Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

106 SHINICHI MOCHIZUKI
(ii) (Mono-analytic Processions) By composing the functor of (i) with the
mono-analyticization operation discussed in Definition 4.1, (iv), one obtains a
natural functor
category of
D-Θ-bridges
and isomorphisms of
D-Θ-bridges
→
category of processions
of D ⊢ -prime-strips
and morphisms of
processions
† φ Θ ↦→ Prc( † D ⊢ J )
whose output data satisfies the same indeterminacy properties with respect to labels
as the output data of the functor of (i).
Proof.
Assertions (i), (ii) follow immediately from the definitions. ○
The following result is an immediate consequence of our discussion.
Corollary 4.12. (Étale-pictures of Base-ΘNF-Hodge Theaters) Relative
to a fixed collection of initial Θ-data:
(i) Consider the [composite] functor
† HT D-ΘNF ↦→ † D > ↦→ † D ⊢ >
— from the category of D-ΘNF-Hodge theaters and isomorphisms of D-ΘNF-Hodge
theaters to the category of D ⊢ -prime-strips and isomorphisms of D ⊢ -prime-strips —
obtained by assigning to the D-ΘNF-Hodge theater † HT D-ΘNF the mono-analyticization
[cf. Definition 4.1, (iv)] † D ⊢ > of the D-prime-strip † D > that appears as the
codomain of the underlying D-Θ-bridge [cf. Definition 4.6, (ii)] of † HT D-ΘNF .
If † HT D-ΘNF , ‡ HT D-ΘNF are D-ΘNF-Hodge theaters, then we define the base-
NF-, or D-NF-, link
† HT D-ΘNF D
−→
‡ HT D-ΘNF
from † HT D-ΘNF to ‡ HT D-ΘNF to be the full poly-isomorphism
† D ⊢ >
∼
→ ‡ D ⊢ >
between the D ⊢ -prime-strips obtained by applying the functor discussed above to
† HT D-ΘNF , ‡ HT D-ΘNF .
(ii) If
...
D
−→ (n−1) HT D-ΘNF D
−→ n HT D-ΘNF D
−→ (n+1) HT D-ΘNF D
−→ ...