Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

96 SHINICHI MOCHIZUKI
Definition 4.6.
(i) We define a base-NF-bridge, orD-NF-bridge, [relative to the given initial
Θ-data] to be a poly-morphism
† † φ NF
D J −→
† D ⊚
—where † D ⊚ is a category equivalent to D ⊚ ; † D J = { † D j } j∈J is a capsule of D-
prime-strips, indexed by a finite index set J — such that there exist isomorphisms
D ⊚ → ∼ † D ⊚ ∼
, D → † D J , conjugation by which maps φ NF
↦→ † φ NF
. We define a(n)
[iso]morphism of D-NF-bridges
to be a pair of poly-morphisms
( † † φ NF
D J −→ † D ⊚ ) → ( ‡ D J ′
‡ φ NF
−→ ‡ D ⊚ )
† D J
∼
→ ‡ D J ′;
† D ⊚ ∼
→ ‡ D ⊚
—where † ∼
D J → ‡ D J
′ is a capsule-full poly-isomorphism [cf. §0]; † D ⊚ → ‡ D ⊚ is a
poly-morphism which is an Aut ɛ ( † D ⊚ )- [or, equivalently, Aut ɛ ( ‡ D ⊚ )-] orbit [cf. the
discussion of Example 4.3, (i)] of isomorphisms — which are compatible with † φ NF
‡ φ NF
. There is an evident notion of composition of morphisms of D-NF-bridges.
(ii) We define a base-Θ-bridge, or D-Θ-bridge, [relative to the given initial
Θ-data] to be a poly-morphism
† † φ Θ
D J −→ † D >
—where † D > is a D-prime-strip; † D J = { † D j } j∈J is a capsule of D-prime-strips,
∼
indexed by a finite index set J — such that there exist isomorphisms D > → † D > ,
∼
→ † D J , conjugation by which maps φ Θ ↦→ † φ Θ . We define a(n) [iso]morphism
D
of D-Θ-bridges
,
( † † φ Θ
D J −→ † D > ) → ( ‡ D J ′
‡ φ Θ
−→ ‡ D > )
to be a pair of poly-morphisms
† D J
∼
→ ‡ D J ′;
† D >
∼
→ ‡ D >
—where † ∼
D J → ‡ D J ′ is a capsule-full poly-isomorphism; † ∼
D > → ‡ D > is the full
poly-isomorphism — which are compatible with † φ Θ , ‡ φ Θ . There is an evident
notion of composition of morphisms of D-Θ-bridges.
(iii) We define a base-ΘNF-Hodge theater, orD-ΘNF-Hodge theater, [relative
to the given initial Θ-data] to be a collection of data
† HT D-ΘNF =( † D ⊚ † φ NF
←− † † φ Θ
D J −→ † D > )
—where † φ NF
isomorphisms
is a D-NF-bridge; † φ Θ is a D-Θ-bridge — such that there exist
D ⊚ ∼
→ † D ⊚ ; D
∼
→ † D J ; D >
∼
→ † D >