Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

32 SHINICHI MOCHIZUKI
Section 0: Notations and Conventions
Monoids and Categories:
We shall use the notation and terminology concerning monoids and categories
of [FrdI], §0.
We shall refer to an isomorphic copy of some object as an isomorph of the
object.
If C and D are categories, then we shall refer to as an isomorphism C→Dany
isomorphism class of equivalences of categories C→D. [Note that this termniology
differs from the standard terminology of category theory, but will be natural in the
context of the theory of the present series of papers.] Thus, from the point of view
of “coarsifications of 2-categories of 1-categories” [cf. [FrdI], Appendix, Definition
A.1, (ii)], an “isomorphism C→D” is precisely an “isomorphism in the usual sense”
of the [1-]category constituted by the coarsification of the 2-category of all small
1-categories relative to a suitable universe with respect to which C and D are small.
Let C be a category; A, B ∈ Ob(C). Then we define a poly-morphism A → B
to be a collection of morphisms A → B [i.e., a subset of the set of morphisms
A → B]; if all of the morphisms in the collection are isomorphisms, then we shall
refer to the poly-morphism as a poly-isomorphism; if A = B, then we shall refer
to a poly-isomorphism A → ∼ B as a poly-automorphism. We define the full
poly-isomorphism A → ∼ B to be the poly-morphism given by the collection of all
isomorphisms A → ∼ B. The composite of a poly-morphism {f i : A → B} i∈I with a
poly-morphism {g j : B → C} j∈J is defined to be the poly-morphism given by the
set [i.e., where “multiplicities” are ignored] {g j ◦ f i : A → C} (i,j)∈I×J .
Let C be a category. We define a capsule of objects of C to be a finite collection
{A j } j∈J [i.e., where J is a finite index set] of objects A j of C; if|J| denotes the
cardinality of J, then we shall refer to a capsule with index set J as a |J|-capsule;
also, we shall write π 0 ({A j } j∈J ) def
= J. Amorphism of capsules of objects of C
{A j } j∈J →{A ′ j ′} j ′ ∈J ′
is defined to consist of an injection ι : J↩→ J ′ , together with, for each j ∈ J, a
morphism A j → A ′ ι(j)
of objects of C. Thus, the capsules of objects of C form a
category Capsule(C). A capsule-full poly-morphism
{A j } j∈J
∼
→{A
′
j ′} j′ ∈J ′
between two objects of Capsule(C) is defined to be the poly-morphism associated
to some [fixed] injection ι : J ↩→ J ′ which consists of the set of morphisms of
∼
Capsule(C) given by collections of [arbitrary] isomorphisms A j → A
′
ι(j)
,forj ∈
J. A capsule-full poly-isomorphism is a capsule-full poly-morphism for which the
associated injection between index sets is a bijection.
If X is a connected noetherian algebraic stack which is generically scheme-like,
then we shall write
B(X)