Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 141
whose output data satisfies the same indeterminacy properties with respect to labels
as the output data of the functor of (i).
(iii) The functors of (i), (ii) are compatible, respectively, with the functors of
Proposition 4.11, (i), (ii), relative to the functor [i.e., determined by the functorial
algorithm] of Proposition 6.7, in the sense that the natural inclusions
S j = {1,... ,j} ↩→ S± t = {0, 1,... ,t− 1}
[cf. the notation of Proposition 4.11] — where j =1,... ,l ,andt def
= j +1 —
determine natural transformations
† φ Θ±
± ↦→
† φ Θ±
± ↦→
(
)
Prc( † D T ) ↩→ Prc( † D T )
)
(Prc( † D ⊢ T
) ↩→ Prc( † D ⊢ T )
from the respective composites of the functors of Proposition 4.11, (i), (ii), with the
functor [determined by the functorial algorithm] of Proposition 6.7 to the functors
of (i), (ii).
Proof.
Assertions (i), (ii), (iii) follow immediately from the definitions. ○
The following result is an immediate consequence of our discussion.
Corollary 6.10. (Étale-pictures of Base-Θ ±ell -Hodge Theaters) Relative
to a fixed collection of initial Θ-data:
(i) Consider the [composite] functor
† HT D-Θ±ell ↦→ † D > ↦→ † D ⊢ >
— from the category of D-Θ ±ell -Hodge theaters and isomorphisms of D-Θ ±ell -
Hodge theaters to the category of D ⊢ -prime-strips and isomorphisms of D ⊢ -primestrips
— obtained by assigning to the D-Θ ±ell -Hodge theater † HT D-Θ±ell
the monoanalyticization
[cf. Definition 4.1, (iv)] † D ⊢ > of the D-prime-strip † D > associated,
via the functorial algorithm of Proposition 6.7, to the underlying D-Θ ± -bridge
of † HT D-Θ±ell .If † HT D-Θ±ell , ‡ HT D-Θ±ell
are D-Θ ±ell -Hodge theaters, thenwe
define the base-Θ ±ell -, or D-Θ ±ell -, link
† HT D-Θ±ell D
−→
‡ HT D-Θ±ell
from † HT D-Θ±ell to ‡ HT D-Θ±ell to be the full poly-isomorphism
† D ⊢ >
∼
→ ‡ D ⊢ >
between the D ⊢ -prime-strips obtained by applying the functor discussed above to
† HT D-Θ±ell , ‡ HT D-Θ±ell .