Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

140 SHINICHI MOCHIZUKI
/ ± ↩→ / ± / ± ↩→ / ± / ± / ± ↩→ ... ↩→ / ± / ± / ± ... / ±
Fig. 6.2: An l ± -procession of D-prime-strips
Proposition 6.9. (Processions of Base-Prime-Strips) Relative to a fixed
collection of initial Θ-data:
(i) (Holomorphic Processions) Given a D-Θ ± -bridge † φ Θ±
± : † D T → † D ≻ ,
with underlying capsule of D-prime-strips † D T , write Prc( † D T ) for the l ± -procession
of D-prime-strips [cf. Fig. 6.2, where each “/ ± ” denotes a D-prime-strip] determined
by considering the [“sub”]capsules of the capsule † D |T | of Definition 6.4, (i),
corresponding to the subsets S ± 1 ⊆ ...⊆ S± def
t = {0, 1, 2,... ,t−1} ⊆...⊆ S ± l
= |F ± l |
[where, by abuse of notation, we use the notation for nonnegative integers to denote
the images of these nonnegative integers in |F l |], relative to the bijection |T | →|F ∼ l |
determined by the F ± l
-group structure of T [cf. Definition 6.4, (i)]. Then the assignment
† φ Θ±
± ↦→ Prc( † D T ) determines a [well-defined, up to a unique isomorphism!]
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 T )
whose output data satisfies the following property: there are precisely n possibilities
for the element ∈|F l | to which a given index of the index set of the n-capsule
that appears in the procession constituted by this output data corresponds, prior to
the application of this functor. That is to say, by taking the product, over elements
of |F l |, of cardinalities of “sets of possibilies”, one concludes that
by considering processions — i.e., the functor discussed above, possibly
pre-composed with the functor † HT D-Θ±ell ↦→ † φ Θ±
± that associates to a
D-Θ ±ell -Hodge theater its associated D-Θ ± -bridge — the indeterminacy
consisting of (l ± ) (l±) possibilities that arises in Proposition 6.8, (ii), is
reduced to an indeterminacy consisting of a total of l ± ! possibilities.
(ii) (Mono-analytic Processions) By composing the functor of (i) with the
mono-analyticization operation discussed in Definition 4.1, (iv), one obtains a
[well-defined, up to a unique isomorphism!] 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 ⊢ T )