Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

72 SHINICHI MOCHIZUKI
Definition 3.6. Fix a collection of initial Θ-data (F/F, X F ,l,C K , V, V bad
mod ,ɛ)
as in Definition 3.1. In the following, we shall use the various notations introduced
in Definition 3.1 for various objects associated to this initial Θ-data. Then we define
aΘ-Hodge theater [relative to the given initial Θ-data] to be a collection of data
that satisfies the following conditions:
† HT Θ =({ † F v
} v∈V , † F ⊩ mod)
(a) If v ∈ V non ,then † F v
is a category which admits an equivalence of categories
† ∼
F →Fv v
[where F v
is as in Examples 3.2, (i); 3.3, (i)]. In particular,
† F v
admits a natural Frobenioid structure [cf. [FrdI], Corollary 4.11,
(iv)], which may be constructed solely from the category-theoretic structure
of † F v
.Write † D v , † Dv ⊢ , † Dv Θ , † Fv ⊢ , † Fv
Θ for the objects constructed
category-theoretically from † F v
that correspond to the objects without a
“†” discussed in Examples 3.2, 3.3 [cf., especially, Examples 3.2, (vi); 3.3,
(iii)].
(b) If v ∈ V arc ,then † F v
is a collection of data ( † C v , † D v , † κ v )—where † C v
is a category equivalent to the category C v of Example 3.4, (i); † D v is an
Aut-holomorphic orbispace; and † κ v : O ⊲ ( † C v ) ↩→ A† D v
is an inclusion
of topological monoids, which we shall refer to as the Kummer structure
on † C v — such that there exists an isomorphism of collections of data
† ∼
F →Fv v
[where F v
is as in Example 3.4, (i)]. Write † Dv ⊢ , † Dv Θ , † Fv ⊢ ,
† Fv Θ for the objects constructed algorithmically from † F v
that correspond
to the objects without a “†” discussed in Example 3.4, (ii), (iii).
(c) † F ⊩ mod
is a collection of data
( † C ⊩ mod, Prime( † C ⊩ mod) ∼ → V, { † F ⊢ v } v∈V , { † ρ v } v∈V )
—where † Cmod ⊩ is a category which admits an equivalence of categories
† Cmod
⊩ ∼
→Cmod ⊩ [which implies that † Cmod ⊩ admits a natural category-theoretically
constructible Frobenioid structure — cf. [FrdI], Corollary 4.11,
(iv); [FrdI], Theorem 6.4, (i)]; Prime( † Cmod ⊩ ) → ∼ V is a bijection of sets,
where we write Prime( † Cmod ⊩ ) for the set of primes constructed from the
category † Cmod ⊩ [cf. [FrdI], Theorem 6.4, (iii)]; † Fv ⊢ is as discussed in
(a), (b) above; † ∼
ρ v :Φ† C ⊩ → Φ
mod ,v
rlf
† C
[where we use notation as in the
v
⊢
discussion of Example 3.5, (i)] is an isomorphism of topological monoids.
Moreover, we require that there exist an isomorphism of collections of data
† F ⊩ ∼
mod
→ F ⊩ mod [where F⊩ mod is as in Example 3.5, (ii)]. Write † F ⊩ tht , † F ⊩ D
for the objects constructed algorithmically from † F ⊩ mod
that correspond to
the objects without a “†” discussed in Example 3.5, (ii), (iii).
Remark 3.6.1. When we discuss various collections of Θ-Hodge theaters, labeled
by some symbol “□” inplaceofa“†”, we shall apply the notation of Definition 3.6