Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

58 SHINICHI MOCHIZUKI
— which we think of as mapping Ob(Dv ⊢ ) ∋ A ↦→ A Θ def
= Ÿ × A ∈ v Ob(DΘ v )—we
have natural isomorphisms
O ⊲ C ⊢ v
(−)
∼
→ O ⊲ C
(−);
v
Θ
O × C ⊢ v
(−)
∼
→ O × (−)
Cv
Θ
[where O ⊲ (−), O × (−) are the monoids associated to the Frobenioid C ⊢ Cv
⊢ Cv
⊢ v
[FrdI], Proposition 2.2] which are compatible with the assignment
as in
q
v
| TA
↦→ Θ v
| TA Θ
and the natural isomorphism [i.e., induced by the natural projection A Θ = Ÿ × v
A → A] O × (T A ) →O ∼ × (T A Θ). In particular, we conclude that the monoid O ⊲ (−)
Cv
Θ
determines — in a fashion consistent with the notation of [FrdI], Proposition 2.2!
—ap v -adic Frobenioid with base category given by Dv Θ [cf. [FrdII], Example 1.1,
(ii)]
Cv Θ (⊆ F ÷ ) v
— which may be thought of as a subcategory of F ÷ , and which is equipped with a
v
μ 2l (−)-orbit of characteristic splittings [cf. [FrdI], Definition 2.3]
τ Θ v
determined by Θ v
. Moreover, we have a natural equivalence of categories
C ⊢ v
∼
→C Θ v
that maps τ ⊢ v
to τ Θ v . This fact may be stated more succinctly by writing
F ⊢ v
∼
→F Θ v
—wherewewriteFv
⊢ def
=(Cv ⊢ ,τv ⊢ ); Fv
Θ def
=(Cv Θ ,τv Θ ). In the following, we shall refer
toapairsuchasFv ⊢ or Fv Θ consisting of a Frobenioid equipped with a collection
of characteristic splittings as a split Frobenioid.
(vi) Here, it is useful to recall [cf. Remark 3.2.1 below] that:
(a) the subcategory D ⊢ v ⊆D v may be reconstructed category-theoretically
from D v [cf. [AbsAnab], Lemma 1.3.8];
(b) the category Dv
Θ may be reconstructed category-theoretically from D v [cf.
(a); the discussion at the beginning of [EtTh], §5];
(c) the category Dv ⊢ (respectively, Dv Θ )maybereconstructed category-theoretically
from Cv
⊢ (respectively, Cv Θ ) [cf. [FrdI], Theorem 3.4, (v); [FrdII], Theorem
1.2, (i); [FrdII], Example 1.3, (i); [AbsAnab], Theorem 1.1.1, (ii)];