Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 33
for the category of finite étale coverings of X [and morphisms over X]; if A is a
noetherian [commutative] ring [with unity], then we shall write B(A) def
= B(Spec(A)).
Thus, [cf. [FrdI], §0] the subcategory of connected objects B(X) 0 ⊆B(X) may
be thought of as the subcategory of connected finite étale coverings of X [and
morphisms over X].
Let Π be a topological group. Then let us write
B temp (Π)
for the category whose objects are countable [i.e., of cardinality ≤ the cardinality
of the set of natural numbers], discrete sets equipped with a continuous Π-action
and whose morphisms are morphisms of Π-sets [cf. [SemiAnbd], §3]. If Π may be
written as an inverse limit of an inverse system of surjections of countable discrete
topological groups, then we shall say that Π is tempered [cf. [SemiAnbd], Definition
3.1, (i)]. A category C equivalent to a category of the form B temp (Π), where Π is a
tempered topological group, is called a temperoid [cf. [SemiAnbd], Definition 3.1,
(ii)]. Thus, if C is a temperoid, then C is naturally equivalent to (C 0 ) ⊤ [cf. [FrdI],
§0]. Moreover, one can reconstruct the topological group Π, up to inner automorphism,
category-theoretically from B temp (Π) or B temp (Π) 0 [i.e., the subcategory of
connected objects of B temp (Π)]; in particular, for any temperoid C, it makes sense
to write
π 1 (C), π 1 (C 0 )
for the topological groups, up to inner automorphism, obtained by applying this
reconstruction algorithm [cf. [SemiAnbd], Remark 3.2.1].
In this context, if C 1 , C 2 are temperoids, then it is natural to define a morphism
C 1 →C 2
to be an isomorphism class of functors C 2 →C 1 that preserves finite limits and
countable colimits. [Note that this differs — but only slightly! — from the definition
given in [SemiAnbd], Definition 3.1, (iii).] In a similar vein, we define a morphism
C 0 1 →C 0 2
to be a morphism (C1) 0 ⊤ → (C2) 0 ⊤ [where we recall that we have natural equivalences
∼
of categories C i → (C
0
i ) ⊤ for i =1, 2]. One verifies immediately that an “isomorphism”
relative to this terminology is equivalent to an “isomorphism of categories”
in the sense defined at the beginning of the present discussion of “Monoids and
Categories”. Finally, if Π 1 ,Π 2 are tempered topological groups, then we recall
that there is a natural bijective correspondence between
(a) the set of continuous outer homomorphisms Π 1 → Π 2 ,
(b) the set of morphisms B temp (Π 1 ) →B temp (Π 2 ), and
(c) the set of morphisms B temp (Π 1 ) 0 →B temp (Π 2 ) 0
— cf. [SemiAnbd], Proposition 3.2.