Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

124 SHINICHI MOCHIZUKI
i =1, 2], as well as with the respective “’s”. It is immediate that any morphism
of NF-bridges induces a morphism between the associated D-NF-bridges. There is
an evident notion of composition of morphisms of NF-bridges.
(ii) We define a Θ-bridge [relative to the given initial Θ-data] to be a collection
of data
( ‡ ‡ ψ Θ
F J −→ ‡ F >
‡ HT Θ )
as follows:
(a) ‡ F J = { ‡ F j } j∈J is a capsule of F-prime-strips, indexed by a finite index
set J. Write ‡ D J = { ‡ D j } j∈J for the associated capsule of D-prime-strips
[cf. Remark 5.2.1, (i)].
(b) ‡ HT Θ is a Θ-Hodge theater.
(c) ‡ F > is the F-prime-strip tautologically associated to ‡ HT Θ [cf. the
discussion preceding Example 5.4]; we use the notation “” todenote
this relationship between ‡ F > and ‡ HT Θ . Write ‡ D > for the D-primestrip
associated to ‡ F > [cf. Remark 5.2.1, (i)].
(d) ‡ ψ Θ = { ‡ ψ Θ j } j∈J is the collection of poly-morphisms ‡ ψ Θ j : ‡ F j → ‡ F >
determined [i.e., as discussed in Remark 5.3.1] by a D-Θ-bridge ‡ φ Θ =
{ ‡ φ Θ j : ‡ D j → ‡ D > } j∈J .
Thus, one verifies immediately that any Θ-bridge as above determines an associated
D-Θ-bridge ( ‡ φ Θ : ‡ D J → ‡ D > ). We define a(n) [iso]morphism of Θ-bridges
( 1 1 ψ Θ
F J1 −→ 1 F >
to be a collection of arrows
1 HT Θ ) → ( 2 2 ψ Θ
F J2 −→ 2 F >
2 HT Θ )
1 F J1
∼
→ 2 F J2 ;
1 F >
∼
→ 2 F > ;
1 HT Θ ∼
→ 2 HT Θ
—where 1 ∼
F J1 → 2 F J2 is a capsule-full poly-isomorphism [cf. §0]; 1 ∼
F > → 2 F > is
a full poly-isomorphism; 1 HT Θ ∼
→ 2 HT Θ is an isomorphism of Θ-Hodge theaters
[cf. Remark 3.6.2] — which are compatible [in the evident sense] with the i ψ Θ [for
i =1, 2], as well as with the respective “’s” [cf. Corollary 5.6, (i), below]. It
is immediate that any morphism of Θ-bridges induces a morphism between the
associated D-Θ-bridges. There is an evident notion of composition of morphisms
of Θ-bridges.
(iii) We define a ΘNF-Hodge theater [relative to the given initial Θ-data] to be
a collection of data
‡ HT ΘNF =( ‡ F ⊛
‡ F ⊚ ‡ ψ NF
←− ‡ ‡ ψ Θ
F J −→ ‡ F >
‡ HT Θ )
—wherethedata( ‡ F ⊛ ‡ F ⊚ ←− ‡ F J ) forms an NF-bridge; the data
( ‡ F J −→ ‡ F > ‡ HT Θ )formsaΘ-bridge — such that the associated data
{ ‡ φ NF
, ‡ φ Θ } [cf. (i), (ii)] forms a D-ΘNF-Hodge theater. A(n) [iso]morphism of
ΘNF-Hodge theaters is defined to be a pair of morphisms between the respective