, it is an isomorphism.
⊓⊔
Lemma 5.2.6. The map p ∗ : Ext ∗ RC e(RC, RC) → Ext∗ RC(R, R)
is equal to φ C , induced by − ⊗ RC R.
Proof. Suppose M ∈ RC e -mod. We show LK p M ∼ = M ⊗ RC R
as RC-modules. Let x ∈ Ob C. Then
1 x · LK p M = (LK p M)(x) = lim −→p/x
M.
Because p/x ∼ = (Id C /x) × C op ≃ {(x, 1 x )} × C op , we have
lim M ∼ = lim
−→p/x −→IdC
M
/x×C
∼ op = lim −→{(x,1x
M
)}×C
∼ op = lim −→C opM(x, −)
∼ = lim −→C
op1 x · M
∼ = R ⊗RC op (1 x · M)
∼ = 1x · M ⊗ RC R.
In particular it implies LK p (RC e ) ∼ = RC e ⊗ RC R ∼ = RC ⊗ R
R. Thus if ˜P∗ → RC → 0 is the projective resolution in
Section 5.1.1, it splits when regarded as a complex of right RCmodules.
Consequently LK p
∼ = − ⊗RC R maps any projective
resolution of RC to an exact sequence of left RC-modules
LK p ˜P∗ → LK p (RC) ∼ = R → 0,

200 5 Hochschild cohomology
which is a projective resolution. Hence LK p induces a chain
map
Hom RC e( ˜P∗ , RC) → Hom RC (LK p ˜P∗ , R),
and the ring homomorphism
p ∗ = φ C : Ext ∗ RC e(RC, RC) → Ext∗ RC(R, R).
Lemma 5.2.7. Let C be a small category. Then ∇ ∗ is a ring
homomorphism and there is another ring homomorphism ɛ ∗
Ext ∗ RC e(RC, RC) ↠ Ext∗ RF (C)(R, R),
such that ɛ ∗ ∇ ∗ ∼ = 1 and p ∗ ∼ = t ∗ ɛ ∗ . It means ∇ ∗ and ∇ ∗ (t ∗ ) −1
are injective ring homomorphisms while p ∗ and ɛ ∗ = (t ∗ ) −1 p ∗
are surjective ring homomorphisms.
Proof. We prove the map ∇ ∗ is a ring homomorphism. Then
from p ∗ ∇ ∗ = t ∗ we get [(t ∗ ) −1 p ∗ ]∇ ∗ = 1 Ext
∗
RF (C)
(R,R) and thus
we can define ɛ ∗ = (t ∗ ) −1 p ∗ which is a surjective ring homomorphism.
Take the bar resolution B∗
F (C) → R → 0. We have seen that
LK ∇ B∗
F (C) → LK ∇ R ∼ = RC → 0 is a projective resolution of
the RC e -module RC. Let f, g be two cocycles representing two
cohomology classes. Then we construct the following diagram
B F (C)
∗
LK ∇ B F (C)
∗
D F (C)
B F (C)
∗
LK ∇ D F (C)
LK ∇ (B F (C)
LK ∇ B∗
F (C) LK
D Ce ∇ B∗
F (C)
∗
Θ ∇
ˆ⊗B F (C)
∗
ˆ⊗B F (C)
∗
⊗ RC LK ∇ B F (C)
∗
f⊗g
⇓LK ∇
) LK ∇(f⊗g)
LK ∇ (f)⊗LK ∇ (g)
R ˆ⊗R
⊓⊔
∼ =
LK ∇ (R ˆ⊗R) ∼ =
Θ 0
∼ =
LK ∇ (R) ⊗ RC LK ∇ (R)
The first row represents f ∪ g, the cup product of f and g.
The left Kan extension LK ∇ maps it to the second row of
R
RC
∼ =
RC

5.2 Hochschild (co)homology of category algebras 201
RC e -modules which represents the image of the cup product,
and we want to show it gives rise to the cup product of
LK ∇ (f) and LK ∇ (g) as Hochschild cohomology classes. Since
we have LK ∇ (D F (C) ) : LK ∇ B∗
F (C) → LK ∇ (B∗
F (C) ˆ⊗B∗ F (C) ),
and LK ∇ (B∗
F (C) ) and LK ∇ (B∗
F (C) ) ⊗ RC LK ∇ (B∗
F (C) ) are chain
homotopy equivalent as both of them are projective resolutions
of RC, we can construct chain maps D Ce and Θ ∇ , unique
up to chain homotopy, such that the above diagram is commutative.
Because the Hochschild diagonal approximation map
always exists and is unique up to chain homotopy, independent
of the choice of a projective resolution of RC [67], D Ce will
serve as the Hochschild diagonal approximation map. Then
since the lower two rows form a commutative diagram, we
know they represent the same cohomology class, i.e. the cup
product LK ∇ (f) ∪ LK ∇ (g), in Ext ∗ RCe(RC, RC). ⊓⊔
The surjective ring homomorphism
ɛ ∗ = (t ∗ ) −1 p ∗ : Ext ∗ RC e(RC, RC) → Ext∗ RF (C)(R, R)
is the composite of
Ext ∗ RC e(RC, RC) ∼= →Ext ∗ RF (C)(R, Res ∇ RC)↠Ext ∗ RF (C)(R, R).
Remark 5.2.8. Slightly modifying the previous argument, we
can also demonstrate the action of Ext ∗ RC e(RC, RC) on Ext∗ RCe(RC, M)
alternatively via ˆ⊗ on RF (C)-mod. For any M ∈ RC e -mod,
one gets
Ext ∗ RC e(RC, M) ∼ = Ext ∗ RF (C)(R, Res ∇ M).
It was also shown that the RF (C)-module Res ∇ RC natural
splits as R ⊕ N C for some N C ∈ RF (C)-mod. This provides
a surjective homomorphism ρ : Res τ RC ˆ⊗Res ∇ M → Res ∇ M,
and hence a map

202 5 Hochschild cohomology
ρ ∗ : Ext ∗ RF (C)(R, Res ∇ RC ˆ⊗Res ∇ M) → Ext ∗ RF (C)(R, Res ∇ M).
The latter fits into the following commutative diagram
Ext ∗ RF (C) (R, Res ∇RC) ⊗ Ext ∗ RF (C) (R, Res ∇M)
∪
Ext ∗ RC e(RC, RC) ⊗ Ext∗ RCe(RC, M)
Ext ∗ RF (C) (R, Res ∇RC ˆ⊗Res ∇ M)
ρ ∗
Ext ∗ RF (C) (R, Res ∇M)
Ext ∗ RC
∪
e(RC, M),
which reduces to [67, Proposition 3.1] when C = G is a
group. The top row is the so-called cup product with respect
to the pairing ρ. Since Ext ∗ RF (C)(R, R) is a direct summand
of Ext ∗ RF (C)(R, Res ∇ RC), it also exhibits the action of
Ext ∗ RC(R, R) on Ext ∗ RCe(RC, M), via its identification with
Ext ∗ RF (C)(R, R).
Note that when C is an abelian group, we obtain an isomorphism
[36, ]
Ext ∗ RC e(RC, RC) ∼ = Ext ∗ RF (C)(R, Res ∇ (RC)) ∼ = RC⊗ R Ext ∗ RC(R, R).
Finally we comment on Hochschild homology. It is a direct
consequence of Lemma 5.2.3. Its proof is entirely analogues
to that of Corollary 4.3.4 (1). We do not have a counterpart
for Corollary 4.3.4 (2) because we do not know the structure
of (y, x)\∇ ( for posets ).
Theorem 5.2.9. Suppose N ∈ RC e -mod. Then we have
Particularly
Tor RCe
∗
Thus Tor RC
∗
Tor RCe
∗
Tor RCe
∗
(RC, RC) ∼ = Tor
(N, RC) ∼ = Tor
RF (C)
∗
RF (C)
∗
(Res ∇ N, R).
(Res ∇ RC, R) ∼ = Tor
RF (C)
∗
(R, R)⊕Tor
(R, R) is isomorphic to a direct summand of
(RC, RC).
RF (C)
∗

5.2 Hochschild (co)homology of category algebras 203
Remark 5.2.10. Suppose u : D → C is an equivalence. By
Proposition 2.3.9 we naturally obtain another equivalence
F (u) : F (D) → F (C). Because we have a commutative diagram
F (D) F (u)
F (C)
∇
D e u e
C e
we can deduce that the Hochschild (co)homology of RD and
RC are isomorphic. For instance, given any M ∈ RC e -mod,
we have
Tor RCe
∗
(RC, RC) ∼ RF (C)
= Tor∗ (Res ∇ RC, R)
∼ = Tor
RF (D)
∇
∗ (Res F (u) Res ∇ RC, R)
∼ = Tor
RF (D)
∗ (Res ∇ Res u eRC, R)
∼ = Tor
RD e
∗ (Res u eRC, RD).
Finally using Hochschild (co)homology, we can establish the
following useful isomorphisms between ordinary (co)homology.
Part (1) extends Proposition 3.3.10.
Theorem 5.2.11. Suppose k is a field and C is a small category.
Let M, N ∈ kC-mod and M ′ ∈ mod-kC. Then we
have
1. Ext ∗ kC(M, N) ∼ = Ext ∗ kF (C)(k, Res ∇ Hom k (M, N)); and
2. Tor kC
∗ (M ′ , N) ∼ kF (C)
= Tor∗
(Res ∇ (N ⊗ k M ′ ), k).
Proof. By Proposition 5.1.4, Ext ∗ kC(M, N) ∼ = Ext ∗ kC e(kC, Hom k(M, N))
From Theorem 5.2.2, Ext ∗ kC e(kC, Hom k(M, N)) ∼ = Ext ∗ kF (C)(k, Res ∇ Ho
Similarly by Proposition 5.1.4 and Theorem 5.2.9, we get
the isomorphism for homology. Note that it is natural to give
Res ∇ (N ⊗ M ′ ) a right RF (C)-module structure. ⊓⊔

204 5 Hochschild cohomology
The above isomorphisms generalize the well known isomorphisms
in group cohomology Ext ∗ kG(M, N) ∼ = Ext ∗ kG(k, Hom k (M, N))
and Tor kG
∗ (M, N) ∼ = Tor kG
∗ (N ⊗ k M, k).
Proposition 3.3.10 does not generalize to cohomology. Let
B∗ C → k → 0 be the bar resolution. Then the lefting B∗
F (C) →
Res t B∗ C induces
Hom kC (B C ∗, RK t Res ∇ Hom k (M, N)) ∼ = Hom kF (C) (Res t B C ∗, Res ∇ Hom k (M
→ Hom kF (C) (B F (C)
∗ , Res ∇ Hom R (M
and thus
Ext ∗ kC(k, RK t Res ∇ Hom k (M, N)) → Ext ∗ kF (C)(k, Res ∇ Hom R (M, N))
∼ = Ext
∗
kC (M, N).
These maps have no reason to be isomorphisms in general.
Similarly we have a map
Tor kC
∗ (M ′ , N) ∼ kF (C)
= Tor∗ (Res ∇ (N ⊗ k M ′ ), k)
→ Tor kC
∗ (LK t Res ∇ (N ⊗ k M ′ ), k),
by Corollary 4.3.4 (1) and Lemma 4.4.1.
5.2.3 EI categories
In this section, we assume our categories are EI. The purpose
is to compare Hochschild (co)homology of C with that of an
automorphism group of an object.
Suppose A is the full subcategory of C which consists of all
objects and all isomorphisms in C. The category A is a disjoint
union of finitely many finite groups. Its category algebra
RA = ⊕ x∈Ob C RAut C(x) is an RC e -module, a direct sum of
atomic modules supported on minimal objects in C RC . There
is a surjective RC e -morphism π : RC → RA, with kernel written
as ker π. Considered as a functor ker π ⊂ RC takes non-

5.2 Hochschild (co)homology of category algebras 205
zero values only at (x, y) for which there exists a C-morphism
from y to x and x ≁ = y. Since there is an inclusion functor
i : A → C, we have maps between ordinary (co)homology i ∗ :
H ∗ (C, R) → H ∗ (A, R) and i ∗ : H∗(A, R) → H∗(C, R). Here we
show there are maps between their Hochschild (co)homology
as well.
The short exact sequence of RC e -modules
induces a long exact sequence
0 → ker π → RC π →RA → 0
· · · → Ext n RC e(RC, ker π) → Extn RC e(RC, RC) ˜π →Ext n RC e(RC, RA) η → · · ·
By Proposition 4.2.1, one can see Ext ∗ RCe(RC, RA) is naturally
isomorphic to
Ext ∗ RAe(RA, RA),
which is isomorphic to the direct product of the Hochschild
cohomology rings of the automorphism groups of objects in
C: ∏
Ext ∗ RAut C (x) e(RAut C(x), RAut C (x)),
[x]⊂Ob C
where [x] stands for the isomorphism class of x ∈ Ob C. The
following map will still be written as ˜π
˜π : Ext ∗ RC e(RC, RC) → Ext∗ RAe(RA, RA).
We show ˜π can be identified with the algebra homomorphism
induced by − ⊗ RC RA, where as a left RC-module RA is a
direct sum of atomic modules.
φ A : Ext ∗ RC e(RC, RC) → Ext∗ RC e(RA, RA) ∼ = Ext ∗ RAe(RA, RA).
Hence we do not need to distinguish the maps φ A and ˜π.
Lemma 5.2.12. The following diagram is commutative

206 5 Hochschild cohomology
Ext ∗ RC e(RC, RC) ˜π
Ext ∗ RCe(RC, RA)
φ A
∼
=
Ext ∗ RC e(RA, RA)
∼ =
Ext ∗ RAe(RA, RA).
Proof. This can be seen on the cochain level. Suppose R ∗ →
RC → 0 is the minimal projective resolution of the RC e -
module RC. Then Ext ∗ RCe(RC, RC) is the cohomology of the
cochain complex Hom RC e(R ∗ , RC). The tensor product −⊗ RC
RA induces a map
Hom RC e(R ∗ , RC) → Hom RC−RA (R ∗ ⊗ RC RA, RA).
Since B∗
RC ⊗ RC RA is a projective resolution of the RC-RAmodule
RA, R ∗ ⊗ RC RA is also a projective resolution of RA.
Moreover because R ∗ is minimal, it is supported on C RC . It
implies R ∗ ⊗ RC RA is also supported on C RC . But the RC e -
module RA is supported on minimal objects of C RC , we have
Hom RC−RA (R ∗ ⊗ RC RA, RA) ∼ = Hom RA e(Res i (R ∗ ⊗ RC RA), RA)
∼ = HomRA e(Res i R ∗ , RA),
which gives rise to φ A . Here i : A e → C e is the inclusion,
induced by another inclusion, also written as i : A → C. On
the other hand ˜π is exactly given by the same chain map
Hom RC e(R ∗ , RC) → Hom RC e(R ∗ , RA) ∼ = Hom RA e(Res i R ∗ , RA).
Thus we are done.
We have the following commutative diagram, involving four
cohomology rings.
Theorem 5.2.13. Let C be an EI-category. Then we have
the following commutative diagram
⊓⊔

5.2 Hochschild (co)homology of category algebras 207
Ext ∗ RC e(RC, RC) φ A =˜π
φ C
Ext ∗ RAe(RA, RA)
φ A
Ext ∗ RC(R, R)
Res i
Ext ∗ RA(R, R).
Proof. We prove it on the cochain level. Let R ∗ → RC → 0
be the minimal projective resolution of the RC e -module RC.
Then we have the following commutative diagram
Hom RC e(R ∗ , RC) −⊗ RCRA
Hom RC−RA (R ∗ ⊗ RC RA, RA)
−⊗ RC R
Hom RC (R ∗ ⊗ RC R, R)
−⊗ RA R
Hom RC (R ∗ ⊗ RC R, R)
Hom RC (R ′ ∗, R)
Res i
Hom RA (Res i R ′ ∗, R)
Hom RA (R ′′
∗, R),
in which R ′ ∗ → R → 0 and R ′′
∗ → R → 0 are the projective
resolutions of the trivial RC- and RA-modules satisfying
the following commutative diagrams of RC-modules and RAmodules,
respectively,
R ′ ∗
R 0 R ′′
∗
R ∗ ⊗ RC R R 0 and Res i R ′ ∗
R
0
R 0.
In the main diagram, upper left cochain complex computes
Ext ∗ RC e(RC, RC), upper right corner computes Ext∗ RAe(RA, RA)
by Lemma 5.2.12, lower left corner computes Ext ∗ RC(R, R) and
lower right corner computes Ext ∗ RA(R, R). Hence our statement
follows.
⊓⊔

208 5 Hochschild cohomology
We note that in the theorem the category A may be replaced
by any full subcategory of it. Especially, we have a
commutative diagram for each Aut C (x) ⊂ A
Ext ∗ RC e(RC, RC)φ RAut C (x)
Ext ∗ RAut C (x) e(RAut C(x), RAut C (x))
φ C
φ AutC (x)
Ext ∗ RC(R, R)
Res i
Ext ∗ RAut C (x)(R, R).
5.3 Examples of the Hochschild cohomology rings of categories
In this section we calculate the Hochschild cohomology rings
for four finite EI-categories, with base field k of characteristic
2.
5.3.1 The category E 0
In [87] we presented an example, by Aurélien Djament, Laurent
Piriou and the author, of the mod-2 ordinary cohomology
ring of the following category E 0
h
g
x
1 x
α
β
gh
y {1 y }
,
where g 2 = h 2 = 1 x , gh = hg, αh = βg = α, and αg = βh =
β. The ordinary cohomology ring Ext ∗ kE 0
(k, k) is a subring of
the polynomial ring H ∗ (Z 2 × Z 2 , k) ∼ = k[u, v], removing all
u n , n ≥ 1, and their scalar multiples. It has no nilpotents and
is not finitely generated. By Theorem 2.3.4, it implies that the
Hochschild cohomology ring Ext ∗ kE e 0 (kE 0, kE 0 ) is not finitely
generated either. We compute its Hochschild cohomology ring
using Proposition 2.3.5.

5.3 Examples of the Hochschild cohomology rings of categories 209
The category of factorizations in E 0 , F (E 0 ), has the following
shape
[α]
[β]
[1 x ]
[1 y ]
[h]
[gh]
[g]
in which [1 x ] ∼ = [h] ∼ = [g] ∼ = [gh] and [α] ∼ = [β]. For the purpose
of computation, we use the skeleton F ′ (E 0 ) of F (E 0 ) (which
is equivalent to F (E 0 ) hence the two category algebras and
their module categories are Morita equivalent)
{(α,1 op
x ),(α,h op ),(β,g op ),(β,(gh) op )}
{(1 y ,1 op
x )}
[α]
[1 x ]
[1
y ].
{(1 x ,1 op
x ),(h,h op ),(g,g op ),(gh,(gh) op )}
{(1 y ,α op )}
{(1 y ,1 op
y )}
In the above category, next to each arrow is the set of homomorphisms
in F ′ (E 0 ) from one object to another. The module
N E0 ∈ kF ′ (E 0 )-mod (see Proposition 2.3.5) takes the following
values
N C ([1 x ]) = k{1 x + h, g + gh, 1 x + g} , N C ([h]) = k{1 x + h, g + gh, 1 x
N C ([g]) = k{1 x + h, g + gh, 1 x + g} , N C ([gh]) = k{1 x + h, g + gh, 1 x
N C ([α]) = k{α + β}
, N C ([β]) = k{α + β},
N C ([1 y ]) = 0.
,

210 5 Hochschild cohomology
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)
and S [1x ],k(g+gh) are simple kF ′ (E 0 )-modules such that S [1x ],k(1 x +h)([1 x ])
k(1 x + h) and S [1x ],k(g+gh)([1 x ]) = k(g + gh), and k ′ 1 x +g is a
kF ′ (E 0 )-module such that k ′ 1 x +g([1 x ]) = k(1 x +g), k ′ 1 x +g([α]) =
k(α + β) and k ′ 1 x +g([1 y ]) = 0. Note that S [1x ],k(1 x +h)([1 x ]) =
k(1 x + h), S [1x ],k(g+gh)([1 x ]) = k(g + gh) and k ′ 1 x +g([1 x ]) =
k(1 x + g) are all isomorphic to the trivial kAut F ′ (E 0 )([1 x ])-
module, and have the same trivial ring structure in the sense
that the product of any two elements is zero. Hence we have
(along with the result quoted in Section 2.4, paragraph two)
Ext ∗ kF ′ (E 0 )(k, S [1x ],k(1 x +h)) ∼ = k(1 x + h) ⊗ k Ext ∗ kAut F ′ (E0 ) ([1 x])(k, k)
and
Ext ∗ kF ′ (E 0 )(k, S [1x ],k(g+gh)) ∼ = k(g + gh) ⊗ k Ext ∗ kAut F ′ (E0 ) ([1 x])(k, k)
as rings, in which k(1 x + h) and k(g + gh) are concentrated
in degree zero in each ring. From the structure of F (E 0 ), one
has Aut F ′ (E 0 )([1 x ]) ∼ = Z 2 × Z 2 .
For computing Ext ∗ kF ′ (E 0 )(k, k ′ 1 x +g), we use the following
short exact sequence of kF (E 0 )-modules
0 → k ′ 1 x +g → k → S [1y ],k → 0.
It induces a long exact sequence in which one can find
Ext 0 kF ′ (E 0 )(k, S [1y ],k) = k and Ext n kF ′ (E 0 )(k, S [1y ],k) = 0 if n ≥ 1.
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)
for each n ≥ 1. Hence as a ring
Ext ∗ kF ′ (E 0 )(k, k ′ 1 x +g) ∼ = k(1 x +g)⊗ k Ext ∗>0
kF ′ (E 0 ) (k, k) ∼ = k(1 x +g)⊗ k Ext ∗>0
kE 0
(
All in all, we have
Ext 0 kE e 0 (kE 0, kE 0 ) ∼ = Ext 0 kE 0
(k, k) ⊕ k(1 x + h) ⊕ k(g + gh),

5.3 Examples of the Hochschild cohomology rings of categories 211
and if n ≥ 1
Ext n kE e 0 (kE 0, kE 0 )
∼ = Ext
n
kE0 (k, k) ⊕ {k(1 x + g) ⊗ k Ext n kE 0
(k, k)}
⊕{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,
Combining all the information we obtained, the surjective ring
homomorphism
φ E0 : Ext ∗ kE e 0 (kE 0, kE 0 ) ↠ Ext ∗ kE 0
(k, k)
has its kernel consisting of all nilpotents. Consequently this
Hochschild cohomology ring modulo nilpotents is not finitely
generated, against the finite generation conjecture in []. We
comment that the category algebra kE 0 is not a self-injective
algebra (hence is not Hopf, by []). Nicole Snashall points out
to the author that this algebra is Koszul since both kE 0 and
Ext ∗ kE 0
(kE 0 , kE 0 ) as graded algebras are generated in degrees
zero and one, where kE 0 = kE 0 /Rad(kE 0 ) ∼ = S x,k ⊕ S y,k .
5.3.2 The category E 1
The following category E 1 has a terminal object and hence is
contractible:
h
x
1 x
α
y {1 y } ,
where h 2 = 1 x and αh = α. The contractibility implies the
ordinary cohomology ring is simply the base field k. In this
case F (E 1 ) is the following category

212 5 Hochschild cohomology
[1 x ]
(h,1 op
y )
(α,Aut E1 (x) op )
(1 x ,1 op
x )
(h,h op )
(1 x ,h op )
(h,1 op
x )
[α]
[h]
(1 x ,h op )
(1 x ,1 op
y )
(h,1 op
x )
(1 y ,α op )
(α,Aut E1 (x) op )
[1 y ]
(1 y ,1 op
y )
We calculate its Hochschild cohomology ring. By proposition
2.3.5, we only need to compute Ext ∗ kF (E 1 )(k, N E1 ), where N E1
has the following value at objects of F (E 1 )
N E1 ([1 x ]) = k{1 x + h} , N E1 ([h]) = k{1 x + h},
N E1 ([1 y ]) = 0 , N E1 ([α]) = 0.
One can easily see that N E1 = S [1x ],k(1 x +h) is a simple module of
dimension one with a specified value k(1 x + h) at [1 x ]. Since
[1 x ] ∼ = [h] ∈ Ob F (E1 ) are minimal objects, using quoted
result in Section 2.4 paragraph two, we get
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
(
which is isomorphic to k(1 x + h) ⊗ k k[u]. Here k[u] is a polynomial
algebra with an indeterminant u at degree one and
k(1 x + h) is at degree zero. Thus
Ext ∗ kE e 1 (kE 1, kE 1 ) ∼ = Ext ∗ kE 1
(k, k)⊕Ext ∗ kF (E 1 )(k, N E1 ) ∼ = k⊕{k(1 x +h)⊗ k k
The kernel of φ E1
cohomology ring.
consists of all nilpotents in the Hochschild
5.3.3 The category E 2
The following category has its classifying space homotopy
equivalent to the join, BZ 2 ∗ BZ 2 = Σ(BZ 2 ∧ BZ 2 ) =

5.3 Examples of the Hochschild cohomology rings of categories 213
Σ[B(Z 2 × Z 2 )/(BZ 2 ∨ BZ 2 )], of the classifying spaces of the
two automorphism groups:
h
x
1 x
α
g
1 y
y ,
where h 2 = 1 x , αh = α = gα and g 2 = 1 y . As direct consequences,
its ordinary cohomology groups are equal to k, 0, 0
at degrees zero, one and two, and k n−2 at each degree n ≥ 3,
and furthermore the cup product in this ring is trivial [87].
We compute its Hochschild cohomology ring. The category
F (E 2 ) is as follows
(1 x ,1 op
x )
(h,h op )
[1 x ]
(α,Aut E2 (x) op )
(α,Aut E2 (x) op )
(1 x ,h op )
[h]
(Aut E2 (x),Aut E2 (y) op )
(h,1 op
x )
[α]
(1 y ,1 op
y )
(Aut E2 (y),α op )
(Aut E2 (y),α op )
[1 y ]
(g,g op )
[g]
(1 y ,g op )
(g,1 op
y )
By Proposition 2.3.5, we need to compute Ext ∗ kE 2
(k, N E2 ). In
this case we have
N E2 ([1 x ]) = k{1 x + h} , N E2 ([h]) = k{1 x + h},
N E2 ([1 y ]) = k{1 y + g} , N E2 ([g]) = k{1 y + g},
N E2 ([α]) = 0.
It means N E2 = S [1x ],k(1 x +h) ⊕ S [1y ],k(1 y +g) and thus by Proposition
2.2.5
Ext ∗ kE 2
(k, N E2 ) ∼ = Ext ∗ kAut F (E2 )([1 x ])(k, k(1 x + h)) ⊕ Ext ∗ kAut F (E2 )([1 y ])(k, k(1
∼ = {k(1x + h) ⊗ k Ext ∗ kZ 2
(k, k)} ⊕ {k(1 y + g) ⊗ k Ext ∗ kZ 2
Hence

214 5 Hochschild cohomology
Ext ∗ kE e 2 (kE 2, kE 2 ) ∼ = Ext ∗ kE 2
(k, k)⊕{k(1 x +h)⊗ k k[u]}⊕{k(1 y +g)⊗ k k[v]}
where k[u] and k[v] are two polynomial algebras with indeterminants
in degree one. Both the Hochschild and ordinary
cohomology rings modulo nilpotents are isomorphic to the
base field k.
5.3.4 The category E 3
The following category has a classifying space homotopy
equivalent to that of Aut E3 (x) ∼ = Z 2 (by Quillen’s Theorem A
[], or see [86])
h
x
1 x
α
β
y {1 y }
,
where h 2 = 1 x and αh = β. We compute its Hochschild
cohomology ring. The category F (E 3 ) is as follows (not all
morphisms are presented since only its skeleton is needed)
(1 y ,1 op
x )
(α,1 op
x ),(β,h op )
(h,1 op
[α]
(1 y ,h op )
[β]
(1 y ,1 op
x )
x ) [1 x ]
[1 y ]
(1 x ,h op )
(1 x ,h op )
[h]
(h,1 op
x )
(1 y ,β op )
.
The module N E3 takes the following values
(1 y ,1 op
y )
N E2 ([1 x ]) = k{1 x + h} , N E2 ([h]) = k{1 x + h},
N E2 ([1 y ]) = 0 , N E2 ([α]) = k{α + β},
N E2 ([α]) = k{α + β}.

5.3 Examples of the Hochschild cohomology rings of categories 215
Thus N E3 fits into the following short exact sequence of
kF (E 3 )-modules
0 → N E3 → k → S [1y ],k → 0.
Just like in our first example, using the long exact sequence
coming from it, we know Ext 0 kE 2
(k, N E3 ) = 0 and Ext ∗>0
kE 2
(k, N E3 ) ∼ =
k(1 x +h)⊗ k Ext ∗>0
kF (E 3 ) (k, k) ∼ = k(1 x +h)⊗ k Ext ∗>0
kE 3
(k, k). Hence
Ext ∗ kE e 3 (kE 3, kE 3 ) ∼ = Ext ∗ kE 3
(k, k)⊕{k(1 x +h)⊗ k Ext ∗>0
kE 3
(k, k)}.
The kernel of φ E3
cohomology ring.
contains all nilpotents in the Hochschild

Chapter 6
Connections with group representations and cohomology
Abstract We study various local categories of finite groups.
The purpose is to compare representations and cohomology
of groups and of these categories. The transporter categories
play a key role to bridge up these two concepts. In fact a
group and all its subgroups are some sort of transporter categories.
We pay attention to the transporter categories and
try to demonstrate their similarities with and differences from
groups, in terms of homological properties. We shall apply
tools developed in the preceding two chapters to investigate
transporter categories, a special kind of finite EI categories
which partially motivate modern research on category representations
and cohomology. In this chapter all categories are
finite and all modules are finitely generated.
6.1 Local categories
Let G be a finite group and p a positive prime that divides
|G|, the order of G. We study various collections of subgroups
of G and resulting categories. Finite categories naturally appear
in both group representation and homotopy theory. For
a finite group G and a positive prime p that divides the order
of G, various posets of p-subgroups have been investigated intensively
to establish connections between representations of
G and those of its local subgroups [61, 7, 76, 80, 81, 69]. Over
the past two decades, mathematicians realized that categories

218 6 Connections with group representations and cohomology
build upon those previously mentioned posets should play an
important role in modular representation theory [60]. Indeed,
to any such poset P, it naturally comes with a G-action.
There is a Grothendieck construction on P which results in
a category G ∝ P, called a transporter category, containing
P as a subcategory. A transporter category admits some interesting
quotient categories, namely Brauer categories, Puig
categories and orbit categories [75]. For many good reasons,
we shall call any quotient category of a transporter category a
local category, as it reveals some p-local information about G.
Representations and cohomology of these local categories are
currently in the center of group representation theory. In homotopy
theory of classifying spaces, the geometric realization
of the nerve of a transporter category is a Borel construction,
EG × G BP. It was shown by several authors, especially
Dwyer [17], that transporter categories play a key role in homology
decompositions of classifying spaces. Recently, Broto,
Levi and Oliver [10, 11] developed a theory of p-local finite
groups, which is motivated by the use of local categories associated
with BG (or its p-completion). In their terminology,
a p-local finite group is a triple (S, F, L) such that S is a
finite p-group, F is a finite category called a fusion system
on S, and L is an extension of a full subcategory of F. The
(p-completion of) classifying space of L behaves like the classifying
space of a finite group. In this theory, cohomology of
small categories is an essential ingredients. See the new book
[2] for fusion systems in group theory, homotopy theory and
representation theory.

6.1 Local categories 219
6.1.1 G-categories
Definition 6.1.1. Consider G as a category with one object
•. A G-category is a functor F : G → Cat.
In other words, a G-category is a category C, equipped with
a group homomorphism G = Aut G (•) → Aut Cat (C). The
simplest example is a point with trivial action by G. Recall
that a set is regarded as a poset with trivial relations, and a
poset is regarded as a category.
Example 6.1.2. Suppose H ⊂ G is a subgroup. Consider the
discrete set of left cosets G/H = {gH ∣ ∣ g ∈ G}. Then G acts
on it by left multiplication, permuting these cosets.
Example 6.1.3. Suppose H ⊂ G is a subgroup. Then the discrete
set of conjugacy class G H = { g H ∣ ∣ g ∈ G} is also a
G-category with G acting by conjugations.
The above two examples are the examples of G-sets.
Definition 6.1.4. A collection of subgroups of G is a set of
subgroups of G, closed under conjugations by elements in G.
Example 6.1.5. A collection of subgroups of G is naturally
a poset with inclusions as relations. Hence any collection of
subgroups is a G-poset.
There are various interesting collections of subgroups of G.
Example 6.1.6.1. The collection of all p-subgroups of G, denoted
by S e p.
2. The collection of all non-identity p-subgroups of G, denoted
by S p . Below is a concrete example of the poset of all nontrivial
2-subgroups of Σ 4 , S 2 (Σ 4 ):

220 6 Connections with group representations and cohomology
· · ·
· · ·
D 8 D 8 D 8
C 2 × C 2
C 4
V
C 2
C 2
Here D 8 is the dihedral group of order 8, V is a Kleine four
group and those C are cyclic groups with order specified
in the subscripts. Due to the size, we do not record the full
poset. In deed we only write down the subgroups of one of
the three D 8 . However, in order to obtain the full poset, one
just needs to copy the same thing under the leftmost D 8 and
put it for each omitted part, where the dots appear. Note
that V is a subgroup of all three D 8 .
3. The collection of all non-identity elementary p-subgroups of
G, denoted by A p .
4. The collection of all p-radical subgroups of G, denoted by
B p .
Moreover, there are various refinements of some of the above
posets.
Example 6.1.7.1. Let b be a p-block of kG. The the b-Brauer
pairs S b form a G-poset. When b is the principal block, it is
isomorphic to S e p.
2. Let A be an interior G-algebra. Then the pointed subgroups
of G form a G-poset.
C 2
C 2
C 2
6.1.2 Homology representations of kG
Suppose C is a small G-category. Its nerve N ∗ C is a simplicial
set, from which one can construct a chain complex C ∗ (C, k)

6.1 Local categories 221
such that, for each i > 0, C i (C, k) is a k-vector space with a
basis the set of i-chains of morphisms in C, while C 0 (C, k) =
k Ob C. Then one can see that G acts on each C i (C, k) and
C i (C, k) by permutating its base elements. Consequently we
obtains G-modules Hi(C, k) and H i (C, k).
Definition 6.1.8. Let C be a small category. Assume the normalized
complex C † ∗(C, k) is finite. Then the Euler characteristic
χ(C) = χ(C, k) is defined to be ∑ i≥0 (−1)i dim k C † i (C, k).
Note that C † ∗(C, k) being finite is equivalent to BC being
a finite CW complex. Since ∑ i≥0 (−1)i dim k C † i (C, k) =
∑
i≥0 (−1)i dim k Hi(C, k), χ(C, k) is invariant under homotopy
equivalence.
Example 6.1.9.1. In Example 6.1.2, C ∗ (G/H) is a stalk complex
C ∗ (G/H) = C 0 (G/H) ∼ = k ↑ G H with χ(G/H) = [G :
H].
2. In Example 6.1.3, C ∗ ( G H) is a stalk complex C ∗ ( G H) =
C 0 ( G H) ∼ = k ↑ G N G (H) with χ(G H) = [G : N G (H)].
3. In Example 6.1.4 (1), Sp e is not a stalk complex. However
since Sp e has an initial object and thus is contractible,
χ(Sp) e = 1.
4. The inclusions A p ⊂ S p and B p ⊂ S p induce homotopy
equivalences, by using Quillen’s Theorem A. Hence
χ(S p , k) = χ(A p , k) = χ(B p , k).
6.1.3 Transporter categories as Grothendieck constructions
Various G-posets of subgroups of G demonstrate certain local
structural information of G. However it is not complete. For
example given H ∈ G and the discrete set G H, any two nonidentical
objects are conjugate in G while this relationship is

222 6 Connections with group representations and cohomology
not seen in G H. Thus we want a category that presents all intrinsic
connections among any chosen collection of subgroups
of G.
Definition 6.1.10. Let G be a group and P a G-poset. We
define the transporter category on P to be a Grothendieck
construction G ∝ P := Gr G P. More precisely, G ∝ P has
the same objects as P, that is, Ob(G ∝ P) = Ob P. For
x, y ∈ Ob(G ∝ P), a morphism from x to y is a pair (g, gx ≤
y) for some g ∈ G.
In the literature the transporter categories are mostly considered
as auxiliary constructions before passing to various
quotient categories of them. Here we want to stress on the perhaps
unique property, among various categories constructed
from a group, that transporter categories admit natural functors
to the group itself. It singles out this particular type of
categories and is the starting point of this chapter. Here in
order to emphasize the similarities and connections between
transporter categories and subgroups, we follow a definition
which is well known to some algebraic topologists. The symbol
G ∝ P is not standard and is used because this particular
Grothendieck construction resembles a semidirect product,
yet is different.
This neat but seemingly abstract definition can be easily
seen to give the usual transporter categories. For example,
when P is the poset of non-trivial p-subgroups, we get
G ∝ P = Tr p (G), the p-transporter category of G. The advantage
of taking our approach is shown by the upcoming
examples, where each subgroup of G is identified as a transporter
category, up to a category equivalence.
From Definition 6.1.10 one can easily see that there is a
natural embedding ι P : P ↩→ G ∝ P via (x ≤ y) ↦→ (e, x ≤
y). On the other hand, the transporter category admits a

6.1 Local categories 223
natural functor π P : G ∝ P → G, given by x ↦→ • and
(g, gx ≤ y) ↦→ g. Thus we always have a sequence of functors
P ι P
↩→G ∝ P π P
→G
such that π P ◦ ι P (P) is the trivial subgroup or subcategory
of G. For convenience, in the rest of this chapter we often
neglect the subscript P and write ι = ι P , π = π P .
Example 6.1.11.1. If G acts trivially on P, then G ∝ P =
G × P.
2. Let G be a finite group and H a subgroup. We consider
the set of left cosets G/H which can be regarded as a G-
poset: G acts via left multiplication. The transporter category
G ∝ (G/H) is a connected groupoid whose skeleton
is isomorphic to H. In this way one can recover all subgroups
of G, up to category equivalences. Consequently we
have k(G ∝ G/H) ≃ kH as well as a homotopy equivalence
B(G ∝ G/H) ≃ BH.
3. From Example 6.1.6 (2) we build a concrete transporter category
Σ 4 ∝ S 2 (Σ 4 ) :
C 2
4
8
C 2 × C 2
8
4
4
8
C 2
4
8
8
· · ·
· · ·
8
D 8
8
8 D 8
8
8 D 8
8 24 24 24
C 4 8 V 24
8
C 2
8
24
8
8
24
C 2
Here the numbers are the numbers of morphisms from one
object to another. Note that again this is part of the whole
category. However, it contains a skeleton so we know what
is missing.
8
24
8
8
C 2
8

224 6 Connections with group representations and cohomology
Remark 6.1.12. One can check directly that if Hom G∝P (x, y) ≠
∅ then both Aut G∝P (x) and Aut G∝P (y) act freely on Hom G∝P (x, y).
Another notable structural fact is that G ∝ P is a category
with subobjects, which means each morphism can be
uniquely factorized as an isomorphism followed by a morphism
in P (regarded as a subcategory of G ∝ P). In fact we
have (g, gx ≤ y) = (e, gx ≤ y) ◦ (g, gx = gx) as shown in the
diagram
y
(g,gx≤y)
x
(e,gx≤y)
(g,gx=gx) gx.
Definition 6.1.13. We call B(G ∝ P) the Borel construction
on P.
For any G-space X, one can define a Borel construction
EG × G X. In our definition, we actually have B(G ∝ P) ≃
EG × G BP. It explains the concept. In particular B(G ∝
(G/H)) ≃ BH. Forming the transporter category over a G-
poset eliminates the G-action, as an algebraic analogy of introducing
a Borel construction over a G-space.
6.1.4 Local categories
Our concept of a transporter category is quite general because
each subgroup H of G can be recovered as a transporter category
G ∝ (G/H), for the G-poset G/H, up to a category
equivalence. Thus it makes sense if we deem transporter categories
as generalized subgroups for a fixed finite group.
Definition 6.1.14. Any quotient category C of G ∝ P is
called a local category of G. When P consists of p-subgroups
of G, for a prime p ∣ ∣ |G|, we also call such a quotient category
a p-local category.

6.1 Local categories 225
A local category is connected with the group by the following
diagram
G ∝ P
π
G
C
Transporter categories were implicitly considered by Mark
Ronan and Steve Smith [65] in the 1980s for constructing
group modules, and later on played a key role in Bill Dwyer’s
work [17] on homology decomposition of classifying spaces.
Dwyer used this diagram to establish connections among various
homotopy colimits (e.g. classifying spaces), while Ronan
and Smith constructed kG-modules via representations
of G ∝ P (using the language of G-presheaves on P).
Example 6.1.15.1. The p-transporter category Tr p (G) = G ∝
S p has all non-identity subgroups as its objects. For any
p-subgroups P, Q, the morphism set is often written as
Hom G (P, Q) = { g P ⊂ Q ∣ ∣ g ∈ G}. In particular AutG (P ) =
N G (P ), the normalizer.
2. The p-fusion system F p (G) is a quotient category of G ∝ S p ,
given by Hom Fp (G)(P, Q) = Hom G (P, Q)/C G (P ).
3. The p-orbit category O p (G) is a quotient category of G ∝ P,
given by Hom Op (G)(P, Q) = Q\Hom G (P, Q).
Example 6.1.16. Let b be a p-block of kG.
1. The b-transporter category Tr b (G) = G ∝ S b has all nonidentity
subgroups as its objects. For any p-subgroups P, Q,
the morphism set is often written as Hom G∝Sb ((P, e P ), (Q, e Q )) =
{ g (P, e P ) ⊂ (Q, e Q ) ∣ ∣ g ∈ G}. In particular AutG∝Sb (P ) =
N G (P, e P ).
2. The Brauer category, or p-fusion system, F b (G) is a quotient
category of G ∝ S b , given by Hom Fb (G)(P, Q) =
Hom G∝Sb (P, Q)/C G (P ).
ρ

226 6 Connections with group representations and cohomology
3. The b-orbit category O b (G) of F b (G) is a quotient category
of F b (G), given by Hom Ob (G)(P, Q) = Q\Hom Fb (G)(P, Q).
When b is the principal block, then the first two categories
are isomorphic to the first two in Example 6.1.15. The b-orbit
category is quite different from the p-orbit category because it
is commonly believed that in non-principal block situation the
latter construction is better than the former in capturing p-
local information of the block b. This explains why transporter
categories are important but still not enough and so we have
to examine various quotient categories.
6.2 Properties of local categories
In this section, we illustrate the role of transporter categories
in group representations. Then we continue to give several results
on representations of transporter categories, which further
demonstrate close connections between representations of
groups and transporter categories. This section will end with
two applications, of representation theory, to cohomology of
transporter category algebras.
6.2.1 Two diagrams of categories
We have seen that given a G-poset P there exists a diagram
of categories and functors
π
G ∝ P
G
C
for any given local category C. Diagrams similar to this
have been (implicitly) considered by Ronan-Smith [65, 6] and
Dwyer [17]. Here we pursue a direction that is closely related
to the work of Ronan and Smith. The reader is referred to [17]
ρ

6.2 Properties of local categories 227
for Dwyer’s beautiful results on homology decompositions of
classifying spaces of groups.
In practice one often finds that the functor G ∝ P → C
is part of an extension (or an opposite extension) sequence
of categories. (Among examples are various orbit categories,
Brauer categories and Puig categories.) It means that for such
a quotient category C there exists a category K which is a
disjoint union of subgroups of Aut G∝P (x), x running over
Ob P = Ob(G ∝ P), such that we can add K into the picture
K
π
G ∝ P
G
C
and moreover K ↩→ G ∝ P ↠ C satisfies some natural conditions
provided in Section 4.1.. It will helps us to understand
relationship between k(G ∝ P)-mod and kC-mod.
For any M ∈ kG-mod, sometimes we denote by κ M the
restriction Res π M and call it a constant value module.
ρ
6.2.2 Frobenius Reciprocity
Applying classical tools in homological algebra, particularly
the Kan extensions, we obtain a Frobenius Reciprocity between
kG-mod and kC-mod, where kC is the (k-)category
algebra of C. It implies, to some extent, comparing kG-mod
with kC-mod provides an extended context for local representation
theory. This observation illustrates a possible way to
understand group representations and cohomology via those
of suitable categories. From the preceding diagram we obtain
a diagram of module categories

228 6 Connections with group representations and cohomology
LK π ,RK π
k(G ∝ P)-mod
LK ρ ,RK ρ
Res π
Res ρ
kG-mod
kC-mod.
The adjunctions between the restrictions and Kan extensions
have the following consequences.
Proposition 6.2.1 (Frobenius Reciprocity). Suppose P
is a G-poset and C is a quotient category of G ∝ P as in
the preceding diagrams. Let M, N ∈ kG-mod and m, n ∈ kCmod.
Then
1. Hom kG (M, RK π Res ρ n) ∼ = Hom kC (LK ρ Res π M, n);
2. Hom kG (LK π Res ρ m, N) ∼ = Hom kC (m, RK ρ Res π N).
By direct calculations, these particular Kan extensions in
Proposition 6.2.1 are simplified:
•LK π
∼ = lim −→P
, RK π
∼ = lim ←−P
(used by Ronan-Smith. See
Corollary 6.3. for a proof);
•LK ρ
∼ =↑
kC
k(G∝P)
(the induction), RK ρ
∼ =⇑
kC
k(G∝P)
(the coinduction),
since ρ induces an algebra homomorphism k(G ∝
P) → kC when C is a quotient category.
Remark 6.2.2. For any n ∈ kC-mod, we shall write the k(G ∝
P)-module Res ρ n as n because they share the same underlying
vector space. Recall that κ M = Res π M. Then the Frobenius
Reciprocity can be rewritten as
(i’) Hom kG (M, lim ←−P
n) ∼ = Hom kC (κ M ↑ kC
k(G∝P) , n);
(ii’) Hom kG (lim −→P
m, N) ∼ = Hom kC (m, κ N ⇑ kC
k(G∝P) ).
When P = G/H for some subgroup H, we have natural
isomorphisms ←−G/H
lim ∼ = lim ∼
−→G/H
=↑
G
H . Then the above
isomorphisms certainly become the usual adjunctions between
↑ G H and ↓G H (the usual Frobenius Reciprocity) with

6.3 The functor π: group representations via transporter categories 229
C = G ∝ (G/H) and ρ = Id, in light of the Morita equivalence
between kC and kH.
Remark 6.2.3. Our Frobenius reciprocity is different from a
similar result of Ronan-Smith, see [6, 7.2.4], where they (implicitly)
had a diagram of the same shape. However their
C = G ∝ Q, not necessarily a quotient of G ∝ P, is another
transporter category and ρ is induced by a G-map P → Q.
This prohibits us from considering various quotients of transporter
categories. Moreover since a G-map P → Q usually
does not induce an algebra homomorphism from k(G ∝ P)
to k(G ∝ Q), their Kan extensions cannot be interpreted as
induction and coinduction.
The functors ↑ kC
k(G∝P)
and ⇑kC
k(G∝P)
admit interesting interpretations
when G ∝ P → C is part of an extension (or an
opposite extension) sequence of categories. Under the circumstance
↑ kC
k(G∝P)
and ⇑kC
k(G∝P)
on certain k(G ∝ P)-modules can
be very well understood. We shall discuss it in Section 5.
6.3 The functor π: group representations via transporter categories
6.3.1 Homology representations via transporter categories
Suppose P is a G-poset and π is the natural functor from the
transporter category G ∝ P to G, regarded as a category with
one object •. The naive fibre of the functor π, i.e. π −1 (•), is
exactly P. On the other hand, the overcategory π/• provides
the (topological) fibre of the map Bπ in the sense that the
sequence of categories
π/• → G ∝ P → G
corresponds to a fibration after passing to classifying spaces
B(π/•) → B(G ∝ P) → BG.

230 6 Connections with group representations and cohomology
It leads to an action of G = π 1 (BG) on B(π/•), which is
realized by a G-action on the category π/•, described shortly
after Proposition 6.3.1. We shall see that there exists a functor
π/• → P as well as an inclusion P → π/•, inducing a
category equivalence. Interestingly the former is a G-functor,
while the latter is not.
The objects of π/• are of the form (x, h), in which x ∈
Ob(G ∝ P) = Ob P and h ∈ G. A morphism from (x, h)
to (x ′ , h ′ ) is a morphism (g, gx ≤ x ′ ) ∈ Mor(G ∝ P) such
that h ′ g = h or equivalently g = h ′−1 h. It implies that each
object (x, h) ∼ = (x ′ , h ′ ) if and only if x ′ = gx and h ′ = hg −1
for some g ∈ G. Indeed the objects isomorphic to (x, h) are
{(gx, hg −1 ) ∣ ∣ g ∈ G}. Particularly (x, h) ∼ = (hx, e) for the
identity e ∈ G.
There is also an undercategory •\π. One can put it in place
of the overcategory and all our observations stay true. The
objects of •\π are of the form (h, x), in which x ∈ Ob(G ∝
P) = Ob P and h ∈ G. A morphism from (h, x) to (h ′ , x ′ ) is
a morphism (g, gx ≤ x ′ ) ∈ Mor(G ∝ P) such that gh ′ = h.
It implies that each object (h, x) ∼ = (h ′ , x ′ ) if and only if
x ′ = gx and h ′ = g −1 h for some g ∈ G. Indeed the objects
isomorphic to (h, x) are {(g −1 h, gx) ∣ ∣ g ∈ G}. Particularly
(h, x) ∼ = (e, hx) for the identity e ∈ G.
When P = •, G ∝ • ∼ = G and the functor π can be identified
with Id G . Consequently π/• ∼ = Id G /•. The following
result generalizes this special situation and gives a characterization
of the overcategory and undercategory coming from
G ∝ P → G.
Proposition 6.3.1. The category π/• is isomorphic to P ×
(Id G /•), and •\π is isomorphic to P×(•\Id G ). Consequently
π/• ∼ = •\π.

6.3 The functor π: group representations via transporter categories 231
Proof. We establish an isomorphism φ : P × (Id G /•) → π/•
as follows. The objects of P × (Id G /•) are {(x, (•, g)) ∣ g ∈ G, x ∈ Ob P}. We define φ((x, (•, g))) = (g −1 x, g) ∈
Ob(π/•). For a morphism (x 1 ≤ x 2 , g2 −1 g 1) : (x 1 , (•, g 1 )) →
(x 2 , (•, g 2 )) we put
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),
a morphism from (g1 −1 x 1, g 1 ) to (g2 −1 x 2, g 2 ). We can write
down its inverse ψ : π/• → P ×(Id G /•), given by ψ((y, h)) :=
(hy, (•, h)). For any morphism in π/•, (h −1
2 h 1, h −1
2 h 1y 1 ≤ y 2 ) :
(y 1 , h 1 ) → (y 2 , h 2 ), we define ψ((h −1
2 h 1, h −1
2 h 1y 1 ≤ y 2 )) :=
(h 1 y 1 ≤ h 2 y 2 , h2 −1 h 1) : (h 1 y 1 , (•, h 1 )) → (h 2 y 2 , (•, h 2 )).
The isomorphism for •\π can be similarly obtained.
In the end, from Example 1.2.7 (1), we have an isomorphism
Id G /• ∼ = •\Id G and hence the isomorphism between the over
category and undercategory.
⊓⊔
As we have shown in Example 1.2.7 (2) that both •\Id G and
Id G /• are equivalent to •. For instance, there is the canonical
functor pt : Id G /• → • as well as a functor • → Id G /•, given
by • ↦→ (•, e), where e is the identity of G. The reader can
quickly verify that they provide a category equivalence. These
two categories are actually the Cayley graph.
Corollary 6.3.2. There exists a natural embedding P ↩→
π/• (or P ↩→ •\π) making P a skeleton of π/• (or •\π).
Consequently LK π m ∼ = lim −→P
m and RK π m ∼ = lim ←−P
m for any
m ∈ k(G ∝ P)-mod.
Proof. By the preceding proposition, π/• ∼ = P × Id G /•. Since
we know explicitly the equivalence-inducing functors between
Id G /• and •, we can easily translate them for π/•.
The natural functor P → π/•, given by x ↦→ (x, e) and
x ≤ y ↦→ (e, x ≤ y), is an embedding, sending P to a skeleton

232 6 Connections with group representations and cohomology
of π/•. It is straightforward to check that there is a natural
surjective functor π/• → P, induced by (x, h) ↦→ hx. For any
morphism in π/•, (h −1
2 h 1, h −1
2 h 1y 1 ≤ y 2 ) : (y 1 , h 1 ) → (y 2 , h 2 ),
we define ψ((h −1
2 h 1, h2 −1 h 1y 1 ≤ y 2 )) := h 1 y 1 ≤ h 2 y 2 .
These two functors provide an equivalence between π/• and
P. Same equivalence can be established between P and •\π.
The existence of equivalences between these categories forces
lim m ∼ = lim
−→π/• −→P
m and ←−•\π
lim m ∼ = lim ←−P
m.
⊓⊔
Next we shall show that both π/• and •\π are G-categories.
Moreover we study whether or not the above functors, inducing
equivalences with P, are compatible with G-action. Before
we do so, we need to specify what we mean by this compatibility.
Definition 6.3.3. Let D, C be two G-categories and u : D →
C a functor. We say u is a G-functor if for any g ∈ G and
x ∈ Ob D we have u(gx) = gu(x), and for any α ∈ Mor D,
gu(α) = u(gα).
We first describe the G-actions on Id G /• and on •\Id G . They
are given on objects by u · (•, g ′ ) = (•, ug ′ ) and u · (g ′′ , •) =
(g ′′ u −1 , •), respectively. On morphisms,
while
u · [(•, g 1 ) g−1 2 g 1
−→(•, g 2 )] = (•, ug 1 ) g−1 2 g 1
−→(•, ug 2 )
u · [(g 1, ′ •) g′ 2 g′ −1
1
−→ (g 2, ′ •)] = (g 1u ′ −1 , •) g′ 2 g′ −1
1
−→ (g 2u ′ −1 , •).
Immediately we see that the surjection Id G /• → • is a G-
functor, and by contrast the embedding • ↩→ Id G /• is not a
G-functor, because g• = • ↦→ (•, e) by the embedding while
g(•, e) = (•, g), for any g ∈ G. Hence we do not expect the
embedding P ↩→ π/• to be a G-functor.

6.3 The functor π: group representations via transporter categories 233
Based on the G-actions on Id G /• and •\Id G and Proposition
6.3.1, we can define G-actions on π/• and •\π. For any object
(x, h) ∈ Ob(π/•) and u ∈ G, we have u · (x, h) = (x, uh),
and for any morphism (g, gx ≤ x ′ ) : (x, h) → (x ′ , h ′ ) we
have u · (g, gx ≤ x ′ ) = (g, gx ≤ x ′ ) : (x, uh) → (x ′ , uh ′ ).
Note that g = h ′−1 h = (uh ′ ) −1 (uh). Similarly For any object
(x, h) ∈ Ob(•\π) and u ∈ G, we have u · (x, h) = (x, hu −1 ),
and for any morphism (g, gx ≤ x ′ ) : (x, h) → (x ′ , h ′ ) we
have u · (g, gx ≤ x ′ ) = (g, gx ≤ x ′ ) : (x, hu −1 ) → (x ′ , h ′ u −1 ),
with g = h ′ h −1 = (h ′ u −1 )(hu −1 ) −1 . From our constructions of
G-actions on π/• and •\π, we reach the following statements.
Corollary 6.3.4.1. The isomorphisms π/• ∼ = P ×Id G /• and
•\π ∼ = P × •\Id G are G-isomorphisms.
2. The natural functors π/• ↠ P and •\π ↠ P are G-
functors.
As we mentioned earlier, a G-category C gives rise to a complex
of kG-modules C ∗ (C, k). Furthermore if u : D → C is
a G-functor between two G-categories, then it induces a G-
simplicial map u ∗ : ND ∗ → NC ∗ , and hence a chain map
between complexes of kG-modules u : C ∗ (D, k) → C ∗ (C, k).
Example 6.3.5. If H ⊂ K are subgroups of G, then there
is a G-functor G/H → G/K. Hence we have a chain map
C ∗ (G/H, k) → C ∗ (G/K, k). Since both complexes concentrate
in degree zero, this chain map consists of only one kGmap:
k(G/H) = C 0 (G/H, k) → k(G/K) = C 0 (G/K, k).
Corollary 6.3.6. The G-functors π/• → P and •\π → P
induce chain maps between complexes of kG-modules. The
complex LK π B∗
G∝P ∼ = C∗ (π/•, k) is a projective resolution
of the finite complex of kG-modules C ∗ (P).
Proof. Since LK π preserves projectives, LK π B∗
G∝P ∼ = C∗ (π/•, k)
is a complex of projective kG-modules. The existing G-

234 6 Connections with group representations and cohomology
functor π/• → P gives rise to a chain map of complexes
of kG-modules LK π B∗
G∝P ∼ = C∗ (π/•, k) → C ∗ (P, k). However
since that G-functor is a category equivalence, it induces
an isomorphism between the homology of complexes. ⊓⊔
Remark 6.3.7. Any left kG-module is naturally a right kGmodule.
If we consider C ∗ (P, k) as a complex of right kGmodules
through (−) · g := g −1 · (−), then a projective resolution
can be obtained as C ∗ (•\π, k) ∼ = RK π B∗
G in which
we take the right bar resolution of k ∈mod-k(G ∝ P). If we
regard the complex of right modules C ∗ (•\π, k) as a complex
of left modules via g · (−) := (−) · g −1 , then it is isomorphic
to C ∗ (π/•, k). In fact, the isomorphism comes from the
isomorphism of categories π/• ∼ = •\π.
It should be useful to understand the complexes C ∗ (π/•, k)
and C ∗ (•\π, k).
Example 6.3.8. By Proposition 6.3.1, if P = G/G = •, then
π G/G = Id G and π G/G /• = Id G /• is the Cayley graph, giving
rise to the total space EG whose complex is the bar resolution
B G ∗ . More generally for P = G/H, we have an isomorphism
of complexes of kG-modules C ∗ (π/•) ∼ = B G ∗ ⊗k(G/H), which
provide a projective resolution of k(G/H) = C 0 (G/H) =
C ∗ (G/H).
For any small category C, there is an augmentation map
ɛ : C ∗ (C, k) → k, which is zero on positive degrees and which
maps every base element in C 0 (C, k) = k Ob C to 1 ∈ k.
Corollary 6.3.9.1. The G-isomorphisms π/• ∼ = P × Id G /•
and •\π ∼ = P × •\Id G induce isomorphisms between complexes
of kG-modules.
2. We have equivalences of complexes of kG-modules C ∗ (π/•, k) ≃
C ∗ (P, k) ⊗ C ∗ (Id G /•, k) and C ∗ (•\π, k) ≃ C ∗ (P, k) ⊗

6.3 The functor π: group representations via transporter categories 235
C ∗ (•\Id G , k). Furthermore C ∗ (π/•) ≃ C ∗ (P) ⊗ B∗
G and
C ∗ (•\π) ≃ C ∗ (P) ⊗ B∗ G .
3. The chain map C ∗ (P × Id G /•, k) → C ∗ (Id G /•, k), induced
by P → •, corresponds to the chain map C ∗ (P, k) ⊗
C ∗ (Id G /•, k) → C ∗ (Id G /•, k), induced by the augmentation
map C ∗ (P, k) → k.
Proof. The first statement is a consequence of the preceding
corollary.
Since π/• ∼ = P × Id G /•, we have isomorphism C ∗ (π/•, k) ∼ =
C ∗ (P × Id G /•, k). By Theorem 2.2.31, there is a natural
chain homotopy equivalence C ∗ (P × Id G /•, k) → C ∗ (P, k) ⊗
C ∗ (Id G /•, k), called the Alexander-Whitney map. From its
definition, one can see it is a chain map of kG-modules.
To see the third statement, we draw a commutative diagram
C ∗ (P × Id G /•, k)
≃
C ∗ (P, k) ⊗ C ∗ (Id G /•, k)
ɛ⊗1
C ∗ (Id G /•, k) k ⊗ C ∗ (Id G /•, k)
Suppose (x 0 → · · · → x n ) ⊗ (g 0 → · · · → g n ) in a base
element of C n (P × Id G /•, k). Then the Alexander-Whitney
map sends it to
n∑
(x 0 → · · · → x i ) ⊗ (g i → · · · → g n ),
i=0
whose image under the right vertical map equals ɛ ⊗ 1(x 0 ⊗
(g 0 → · · · → g n )) = g 0 → · · · → g n . This is exactly the image
of (x 0 → · · · → x n ) ⊗ (g 0 → · · · → g n ) under the left vertical
map, which is a projection map.
⊓⊔
The last corollary of Proposition 6.3.1 is actually a consequence
of Corollary 6.3.2. Since it is closely related to the
second main result in this section, we put it here.

236 6 Connections with group representations and cohomology
Corollary 6.3.10. Suppose M ∈ kG-mod and n ∈ k(G ∝
P)-mod. Then LK π (κ M ˆ⊗n) ∼ = M⊗LK π n and RK π (κ M ˆ⊗n) ∼ =
M ⊗ RK π n as kG-modules. In particular LK π (κ M ) ∼ = M ⊗
LK π k and RK π (κ M ) ∼ = M⊗RK π k, where LK π k ∼ = H0(BP, k) ∼ =
H 0 (BP, k) ∼ = RK π k of dimension equal to the number of
connected components of P.
Proof. For the left Kan extension we have
LK π (κ M ˆ⊗n) ∼ = lim −→P
(κ M ˆ⊗n) ∼ = M ⊗ lim −→P
n ∼ = M ⊗ LK π n.
The second isomorphism is true because κ M as a kP-module
admits trivial action. The statement for the right Kan extension
is similar.
⊓⊔
The above corollary implies that LK π κ M
∼ = RKπ κ M , suggesting
the existence of a transfer map, which we will construct
later on.
The next result is a direct generalization of the fact that the
two obvious kG-module structures on P ⊗ M are isomorphic,
for P, M ∈ kG-mod with P projective. It reveals another connection
between representations of groups and of transporter
categories.
Theorem 6.3.11. Let P ∈ k(G ∝ P)-mod be a projective
module and κ M = Res π M for some M ∈ kG-mod. Then
P ˆ⊗κ M is a projective k(G ∝ P)-module. Consequently
ˆ⊗κ M → k ˆ⊗κ M = κ M → 0 is a projective resolution.
B G∝P
∗
Proof. We will prove P ˆ⊗κ M
∼ = P⊗M, with k(G ∝ P) acting
on the latter via left multiplication. To this end, we assume
P = kHom G∝P (x, −). The proof is entirely analogues to the
case when P = •, i.e. when G ∝ • = G.
We define a k-linear map ϕ : kHom G∝P (x, −) ⊗ M →
kHom G∝P (x, −) ˆ⊗κ M as follows. On base elements ϕ((g, gx ≤
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
considered as an element in κ M (x). For any (h, hy ≤ z), we
readily verify
(h, hy ≤ z)ϕ[(g, gx ≤ y)⊗m] = ϕ[(h, hy ≤ z)((g, gx ≤ y)⊗m)].
Thus ϕ is a homomorphism of k(G ∝ P)-modules.
We remind the reader that, following definition, given any
pair of x, y ∈ Ob(G ∝ P) with Hom G∝P (x, y) non-empty,
both Aut G∝P (x) and Aut G∝P (y) act freely on Hom G∝P (x, y).
This implies that kHom G∝P (x, y) is a free kAut G∝P (x)- or
kAut G∝P (y)-module. Consequently ϕ restricts on each y to
the classical kAut G∝P (y)-isomorphism (see [5, 3.1.5])
kHom G∝P (x, y)⊗M → (kHom G∝P (x, −) ˆ⊗κ M )(y) = kHom G∝P (x, y)⊗
Furthermore because as vector spaces
kHom G∝P (x, −) ˆ⊗κ M =
⊕
kHom G∝P (x, y)⊗M = kHom G∝P (x
y∈Ob(G∝P)
the linear map ϕ is actually one-to-one and hence an isomorphism
of k(G ∝ P)-modules.
⊓⊔
For an arbitrary category algebra, the tensor product of a
projective module with a non-trivial module usually does not
stay projective, as one can find a counter-example in [, 2.5].
The key point here is that the structure of G ∝ P allows us
to apply some results on group algebras.
Assume C ∗ is a complex of kG-modules. Then we naturally
obtain a complex of k(G ∝ P)-modules via restriction, written
as κ C∗ .
Lemma 6.3.12. The functor π : G ∝ P → G induces a
natural chain map Π from B∗ G∝P → k → 0 to κ B
G ∗
→ k → 0.
More generally for any M ∈ kG-mod, it naturally induces
a chain map between exact sequences of k(G ∝ P)-modules

238 6 Connections with group representations and cohomology
Π M : {B∗ G∝P ˆ⊗κ M → κ M → 0} → {κ B
G ∗
ˆ⊗κ M = κ B
G ∗ ⊗M → κ M → 0}.
Proof. The reason is that π induces a natural map between
the nerves of these categories, while the bar resolutions are
constructed from the chain complexes from the nerves. More
explicitly the complexes B∗ G∝P → k → 0 and κ B
G ∗
→ k → 0
evaluated at any x ∈ Ob(G ∝ P) are the augmented chain
complexes C ∗ (Id G∝P /x) → k → 0 and B∗
G → k → 0, respectively.
The chain map Π is induced by π[(g, ga ≤ b)] = g.
The construction of Π M for a fixed M ∈ kG-mod is similar.
⊓⊔
6.3.2 On finite generation of cohomology
Subsections 6.4.2 and 6.4.3 contains two applications of results
from Subsection 6.4.1 to transporter category cohomology
and its connection with group cohomology. To study the
functor ρ : G ∝ P → C, we will start a new Section 6.5.
Let G be a finite group and P a finite G-poset. Then there
exists a sequence of functors
P ι →G ∝ P π →G,
where ι is the natural embedding, and whose topological realization
is a fibration
BP
=
Bι
B(G ∝ P) Bπ
≃
BG
≃
BP
EG × G BP
EG × G •
where • is a point fixed by G. In order to study the finite
generation of cohomology rings, we need to recall the
Grothendieck cohomology spectral sequence for a functor
u : D → C

6.3 The functor π: group representations via transporter categories 239
H i (C; H j (\u; N)) ⇒ H i+j (D; N),
where x\u is the undercategory for each x ∈ Ob C and N is an
kD-module which can be regarded as a x\u-module through
the forgetful functor x\u → D. Since G only has one object,
the Grothendieck spectral sequence for π : G ∝ P → G reads
as follows
H i (G; H j (•\π; N)) ⇒ H i+j (G ∝ P; N),
or, for being consistent with our Ext notation,
Ext i kG(k, Ext j k(•\π)
(k, N)) ⇒ Exti+j
k(G∝P)
(k, N),
for any N ∈ k(G ∝ P)-mod, in which •\π has its skeleton is
isomorphic to the poset P. By Section 4., the above spectral
sequence is a module over
Ext i kG(k, Ext j k(•\π)
(k, k)) ⇒ Exti+j
k(G∝P)
(k, k).
Meanwhile we have a morphism between the following Grothendieck
spectral sequences, induced by
P
ι
G ∝ P π
π
G
•
G ∝ • ∼=
G,
from which we obtain a diagram of spectral sequences
Ext i kG(k, Ext j collapses
k•
(k, k))
Ext i kG(k, Ext j k(•\π) (k, k))
Ext i+j
Ext i+j
k(G∝•)
(k, k) .
k(G∝P) (k, k)
More precisely it makes the lower spectral sequence, and hence
Ext i kG(k, Ext j k(•\π)
(k, N)) ⇒ Exti+j
k(G∝P)
(k, N),

240 6 Connections with group representations and cohomology
modules over Ext ∗ k(G∝•)(k, k) = Ext ∗ kG(k, k). We point out
that the group cohomology ring H ∗ (G, k) ∼ = Ext ∗ kG(k, k) acts
on
H ∗ (G ∝ P; N) ∼ = Ext ∗ k(G∝P)(k, N)
via the algebra homomorphism induced by π (or Bπ),
Ext ∗ kG(k, k) ∼ = H ∗ (G, k) → H ∗ (|G ∝ P|, k) ∼ = H ∗ (G ∝ P; k) ∼ = Ext ∗ k(G
Since P has the property that k is of finite projective dimension,
we know H j (•\π; −) ∼ = Ext j k(•\π)
(k, −) vanishes for large
j. Furthermore, the well-known theorem of Evens and Venkov
says that, for each j, the module Ext ∗ kG(k, Ext j k(•\π)
(k, N)) is
a finitely generated over Ext ∗ kG(k, k). Since E ∞ is a subquotient
of E 2 of a cohomology spectral sequence, we have the
following statement.
Lemma 6.3.13. For any N ∈ kTr P (G)-mod, Ext ∗ k(G∝P)(k, N)
is a finitely generated Ext ∗ kG(k, k)- and Ext ∗ k(G∝P)(k, k)-module.
Theorem 6.3.11 helps us to extend this finite generation
property.
Proposition 6.3.14. For any M ∈ kG-mod and n ∈ k(G ∝
P)-mod, Ext ∗ k(G∝P)(κ M , n) is finitely generated over Ext ∗ k(G∝P)(k, k).
Proof. One can easily deduce from Theorem 6.4.11, together
with the internal hom in Section 3.4., an Eckmann-Sharpiro
type isomorphism
Ext ∗ k(G∝P)(κ M , n) ∼ = Ext ∗ k(G∝P)(k, Hom(κ M , n)).
Then we apply the finite generation result that we just quoted.
⊓⊔
The ultimate goal is to establish the finite generation of
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
the existence of an internal hom is not enough because, unless
m is induced from a kG-module, Ext ∗ k(G∝P)(m, n) ≁ =
Ext ∗ k(G∝P)(k, Hom(m, n)). We shall establish the finite generation
theorem, Theorem 6.5.3, in the very last section because
we have to rely on the Hochschild cohomology of transporter
categories.
Example 6.3.15. Let G = C 2 = {g ∣ ∣ g 2 = e} and chark =
2. Then the G-poset S e 2 is {e} → C 2 , and the transporter
category C := C 2 ∝ S e 2 is
e
{e}
e
g
g
C 2 .
There are two simple modules S {e} = S {e},k and S C2 =
S C2 ,k, both of dimension 1. Furthermore there is a short
exact sequence 0 → S C2 → P {e} → S {e} → 0. If we
apply Ext ∗ kC(−, S C2 ) to it, then we get Ext i kC(S {e} , S C2 ) ∼ =
Ext i−1
kC (S {e}, S {e} ) for i > 0 and Ext 0 kC(S {e} , S C2 ) = 0. The
cohomology ring is finitely generated because
Ext ∗ kC(k, k) ∼ = Ext ∗ kC(S {e} , S {e} ) ∼ = Ext ∗ kC 2
(k, k) ∼ = k[x]
it is a polynomial ring with one indeterminant x of degree 1.
We can see that Ext i kC(S {e} , S C2 ) is of dimension 1 for i > 0
and Ext ∗ kC(S {e} , S C2 ) is finitely generated over Ext ∗ kC(k, k). By
contrast,
Ext ∗ kC(k, Hom(S {e} , S C2 )) = 0
since Hom(S {e} , S C2 ) = 0 by direct calculation.
Remark 6.3.16. Using Dan Swenson’s definition [72] of the
internal hom one verifies that
Consequently,
κ Homk (M,N) ∼ = Hom(κ M , κ N ).
e

242 6 Connections with group representations and cohomology
Ext ∗ k(G∝P)(k, κ Homk (M,N)) ∼ = Ext ∗ k(G∝P)(k, Hom(κ M , κ N )) ∼ = Ext ∗ k(G∝P)
Before moving to the next section, we record a connection
between cohomology of transporter categories and equivariant
cohomology, which is perhaps known to the experts.
Proposition 6.3.17. The left Kan extension induces an
isomorphism
λ M : Ext ∗ k(G∝P)(k, κ M ) ∼ = H ∗ G(BP, M),
where H ∗ G (BP, M) is the equivariant cohomology group for
some M ∈ kG-mod.
Proof. Take the bar resolution B∗
G∝P → k → 0 and consider
the complex Hom k(G∝P) (B∗
G∝P , κ M ). The left Kan extension
induces a chain map
Hom k(G∝P) (B∗
G∝P , κ M ) ∼ = Hom kG (LK π B∗ G∝P , M) ∼ = Hom kG (B∗ G ⊗C ∗ (P
But the rightmost term is
Hom kG (B G ∗ ⊗ C ∗ (P), M) ∼ = Hom kG (B G ∗ , Hom k (C ∗ (P), M)),
which gives rise to H ∗ G (BP, M).
⊓⊔
In light of the above proposition, we may introduce the Tate
cohomology of transporter categories as Tate equivariant cohomology.
With Remark 3 in mind, one can further define
negative degree Ext groups Ext ∗ k(G∝P)(κ M , κ N ).
6.3.3 Transfer for ordinary cohomology
Let u : D → C be a functor between small categories. There
is always a restriction res u : H ∗ (C; k) → H ∗ (D; k). However
usually one cannot construct a map in the opposite direction,
unless the two Kan extensions are connected by a natural
transformation. In this section, based on our knowledge about

6.3 The functor π: group representations via transporter categories 243
representations of k(G ∝ P), we establish various transfer
maps, including the Becker-Gottlieb transfer map with respect
to π P : G ∝ P → G. Here we provide an algebraic
alternative to the construction of Becker-Gottlieb [4], and the
core idea is taken from Dwyer-Wilkerson [21, 9.13]). Essentially
our construction incorporates [21, 9.13] in an entirely
representation-theoretic setting. The upshot is that our construction
is analogues to the classical situation, see for instance
[5, 3.6.17]. Keep in mind that Res π and LK π generalize
↓ G H and ↑G H , respectively, used in group cohomology.
In Corollary 6.3.9 (3), we described a chain map
LK π B∗
G∝P ∼ = C∗ (π/•, k) ∼ = C ∗ (P×Id G /•, k) −→ C ∗ (Id G /•, k) ∼ = B∗ G ,
induced by P → •. This chain map gives rise to a cochain
map
Hom kG (B∗ G , k) → Hom kG (LK π B∗
G∝P , k) ∼ = Hom k(G∝P) (B∗ G∝P , k)
and hence the restriction
res π : Ext ∗ kG(k, k) → Ext ∗ k(G∝P)(k, k).
We want to establish a map, called transfer, on the opposite
direction
Ext ∗ k(G∝P)(k, k) → Ext ∗ kG(k, k).
Example 6.3.18. Before we construct the transfer, we examine
the special case for group cohomology. Suppose H ⊂ G is
a subgroup. We want to define a map tr G H : H ∗ (H, k) →
H ∗ (G, k).
For the functor π G/H : G ∝ (G/H) → G. We get
LK πG/H B G ∗ = C ∗ (π G/H /•, k) ∼ = C ∗ ((G/H) × Id G /•, k)
as complexes of kG-modules. The rightmost admits a map
to C ∗ (Id G /•, k), induced by the G-functor pt : G/H → •. If

244 6 Connections with group representations and cohomology
we invoke C ∗ ((G/H) × Id G /•, k) ≃ k(G/H) ⊗ B G ∗ , then this
chain map is identified with the chain map k(G/H) ⊗ B G ∗ →
C ∗ (Id G /•, k), induced by the augmentation
ɛ : k(G/H) = k ↑ G H→ k.
By applying Hom kG (−, k), this chain map induces the restriction
res πG/H : H ∗ (G, k) → H ∗ (G ∝ (G/H), k) ∼ = H ∗ (H, k).
To produce another map in the opposite direction, we notice
that although there is no G-functor • → G/H (unless G =
H), we can always build a kG-map after linearization
It induces a chain map
k = k• → k(G/H) = k ↑ G H .
B G ∗ → k ↑ G H ⊗B G ∗ .
When we apply Hom kG (−, k) to the following and take cohomology
B∗ G → k ↑ G H ⊗B∗
G ɛ⊗1
−→B∗ G ,
since
k ↑ G H ⊗B∗ G ∼ = (k ⊗ B∗ G ↓ G H) ↑ G H= B∗ G ↓ G H↑ G H,
we get Hom kG (B∗ G ↓ G H ↑G H , k) ∼ = Hom kH (B∗ G ↓ G H , k) and consequently
two induced maps
H ∗ (G, k) trG H
←−H ∗ (H, k) resG H
←−H ∗ (G, k).
Since the composite k → k(G/H) = k ↑ G H→ k equals a scalar
multiplication by [G : H], we know tr G H ◦ resG H = [G : H] =
|G/H|. Here the restriction res G H is identified with res π G/H
upon
the isomorphism H ∗ (G ∝ (G/H), k) ∼ = H ∗ (H, k).

6.3 The functor π: group representations via transporter categories 245
Since for arbitrary G ∝ P, we have constructed the restriction
res π : H ∗ (G, k) → H ∗ (G ∝ P, k).
Now we would like to have a generalized transfer map in the
opposite direction. As in the group case, it should be induced
by some B∗ G → LK π B∗
G∝P , or rather
C ∗ (Id G /•, k) → C ∗ (π/•, k) ∼ = C ∗ (P×Id G /•, k) ≃ C ∗ (P, k)⊗C ∗ (Id G /
Indeed, it should be reduced to establishing a map
k = k• = C ∗ (•, k) → C ∗ (P, k).
Note that, although there are many functors from • → P, all
of them have the problem that they are not G-functors. Thus
they will not directly produce chain maps between the above
two complexes of kG-modules. Now we do have to work on the
chain level. If such a chain map exists, then we immediately
obtain a chain map
C ∗ (•, k)⊗C ∗ (Id G /•, k) → C ∗ (P, k)⊗C ∗ (Id G /•, k) ≃ C ∗ (P×Id G /•, k)
As the first attempt, we record the following observation.
Lemma 6.3.19. For any n ≥ 0 and any x 0 → · · · →
x n ∈ NP n , one can construct a chain map c i : C ∗ (•, k) →
C ∗ (P, k), for each 0 ≤ i ≤ n.
Proof. Since C ∗ (•, k) = k, we define the chain maps as
C 0 (•, k) → C 0 (P, k), 1 ↦→ ∑ I
gx i ,
Here I is the G-orbit of x i .
Obviously various linear combinations of the above maps
are still kG-maps.
⊓⊔

246 6 Connections with group representations and cohomology
Example 6.3.20. Suppose x ∈ NP 0 . Then C 0 (•, k) maps isomorphically
to a 1-dimensional submodule of C 0 (P, k), that
is, kO + (x) for O + (x) = ∑ y∈O(x)
y where O(x) is the G-orbit
of x in G ∝ P. Hence we get
C ∗ (•, k)
kO + (x) ɛ
C ∗ (•, k)
C ∗ (•, k)
kO(x)
C ∗ (•, k)
C ∗ (•, k)
C ∗ (P, k) ɛ
C ∗ (•, k)
Here kO(x) is isomorphic to k(G/Stab G (x)) ∼ = k ↑ G Stab G (x) .
Tensoring with C ∗ (Id G /•, k) we get
C ∗ (Id G /•, k)
∼ =
kO + (x) ⊗ C ∗ (Id G /•, k) ɛ
C ∗ (Id G /•, k)
kO(x) ⊗ C ∗ (Id G /•, k) ɛ
C ∗ (Id G /•, k)
C ∗ (Id G /•, k)
C ∗ (Id G /•, k)
C ∗ (P, k) ⊗ C ∗ (Id G /•, k)
ɛ C ∗ (Id G /•, k)
In the central column, the top one is isomorphic to C ∗ (Id G /•, k),
and the middle is isomorphic to C ∗ (π O(x) /•, k). Thus by applying
Hom kG (−, k) and taking cohomology we a commutative
diagram
H ∗ (G, k)
H ∗ (G, k) H ∗ (G, k)
∼ =
H ∗ (G, k) H ∗ (G ∝ O(x), k)
tr O(x)
tr
res
H ∗ (G, k) H ∗ (G ∝ P, k)
∼ =
res O(x)
res π
H ∗ (G, k)
H ∗ (G, k)

6.3 The functor π: group representations via transporter categories 247
The transfer map is the usual one for group cohomology, see
Example 6.3.17. The restrictions are induced by the functors
G ∝ O(x) ↩→ G ∝ P → G. The above diagram reduces to a
sequence of maps
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (G ∝ O(x), k) → H ∗ (G, k),
or equally
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (Stab G (x), k) → H ∗ (G, k).
We can actually take any G-set sitting inside P. The natural
candidates are all the sets NP n , n ≥ 0. For instance if α =
x 0 < · · · < x n ∈ NP n and O(α) denotes the G-orbit of α,
then we will get a sequence of maps
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (G ∝ O(α), k) → H ∗ (G, k),
or equally
H ∗ (G, k) → H ∗ (G ∝ P, k) → H ∗ (Stab G (α), k) → H ∗ (G, k).
Note that Stab G (α) = ⋂ i Stab G(x i ).
The preceding example implies that using naively constructed
chain maps C ∗ (•, k) → C ∗ (P, k) it is unlikely to
obtain a novel H ∗ (G ∝ P, k) → H ∗ (G, k) which does not depend
on the transfer in group cohomology H ∗ (Stab G (α), k) →
H ∗ (G, k). Thus we must try something more complicated.
In the proof of the next theorem, we construct the Becker-
Gottlieb transfer, also seen in Dwyer-Wilkerson.
It seems like many important theorems in group cohomology
share the same nature and live in the same context. Using
the double complex Hom kG (C † ∗(P) ⊗ B G ∗ , k) to obtain Webb’s
Theorem, Dwyer’s sharp decompositions results

248 6 Connections with group representations and cohomology
When P = S p , by Brown’s theorem, χ(P) = 1 in k. By
Quillen’s() Theorem res π : H ∗ (G, k) → H ∗ (G ∝ P, k) is an
isomorphism. What about A p etc
Theorem 6.3.21. Suppose Res π : kG-mod → k(G ∝ P)-
mod is the restriction along π and write κ M = Res π M for
any M ∈ kG-mod. Then we have the following two maps,
restriction and transfer,
Ext ∗ kG(M, N) res P
−→Ext ∗ k(G∝P)(κ M , κ N ) tr P
−→Ext ∗ kG(M, N),
which compose to χ(P; k) · 1, multiplication by the Euler
characteristic of (the order complex of) P.
Proof. We shall construct these two maps. Then in the sequel
we can deduce the statement on their composite. The restriction
is a generalized version of the one shown at the beginning
of this section.
Suppose B∗
G and B∗
G∝P are the bar resolutions of k ∈ kGmod
and k ∈ k(G ∝ P)-mod, respectively. Then B∗
G ⊗ M
is a projective resolution of a fixed M ∈ kG-mod. For
another kG-module N, the cochain complex Hom kG (B∗
G ⊗
M, N) computes Ext ∗ kG(M, N). The exact functor Res π sends
this cochain complex to Hom k(G∝P) (κ B
G ∗ ⊗M, κ N ). Applying
Lemma 6.3.12 we obtain the composite of two chain maps,
unique up to chain homotopy,
Hom kG (B∗ G ⊗M, N) Res π
→ Hom k(G∝P) (κ B
G ∗ ⊗M, κ N ) Π∗ M
→Hom k(G∝P) (B∗
G∝P
These seemingly abstract maps can be written down explicitly,
but we will leave it to the interested reader. The composite
ΠM ∗ ◦ Res π induces a map on cohomology, which we call
the restriction,
res P : Ext ∗ kG(M, N) → Ext ∗ k(G∝P)(κ M , κ N ).
ˆ⊗κ

6.3 The functor π: group representations via transporter categories 249
It is helpful to have a different characterization of the
restriction. In order to do so we use a series of obvious
isomorphisms to rewrite the previously mentioned complex
Hom k(G∝P) (B∗
G∝P ˆ⊗κ M , κ N ). Firstly by adjunction, it is isomorphic
to
Hom kG (LK π (B∗
G∝P ˆ⊗κ M ), N).
Here (ΠM ∗ ◦ Res π)α is mapped to (Λ N ◦ LK π )(ΠM ∗ ◦ Res πα),
where Λ N : LK π κ N → N, determined by the counit of
adjunction, is isomorphic to ¯ɛ ⊗ Id N . The natural map ¯ɛ :
H0(P) → k is induced by the augmentation map ɛ : C 0 (P) →
k (or rather P → •), by Corollary 6.3.10. Secondly from the
same corollary our cochain complex is canonically isomorphic
to
Hom kG (LK π (B∗
G∝P ) ⊗ M, N).
Thirdly by Corollary 6.3.9 LK π B∗ G∝P ≃ B∗ G ⊗ C ∗ (P), the
above complex is
Hom kG (LK π (B G∝P
∗ ) ⊗ M, N) ≃ Hom kG (B G ∗ ⊗ C ∗ (P) ⊗ M, N)
∼ = HomkG (B G ∗ ⊗ M ⊗ C ∗ (P), N).
The observations imply that res P : Ext ∗ kG(M, N) → Ext ∗ k(G∝P)(κ M , κ N
is the same as the map induced by the following chain map
Hom kG (B G ∗ ⊗ M, N) → Hom kG (B G ∗ ⊗ M ⊗ C ∗ (P), N),
which is given by the augmentation ɛ : C ∗ (P) → k. Now we
are ready to build the transfer map. From the cochain complex
Hom kG (B G ∗ ⊗ M ⊗ C ∗ (P), N) we continue to establish a map
which leads back to group cohomology
Θ ∗ : Hom kG (B G ∗ ⊗ M ⊗ C ∗ (P), N) → Hom kG (B G ∗ ⊗ M, N).
The map Θ ∗ is induce by the following chain map Θ : k →
C ∗ (P) of Dwyer-Wilkerson [21, 9.13], as the composite of

250 6 Connections with group representations and cohomology
k a↦→a·Id
→ Hom k (C ∗ (P), C ∗ (P)) ∼= →C ∗ (P) ∧ ⊗C ∗ (P) Id⊗∆
→ C ∗ (P) ∧ ⊗C ∗ (P)⊗C
Here C ∗ (P) ∧ = Hom k (C ∗ (P), k), non-positively graded, is the
k-dual of C ∗ (P). The composite of
Hom k(G∝P) (B∗
G∝P
defines a map
ˆ⊗κ M , κ N ) ∼= →Hom kG (LK π (B G∝P
∗
tr P : Ext ∗ k(G∝P)(κ M , κ N ) → Ext ∗ kG(M, N).
ˆ⊗κ M ), N) →Hom Θ∗
kG
which is called the transfer.
In fact we have the following commutative diagram of
cochain complexes
Hom kG (B∗ G ⊗ M, N) Hom k(G∝P) (B∗ G∝P ˆ⊗κ M , κ N )
Hom kG (B∗ G ⊗ M
=
≃
=
Hom kG (B∗ G ⊗ M, N) Hom kG (B∗ G ⊗ M ⊗ C ∗ (P), N) Hom kG (B∗ G ⊗ M
Upon passing to cohomology, both rows give rise to
Ext ∗ kG(M, N) res P
→Ext ∗ k(G∝P)(κ M , κ N ) tr P
→Ext ∗ kG(M, N).
In the end we prove that the composite res P ◦tr P = χ(P)·1.
Since the normalization C ∗ (P, k) → C † ∗(P, k) is a G-chain
homotopy equivalence, we can replace C ∗ (P, k) by C † ∗(P, k) in
our calculation. The normalized chain complex is finite so we
can find an integer d such that C † d (P, k) ≠ 0 but C† n(P, k) = 0
for all n > d. The following proof is due to Dwyer-Wilkerson
[21, 9.13] too, which shows by direct calculation that
k → C † ∗(P) → k
is a scalar multiplication by χ(P). Write the natural basis of
C † n(P) as {c i n} d n
i=1 for d n = dim k C † n(P) (see Section 6.2.2).
The step-by-step images of 1 ∈ k under Θ are

6.3 The functor π: group representations via transporter categories 251
1 ↦→ Id C
†
∗ (P)
↦→ ∑ d
n=0 (−1)n { ∑ d n
i=1 (ci n) ∧ ⊗ c i n}
↦→ ∑ d
n=0 (−1)n { ∑ d n
= ∑ d
n=0 (−1)n { ∑ d n
↦→ ∑ d
n=0 (−1)n { ∑ d n
↦→ ∑ d
n=0 (−1)n d n
= χ(P).
i=0 (ci n) ∧ ⊗ [c i n ⊗ t(c i n)]}
i=1 [(ci n) ∧ ⊗ c i n] ⊗ t(c i n)}
i=1 t(ci n)}
Here t(c i n) ∈ C † 0 (P) denotes the last object, i.e. the target, of
the n-chain of morphisms c i n ∈ C † n(P).
⊓⊔
Remark 6.3.22. In fact, by Remark 6.3.15, the above restriction
and transfer coincide with
Ext ∗ kG(k, Hom k (M, N)) res P
→Ext ∗ k(G∝P)(k, κ Homk (M,N)) tr P
−→Ext ∗ kG(k, Hom
When M = N = k, our construction is exactly the Becker-
Gottlieb transfer ([4],[21]), because Ext ∗ k(G∝P)(k, κ M ) ∼ = H ∗ G (BP, M).
We emphasize that if either M ∈ kG-mod is not acted trivially
by kG or if G sends a connected component of P to a different
one, then the constantly valued κ M ∈ k(G ∝ P)-mod
is not truly constant since H 0 (G ∝ P; κ M ) ∼ = lim ←−G∝P
κ M
∼ =
lim lim κ ∼ ←−G ←−P
M = (M ⊗ H 0 (P)) G .
Corollary 6.3.23. If χ(P; k) is invertible in k, then res P is
an injective homomorphism.
According to Dwyer, a collection of subgroups of G is a
set of subgroups that is closed under conjugation. If P is a
collection then it is naturally a G-poset in which the relations
are inclusions and G-acts by conjugation. A collection is called
ample if for a fixed prime p and a field k of characteristic p
the restriction res P : Ext ∗ k(G∝P)(k, k) → Ext ∗ kG(k, k) is an

252 6 Connections with group representations and cohomology
isomorphism. There is an extensive discussion on such posets
in [17] or [18]. When P is ample, we get tr P = χ(P)res −1
P .
6.4 The functor ρ: invariants and coinvariants
We have exploited the functor π : G ∝ P → G. Now we
turn to study the other functor ρ : G ∝ P → C, where
C is a quotient category of G ∝ P. In general situation it
seems hard to make group theoretic interpretation of ↑ kC
k(G∝P)
and ⇑ kC
k(G∝P)
. However we can do so when we have certain
quotient categories, which are part of some category extension
sequences in the sense of Hoff.
An extension E of a category C via a category K is a sequence
of functors
K ι
−→E ρ
−→C,
which satisfies certain properties. A sequence K ↩→ E ↠ C
is called an opposite extension if K op ↩→ E op ↠ C op is an
extension.
The advantage of considering π : G ∝ P → C which is part
of an extension (or opposite extension) is that it enables us
to provide a good characterization of the left (or right) Kan
extension. Indeed it is the case for many familiar category
constructions in representation theory and homotopy theory.
Remark 6.4.1. We emphasize that for any quotient category
C of G ∝ P, P is naturally a subcategory of C. Thus for
any kC-module, it makes sense to consider its limits. Indeed
if m ∈ kC-mod then both ←−P
lim m ∼ = lim ←−P
Res ρ m and −→P
lim m ∼ =
lim Res −→P ρm are kG-modules.
Lemma 6.4.2. Let K → E → C a sequence of three EIcategories
and m ∈ kE-mod.

6.4 The functor ρ: invariants and coinvariants 253
1. Suppose K → E → C is an extension. Then LK ρ m ∼ = m K ,
where m K as a functor over C is given by m K (x) = m(x) K(x)
(K(x)-coinvariants of the kAut E (x)-module m(x)), for any
x ∈ Ob C = Ob E = Ob K.
2. Suppose K → E → C is an opposite extension. Then
RK ρ m ∼ = m K , where m K as a functor over C is given
by m K (x) = m(x) K(x) (K(x)-invariants of the kAut E (x)-
module m(x)), for any x ∈ Ob C = Ob E = Ob K.
In the above lemma, there is another way to express the
Kan extensions. Under the same assumptions, they are m K =
H0(K; m) and m K = H 0 (K; m) respectively.
In what follows, we shall apply the above statements to various
local categories of G, in combination with the Frobenius
reciprocity
(i’) Hom kG (M, lim ←−P
n) ∼ = Hom kC (LK ρ κ M , n);
(ii’) Hom kG (lim −→P
m, N) ∼ = Hom kC (m, LK ρ κ N ).
6.4.1 Orbit categories
Suppose P is a collection of subgroups of G on which G acts
by conjugation. Then it forms a G-poset and we can define
an orbit category O P as the quotient category of G ∝ P by
asking
Hom OP (P, Q) = Q\N G (P, Q).
Then we have an extension sequence
S ↩→ G ∝ P ↠ O P ,
where S is the disjoint union of all objects in P, regarded as
a subcategory of G ∝ P.
When m = κ M for some M ∈ kG-mod, (LK ρ κ M )(P ) =
M P for any P ∈ Ob P. We denote such a kO P -module by

254 6 Connections with group representations and cohomology
M S := (κ M ) S = LK ρ κ M . Since giving a morphism (g, g P ≤
Q) is the same as giving a group homomorphism P → Q, the
conjugation induced by g, there is a natural way to construct
a map M P → M Q , identical to the natural map H0(P ; M) =
k ⊗ kP M → k ⊗ kQ M = H0(Q; M). Hence we know how kO P
acts on M S .
Proposition 6.4.3. Let M ∈ kG-mod and n ∈ kO P -mod.
Then
Hom kG (M, lim ←−P
n) ∼ = Hom kOP (M S , n),
where M S is as above.
Corollary 6.4.4. lim ←−P
n ∼ = Hom kOP (kG S , n) and Hom kG (k, lim ←−P
n) ∼ =
Hom kOP (k, n).
As an example we let H be a subgroup of G and P the
subgroups that are conjugate to H. The size of the discrete
poset P is G/N G (H). Note that both G ∝ P and O P are
connected groupoids, the former equivalent to N G (H) and
the latter N G (H)/H. Thus the above isomorphism can be
interpreted as
Hom kG (M, Hom kNG (H)(kG, N)) ∼ = Hom kNG (H)(M, N)
∼ = Homk(NG (H)/H)(k(N G (H)/H) ⊗ kNG
∼=
Homk(NG (H)/H)(M H , N),
where M ∈ kG-mod and N ∈ k(N G (H)/H)-mod.
By looking at the special case we have just mentioned, the
following statement implies that Proposition 5.2 may be useful
in a greater generality.
Lemma 6.4.5. Suppose M is an indecomposable kG-module.
Let P be a p-subgroup and N a projective simple k(N G (P )/P )-
module. Then Hom kNG (P )/P(M P , N) ≠ 0 if and only if there

6.4 The functor ρ: invariants and coinvariants 255
exists a surjective map f : M → N ↑ G N G (P ). In this case
N ↑ G N G (P ) is indecomposable and f(M) ∼ = k ↑ G P .
Proof. If Hom kNG (P )/P(M P , N) ≠ 0, then N has to be a direct
summand of M P because N is projective simple. By adjunction
Hom kG (M, N ↑ G N G (P ) ) ∼ = Hom kG (M, Hom kNG (P )(kG, N)) ∼ = Hom k(NG (P
we know there is a non-trivial map f : M → N ↑ G N G (P ). If we
restrict this map back to N G (P ), the right side is a semisimple
module and thus the image of M ↓ NG (P ) contains at least a
copy of g ⊗ N for some g ∈ G. It forces the map f : M ↠
N ↑ G N G (P ) to be surjective which implies that N ↑G N G (P ) = f(M)
is indecomposable. Furthermore since as a kN G (P )-module
N has P as a vertex, we get f(M) ∣ (N ↓ P ↑ N G(P ) ) ↑ G . But
N ↓ N G(P )
P
is a direct sum of trivial modules. Simultaneously
we obtain f(M) ∼ = k ↑ G P .
The converse is straightforward by the adjunction. ⊓⊔
6.4.2 Brauer categories, fusion and linking systems
Suppose b is a p-block of the group algebra kG and P b is the
set of b-Brauer pairs. Then for any G-subposet P ⊂ P b we can
introduce the Brauer category B P as the quotient category of
G ∝ P such that
Hom BP (P, Q) = Hom G∝P (P, Q)/C G (P ).
This gives us an opposite extension, which means that the
following sequence
C G ↩→ (G ∝ P) op → B op
P

256 6 Connections with group representations and cohomology
is an extension sequence, given that C G is the disjoint union
of all C G (P ), P ∈ Ob P. Dual to the extension situation we
examined before, now we are able to describe the right Kan
extension of modules.
If m = κ M for some M ∈ kG-mod, we denote by M C G the
kF P -module RK ρ κ M . Since a morphism (g, g P ≤ Q) provides
a group homomorphism P → Q and thus induces an injection
c g
−1 : C G (Q) → C G (P ), we obtain an injection M C G(P ) →
M CG(Q) . This leads to the kB P -action on M C G .
Proposition 6.4.6. Let m ∈ kB P -mod and N ∈ kG-mod.
Then
Hom kG (lim −→P
m, N) ∼ = Hom kBP (m, N C G
),
where N C G
is as above.
As an example we assume b is the principal block b 0 and
P is the conjugacy class of a fixed p-subgroup H. Then the
discrete poset P has |G/N G (H)| objects. Both G ∝ P and
B P are connected groupoids, the former equivalent to N G (H)
and the latter N G (H)/C G (H). Thus the above isomorphism
can be interpreted as
Hom kG (kG ⊗ kNG (H) M, N) ∼ = Hom kNG (H)(M, N)
∼ = Homk(NG (H)/C G (H))(M, Hom kNG (H)(k(N G
∼ = Homk(NG (H)/C G (H))(M, N C G(H) ),
where N ∈ kG-mod and M ∈ k(N G (H)/C G (H))-mod.
Corollary 6.4.7. We have Hom kG (lim −→P
m, k) ∼ = Hom kBP (m, k)
for any m ∈ kB P -mod.
Suppose b is a nilpotent block. Then, for every b-Brauer pair
(H, e), N G (H, e)/C G (H) is a p-group.
Let B b = B Pb . If we fix a maximal object (S, e S ) and take
all objects (Q, e Q ) with Q ⊂ S, then the full subcategory of

6.4 The functor ρ: invariants and coinvariants 257
B b , consisting of all these objects, is a fusion system, usually
written as F b or F S . Note that the inclusion F b ⊂ B b is an
equivalence. There is a general theory of p-local finite groups
introduced by Broto, Levi and Oliver. A p-local finite group
consists of three categories (S, F, L), where S is a p-group, F
is an (abstract) fusion system–a finite category whose objects
are subgroups of S. Let us take the full subcategory F c ⊂ F
consisting of F-centric subgroups of S. (If F = F b , the F-
centric subgroups correspond to the so-called self-centralizing
b-Brauer pairs in modular representation theory.) A centric
linking system L c , if it exists, situates in the middle of a
sequence
Z ↩→ L c ↠ F c
which is an opposite extension. Here Z is the disjoint union
of the centers of all F-centric subgroups.
Proposition 6.4.8. Let n ∈ kL c -mod and m ∈ kF c -mod.
Then
Hom kL c(Res ρ m, n) ∼ = Hom kF c(m, n Z ),
where n Z is defined by n Z (P ) = n(P ) Z(P ) .
Let Bb c be the full subcategory of B b for a block b, consisting
of self-centralizing b-Brauer pairs. Since Fb c naturally identifies
with a full subcategory of Bb c which induces an equivalence,
we similarly can consider an opposite extension
Z ↩→ ˜Lc b ↠ B c b.
If ˜Lc b
exists, then so is L c b
, and vice versa. Moreover between
the corresponding extensions there exists a natural embedding
L c b → ˜Lc b
inducing an category equivalence. By taking the
larger category ˜Lc b
(but essentially the same as L c b
), we can
write down
C G /Z ↩→ G ∝ Pb c ↠ ˜Lc b ,

258 6 Connections with group representations and cohomology
another opposite extension. Here C G /Z is the disjoint union
of C G (P )/Z(P ) in which P runs over all F-centric subgroups.
For the sake of convenience, we introduce a notation C G ′ =
C G /Z so that C G ′ (P ) = C G(P )/Z(P ) ∼ = P C G (P )/P for each
P .
When m = κ M for some M ∈ kG-mod, we write the k ˜Lc b
-
module RK ρ κ M as M C′ G . Given a morphism (g, g P ≤ Q)
it induces an injection C G ′ (Q) → C′ G (P ) thus a morphism
M C′ G (P ) → M C′ G (Q) . Hence we get the k ˜Lc b
-action on M C′ G
Proposition 6.4.9. Let m ∈ k(G ∝ Pb c )-mod and N ∈ kGmod.
Then
Hom kG (lim −→P
m, N) ∼ = Hom k ˜Lc b
(m, N C′ G ),
where N C′ G is defined as above.
In particular if m = Res ρ m ′ for some m ′ ∈ kBb c -mod then
Hom kG (lim −→P
m ′ , N) ∼ = Hom kB
c
b
(m ′ , N C G
),
a special situation of Proposition 6.5.3.
Proof. The first part is a direct consequence of Proposition
5.4. As for the special case We need to notice that
lim Res −→P ρm ′ ∼ = lim −→P
m ′ as kG-modules. Then
Hom kG (lim −→P
Res ρ m ′ , N) ∼ = Hom k ˜Lc b
(Res ρ m ′ , N C′ G )
∼ = HomkB c
b
(m ′ , (N C′ G ) Z )
∼ = HomkB c
b
(m ′ , N C G ).
⊓⊔
6.4.3 Puig categories
If we take P A to be the poset of pointed subgroups on an interior
G-algebra A, then analogues to the Brauer category for

6.4 The functor ρ: invariants and coinvariants 259
any G-subposet P ⊂ P A we can introduce the Puig category
L P as a quotient category of G ∝ P such that
Hom LP (P γ , Q δ ) = Hom G∝P (P γ , Q δ )/C G (P ).
Then some results in last section can be obtained accordingly.
If m = κ M for some M ∈ kG-mod, we denote by M C G the
kL P -module RK ρ κ M . Since a morphism (g, g P ≤ Q) provides
a group homomorphism P → Q and thus induces an injection
c g
−1 : C G (Q) → C G (P ), we obtain an injection M C G(P ) →
M CG(Q) . This leads to the kL P -action on M C G .
Proposition 6.4.10. Let m ∈ kL P -mod and N ∈ kG-mod.
Then
Hom kG (lim −→P
m, N) ∼ = Hom kLP (m, N C G
),
where N C G
is as above.
6.4.4 Orbit categories of fusion systems
This method also works for the orbit category of a fusion
system. Suppose F is an abstract fusion system. The one may
define the orbit category O F in a similar fashion as above by
Hom OF (P, Q) = Q\Hom F (P, Q).
Again we obtain an extension sequence
S ↩→ F ↠ O F ,
where S is the disjoint union of objects in F.
Proposition 6.4.11. If m ∈ kF-mod and n ∈ kO F -mod,
then
Hom kF (m, Res ρ n) ∼ = Hom kOF (m S , n).
Moreover for F = Fb c, Hom kG(lim −→P
m, k) ∼ = Hom kOF c(m S , k).
b
Proof. When F = Fb c , we have isomorphisms

260 6 Connections with group representations and cohomology
Hom kG (lim −→P
m, k) ∼ = Hom kF (m, k)
∼ = HomkOF c
b
(m S , k).
⊓⊔
6.5 Hochschild cohomology
In this section we continue to demonstrate the similarities between
transporter categories and groups. The first main assertion
is the the Hochschild cohomology ring of a finite transporter
category is finitely generated. From here we will establish
the finite generation of cohomology. The second is that we
can construct a transfer map between Hochschild cohomology.
Both results are established by passing between Hochschild
cohomology and ordinary cohomology of F (G ∝ P) and
G ∝ P. Thus computing various over and undercategories
for functors from F (G ∝ P) to appropriate categories are the
major auxiliary statements.
6.5.1 Finite generation
In Section 4.2.2 we have seen that the ordinary cohomology
ring of a finite category can be infinitely generated even after
quotient out nilpotent elements. Based on the Theorem
5.2.2, the Hochschild cohomology ring of such a finite category
algebra is not finitely generated either. So the question
reduces to finding out whether or not Ext ∗ kCe(kC, kC) modulo
nilpotents is finitely generated over Ext ∗ kC(k, k) if the latter is
Noetherian. On the first attempt to solve this question, one
may want to check if the Evens-Venkov Theorem on the finite
generation of group cohomology could be generalized to category
cohomology. This is not true since Example 4.2.2 (3)
implies that we can not expect a finite generation property

6.5 Hochschild cohomology 261
of Ext ∗ kC(M, N) over a finitely generated Ext ∗ kC(k, k). Thus
we have to look at particular families of finite categories for
the finite generation property. In what follows, we show finite
transporter categories constructed over a finite group are very
close to what we expect.
To examine the finite generation of Hochschild cohomology
ring, we use the isomorphism
Ext ∗ k(G∝P) e(k(G ∝ P), k(G ∝ P)) ∼ = Ext ∗ kF (G∝P)(k, Res ∇ (k(G ∝ P)))
because it allows us to use the Grothendieck spectral sequence.
Let us introduce a functor ˜π = π ◦ t
t
F (G ∝ P)
G ∝ P
˜π
π
G .
When we investigated the finite generation of H ∗ (G ∝ P, k)
we used the Grothendieck spectral sequence. Based on the fact
that •\π ≃ P, we obtained the finite generation. Naturally
we want to look at the Grothendieck spectral sequence for
˜π. In order to understand the spectral sequence, we have to
know the undercategory •\˜π. Since t induces a homotopy
equivalence, the undercategory •\˜π should be similar to •\π.
If we take P = • in the above diagram, then π is a canonical
isomorphism and •\π ≃ •. By contrast ˜π can be identified
with t G : F (G) → G and we also have •\˜π = •\t G ≃ •.
Lemma 6.5.1. Let t G : F (G) → G be the target functor.
Then we have two isomorphic categories
1. •\˜π ∼ = F (P) × •\t G .
2. ˜π/• ∼ = F (P) × t G /•.
3. •\t G
∼ = tG /• are equivalent to •.
Proof. We only prove Parts 1 and 3. First we prove (1).
The objects in •\˜π are (•, [(g, gx ≤ y)]). A morphism from

262 6 Connections with group representations and cohomology
(h, [(g, gx ≤ y)]) to (h ′ , [(g ′ , g ′ x ′ ≤ y ′ )]) is a morphism
((l 1 , l 1 y ≤ y ′ ), (l 2 , l 2 x ′ ≤ x)) : [(g, gx ≤ y)] → [(g ′ , g ′ x ′ ≤ y ′ )]
such that l 1 h = h ′ and g ′ = l 1 gl 2 . Thus from one object in
•\˜π to another there is at most one morphism. It implies
this finite undercategory is equivalent to a finite poset. Furthermore
we notice that (h, [(g, gx ≤ y)]) is isomorphic to
(e, [(h −1 g, (h −1 g)x ≤ h −1 y)]). But [(h −1 g, (h −1 g)x ≤ h −1 y)]
is isomorphic to [(e, e(h −1 gx) ≤ h −1 y)] in F (G ∝ P). Now
we define a functor •\˜π → F (P) × •\t G by
(h, [(g, gx ≤ y)]) ↦→ [h −1 gx ≤ h −1 y] × (h, [g]).
Its inverse is given by
[x ≤ y] × (h, [g]) ↦→ (h, [(g, g(g −1 hx) ≤ hy)]).
The isomorphism in (3) is easy to write down as (h, [g]) ↦→
([g], h −1 ). Furthermore, any object ([g], h) ∈ Ob(t G /•) is isomorphic
to ([e], e) which has only one endomorphism. ⊓⊔
Now we can state the consequence of Lemma 6.5.1 (1). It is
similar to the main result in Section 6.3.2.
Proposition 6.5.2. Let G be a finite group and P a finite
G-poset. Then for any M ∈ kF (G ∝ P)-mod, Ext ∗ kF (G∝P)(k, M)
becomes a finitely generated Ext ∗ kG(k, k)- and Ext ∗ kF (G∝P)(k, k)-
module. Especially the Hochschild cohomology ring Ext ∗ k(G∝P) e(k(G ∝
P), k(G ∝ P)) is finitely generated.
Proof. We apply the Grothendieck spectral sequence to ˜π.
Since •\˜π is equivalent to a finite poset F (P), H j (•\˜π, M)
vanishes for all j larger than a chosen positive integer. Consequently
the E 2 page of the spectral sequence only has finitely
many non-zero rows in the first quadrant, and thus we have
the finite generation of Ext ∗ kF (G∝P)(k, M).
⊓⊔

6.5 Hochschild cohomology 263
The preceding result enables us to prove a finite generation
theorem. It will be the foundation for developing a support
variety theory over the ring Ext ∗ k(G∝P)(k, k).
Theorem 6.5.3. Suppose M, N are two k(G ∝ P)-modules.
Then the module Ext ∗ k(G∝P)(M, N) is finitely generated over
Ext ∗ k(G∝P)(k, k).
Proof. By Theorem 5.2.11,
Ext ∗ k(G∝P)(M, N) ∼ = Ext ∗ kF (G∝P)(k, Res ∇ Hom k (M, N)).
Hence by Proposition 6.5.2, Ext ∗ kF (G∝P)(k, Res ∇ Hom k (M, N))
is finitely generated over Ext ∗ kF (G∝P)(k, k) ∼ = Ext ∗ k(G∝P)(k, k).
We are done.
⊓⊔
If P = •, we get the usual assertion that Ext ∗ kGe(kG, kG)
is finitely generated over Ext ∗ kG(k, k). If P = S b as in Example
6.1., we have Ext ∗ kG(k, k) acting on Ext ∗ k(G∝S b ) e(k(G ∝
S b ), k(G ∝ S b )) via Ext ∗ k(G∝S b )(k, k). Especially when b = b 0 ,
S b
∼ = Sp and Ext ∗ k(G∝S p )(k, k) ∼ = Ext ∗ kG(k, k).
Corollary 6.5.4. If k is a field with positive characteristic
p ∣ |G|, and b is a p-block, then, for any full subcategory
Tr ⊂ Tr b (G) whose objects are closed under G-conjugation,
Ext ∗ kTr
e(kTr, kTr) is a finitely generated algebra.
Given the principal block b 0 of a group algebra kG, we have
a fusion system F b0 = F p over a fixed Sylow p-subgroup S.
As we mentioned earlier, there exists a centric linking system
L b0 = L c p which is determined by the full subcategory
Tr c p(G) ≤S of the transporter category Tr p (G), consisting of all
p-centric subgroups contained in S [10]. In fact L c p is a quotient
category of Tr c p(G) ≤S by some p ′ -groups. In other words,
if one looks at the canonical functor π : Tr c p(G) ≤S → L c p, each

264 6 Connections with group representations and cohomology
undercategory has the property such that it has a minimal object
whose automorphism group is p ′ and, if one regards this
p ′ -automorphism group as a subcategory, the left Kan extension
along the inclusion is exact. Furthermore the left Kan
extension of the trivial group module is the trivial module of
the undercategory. This implies, by an Eckmann-Shapiro type
result, the cohomology of each undercategory can be reduced
to the cohomology of the automorphism group of the abovespecified
minimal object in it. Consequently, the mod-p cohomology
of each undercategory of π with arbitary coefficients
vanishes in positive degrees. To summarize, since Tr c p(G) ≤S is
equivalent to Tr c p(G) and the Grothendieck spectral sequence
for π collapses, we have an isomorphism
Ext ∗ kTr c p(G)(k, V ) ∼ = Ext ∗ kTr c p(G) ≤S
(k, V ) ∼ = Ext ∗ kL c p (k, RK πV ),
where RK π V = H 0 (\π; V ) ∼ = lim ←−\π
V is the right Kan extension
along π of V . It is similar to [10, Lemma 1.3 (iii)]
in which right modules are considered and thus the left Kan
extension is applied. Especially, the functors G ← Tr c p(G) ←↪
Tr c p(G) ≤S → L c p induce isomorphisms of mod-p ordinary cohomology
rings.
Proposition 6.5.5. Let L c p be the centric linking system
associated to the principal block of a finite group algebra
kG. Then Ext ∗ kL (k, RK πV ) is finitely generated as an
c p
Ext ∗ kL (k, k)-module.
c p
However this is still far from understanding the finite generation
of the Hochschild cohomology ring Ext ∗ k(L c p) e(kLc p, kL c p) ∼ =
Ext ∗ kF (Lp)(k, Res c τ (kL c p)).

6.5 Hochschild cohomology 265
6.5.2 Transfer for Hochschild cohomology
We end this chapter with a transfer map between Hochschild
cohomology. Consider the following commutative diagram of
functors
(G ∝ P) e πe
G e
F (P)
t
∇
F (G ∝ P)
t
F (π)
∇
F (G)
P
G ∝ P π
G
Recall that in Section 6.3.3, based on the bottom row, we
were able to establish a transfer map between ordinary cohomology.
Since the three target functors all induce homotopy
equivalences, the middle row may as well give a transfer map
Ext ∗ kF (G∝P)(−, −) → Ext ∗ kF (G)(−, −), for suitable modules.
Since furthermore the upper right square demonstrates connections
between the ordinary cohomology over kF (G ∝ P)
and kF (G) with the Hochschild cohomology of kF (G ∝ P)
and kF (G), respectively, we automatically wonder if there
would be a transfer between the Hochschild cohomology of
kF (G ∝ P) and kF (G) The answer is yes, and we shall
provide the construction here, which in spirit is similar to the
transfer we built between the ordinary cohomology. A predecessor
of this construction is due to Lincklemann [45]. For G
a group and H a subgroup, he developed a method over symmetric
algebras and defined a transfer map from HH ∗ (kH) to
HH ∗ (kG). Recall that H can be recovered as the transporter
category G ∝ (G/H), our construction generalizes his.
Lemma 6.5.6. Let kG ∈ kG e -mod and k(G ∝ P) ∈ k(G ∝
P) e -mod. Then the kF (G ∝ P)-module Res F (π) Res ∇ kG contains
Res ∇ k(G ∝ P) as a submodule.
t

266 6 Connections with group representations and cohomology
Proof. Let (x, y) ∈ Ob(G ∝ P) e . Then
k(G ∝ P)(x, y) = kHom G∝P (y, x)
= k{(h, hy ≤ x) ∣ ∣ h ∈ G}
⊂ kG
= (Res π ekG)(x, y).
From here we can verify that k(G ∝ P) is a k(G ∝ P) e -
submodule of Res π ekG. Hence the result follows. ⊓⊔
If P = G/H, then G ∝ (G/H) ≃ H and F (G ∝ (G/H)) ≃
H. The above inclusion says that the kH-module kG ↓ G H contains
kH as a submodule. Here kG and kH are acted by G
and H, respectively, by conjugation.
Lemma 6.5.7. Consider the functor F (π) : F (G ∝ P) →
F (G). Then F (π)/− ∼ = F (P) − × Id F (G) /−, where F (P) [g]
denotes a poset, indexed by [g] ∈ Ob F (G), which is canonically
isomorphic to F (P). Consequently
C ∗ (F (π)/−) ∼ = C ∗ (F (P) − × Id F (G) /−)
as complexes of kF (G)-modules. Since F (G) ≃ G,
C ∗ (F (π)/[e]) ∼ = C ∗ (F (P) [e] × Id F (G) /[e]) ≃ C ∗ (F (P)) ⊗ B G ∗
is a projective resolution of the complex of kG-modules
C ∗ (F (P)).
Proof. Suppose [g] is an object in F (G). The the overcategory
F (G)/[g] has objects {([(h, hx ≤ y)], (h 1 , h 2 ))}, in which
[(h, hx ≤ y)] ∈ Ob F (G ∝ P). It implies g = h 1 hh 2 . As a
consequence, hx ≤ y is equivalent to h −1
2 x ≤ g−1 h 1 y. Morphisms
((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x)) are given by

6.5 Hochschild cohomology 267
(v 1 ,v 2 )=F (π)((v 1 ,v 1 y≤y ′ ),(v 2 ,v 2 x ′ ≤x))
[h] = F (π)([(h, hx ≤ y)])
(h 1 ,h 2 )
(h ′ 1 ,h′ 2 )
[h ′ ] = F (π)([(h ′ , h ′ x ′ ≤ y ′ )])
It implies various identities: g = h 1 hh 2 = h ′ 1h ′ h ′ 2, h 1 = h ′ 1v 1
and h 2 = v 2 h ′ 2. We define a functor F (π)/[g] → F (P) ×
Id F (G) /[g] such that on objects
([(h, hx ≤ y)], (h 1 , h 2 )) ↦→ ([h −1
2 x ≤ g−1 h 1 y], ([h], (h 1 , h 2 ))),
and on morphisms ((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x)) ↦→ ((e, e), (v 1 , v 2 )),
because v 1 y ≤ y ′ implies g −1 h 1 y ≤ g −1 h ′ 1y ′ while v 2 x ′ ≤ x implies
h ′ −1 2 x ′ ≤ h2 −1 x. The inverse of this functor is defined by
([x ≤ y], ([h], (h 1 , h 2 ))) ↦→ ([(h, h(h 2 x) ≤ h −1
1 gy)], (h 1, h 2 ))
on objects, and on morphisms is defined by ((e, e), (v 1 , v 2 )) ↦→
((v 1 , v 1 h −1
1 gy ≤ h′ 1−1 gy ′ ), (v 2 , v 2 h ′ 2x ′ ≤ h 2 x)). Thus we obtain
an isomorphism
F (π)/[g] ∼ = F (P) [g] × Id F (G)/[g],
where F (P) [g] denotes a copy of F (P) indexed by the object
[g]. If (l 1 , l 2 ) : [g] → [g ′ ] is a morphism in F (G). Then it
induces a functor F (π)/[g] → F (π)/[g ′ ] given by
and
([(h, hx ≤ y)], (h 1 , h 2 )) ↦→ ([(h, hx ≤ y)], (l 1 h 1 , h 2 l 2 ))
((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x)) ↦→ ((v 1 , v 1 y ≤ y ′ ), (v 2 , v 2 x ′ ≤ x))
Using the isomorphisms F (π)/[g] ∼ = F (P) [g] × Id F (G) /[g] and
F (π)/[g ′ ] ∼ = F (P) [g ′ ] × Id F (G) /[g ′ ], one can see it induces an
isomorphism F (P) [g] → F (P) [g ′ ].
[g]

268 6 Connections with group representations and cohomology
Finally since G is isomorphic to the automorphism group of
[e] in the groupoid F (G), we have an equivalence F (G) ≃ G.
Thus C ∗ (F (P) [e] × Id F (G) /[e]) ≃ C ∗ (F (P)) ⊗ C ∗ (Id F (G) /[e])
is a complex of projective kG-modules. Furthermore because
([e]), it has to be a projective resolution
of the trivial kG-module k. Hence we get the chain
homotopy equivalence as stated.
⊓⊔
C ∗ (Id F (G) /[e]) = B F (G)
∗
This lemma allows us to describe the transfer map in Theorem
5.2.2 in terms of factorization categories:
tr P : Ext ∗ kF (G∝P)(k, k) → Ext ∗ kF (G)(k, k).
Indeed we have a commutative diagram
F (G ∝ P) F (π)
F (G)
t
G ∝ P π
G
Thus we get a commutative diagram of cochain complexes
Hom kF (G∝P) (B F (G∝P)
∗ , k)
∼ =
∼ =
Hom k(G∝P) (LK t B F (G∝P)
∗ , k) ∼=
≃
Hom k(G∝P) (B G∝P
∗ , k) ≃
t
Hom kF (G) (LK F (π) B F (G∝P)
∗ , k)
∼ =
Hom kG (LK˜π B∗ F (G∝P) , k)
≃
Hom kG (C ∗ (P) ⊗ B G ∗ , k)
Hom k(G∝P) (B∗ G∝P , k)
Hom kG (B∗ G , k)
≃
Hom kG (LK t B F (G)
∗ , k)
∼ =
Hom kF (G) (B F (G) , k)

6.5 Hochschild cohomology 269
The isomorphisms are given by adjunctions and the chain homotopy
equivalences are induced by changing projective resolutions.
From the upper left corner to the lower right corner
is the transfer that we want to describe. It factors through
the lowest square, including the chain map which gives rise to
the transfer constructed in Theorem 6.3.21. Note that F (G)
is a groupoid and G is a skeleton of F (G). Comparing with
[45, Section 4], F (G) plays the role of ∆G if one identifies G e
with G × G.
Theorem 6.5.8. There exists a map
htr P : Ext ∗ k(G∝P) e(k(G ∝ P), k(G ∝ P)) → Ext∗ kGe(kG, kG).
Proof. Using Lemma 6.5.6 we have cochain maps
Hom kF (G∝P) (B∗
G∝P , Res F (π) Res ∇ kG) ∼ = Hom kF (G) (LK F (π) B∗
G∝P , Res ∇
∼ = HomkF (G) (C ∗ (F (π)/−), Res ∇
∼ = HomkG (C ∗ (F (π)/[e]), kG)
≃ Hom kG (C ∗ (F (P)) ⊗ B∗ G , kG)
→ Hom kG (B∗ G , kG)
Here the module kG in Hom kG (−, kG) is acted by kG via
conjugations. The last map is induced by k → C ∗ (F (P)),
a chain map constructed in the same way as the one in the
proof of Theorem 6.3.21 k → C ∗ (P). Indeed since t : F (P) →
P is a G-functor, it induces an chain homotopy equivalence
C ∗ (F (P)) ≃ C ∗ (P).
By Lemma 6.5.5 we also have a chain map
Hom kF (G∝P) (B G∝P
∗
, Res ∇ kF (G ∝ P)) → Hom kF (G∝P) (B∗
G∝P , Res F (π)
induced by the inclusion Res ∇ kF (G ∝ P) → Res F (π) Res ∇ kG.
Hence altogether we obtain a chain map
Hom kF (G∝P) (B∗
G∝P , Res ∇ kF (G ∝ P)) → Hom kG (B∗ G , kG).

270 6 Connections with group representations and cohomology
Passing to cohomology we get a map between Hochschild cohomology
by Theorem 5.2.2
Ext ∗ kF (G∝P)(k, Res ∇ kF (G ∝ P)) htr P
∼ =
Ext ∗ kG(k, kG)
∼ =
Ext ∗ k(G∝P) e(kF (G ∝ P), kF (G ∝ P)) htr P
Ext ∗ kG e(kG, kG) ⊓⊔
The above map deserves to be called a transfer since
k ∣ ∣ Res ∇ kF (G ∝ P), k ∣ ∣ kG and k ∣ ∣ Res ∇ kF (G). In the
construction of htr P if we replace the second module in all
Hom − (−, −) and Ext ∗ −(−, −) by k then it is exactly the transfer
created in Theorem 6.3.21 and reinterpreted before Theorem
6.5.7. In other words, we have a commutative diagram
Ext ∗ k(G∝P)(k, k)
∼ =
Ext ∗ kF (G∝P)(k, k)
injection
Ext ∗ kF (G∝P)(k, Res ∇ kF (G ∝ P))
∼ =
tr P
tr P
Ext ∗ kG(k, k)
∼ =
Ext ∗ kF (G)(k, k)
injection
Ext ∗ kF (G)(k, Res ∇ kF (G))
Ext ∗ k(G∝P) e(k(G ∝ P), k(G ∝ P))
htr P
Ext ∗ kGe(kG, kG)
∼ =

References 271
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