Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

100 SHINICHI MOCHIZUKI
(ii) The set of isomorphisms between two D-Θ-bridges (respectively, two
D-ΘNF-Hodge theaters) isof cardinality one.
(iii) Given a D-NF-bridge and a D-Θ-bridge, the set of capsule-full polyisomorphisms
between the respective capsules of D-prime-strips which allow one to
glue the given D-NF- and D-Θ-bridges together to form a D-ΘNF-Hodge theater
forms an F l -torsor.
(iv) Given a D-NF-bridge, there exists a [relatively simple — cf. the discussion
of Example 4.4, (i), (ii), (iii)] functorial algorithm for constructing, up to an
F l
-indeterminacy [cf. (i), (iii)], from the given D-NF-bridge a D-ΘNF-Hodge
theater whose underlying D-NF-bridge is the given D-NF-bridge.
Proposition 4.9. (Symmetries arising from Forgetful Functors) Relative
to a fixed collection of initial Θ-data:
(i) (Base-NF-Bridges) The operation of associating to a D-ΘNF-Hodge theater
the underlying D-NF-bridge of the D-ΘNF-Hodge theater determines a natural
functor
category of
D-ΘNF-Hodge theaters
and isomorphisms of
D-ΘNF-Hodge theaters
→
category of
D-NF-bridges
and isomorphisms of
D-NF-bridges
† HT D-ΘNF ↦→ ( † D ⊚ † φ NF
←− † D J )
whose output data admits a F l
-symmetry which acts simply transitively on
the index set [i.e., “J”] of the underlying capsule of D-prime-strips [i.e., “ † D J ”] of
this output data.
(ii) (Holomorphic Capsules) The operation of associating to a D-ΘNF-
Hodge theater the underlying capsule of D-prime-strips of the D-ΘNF-Hodge theater
determines a natural functor
category of
D-ΘNF-Hodge theaters
and isomorphisms of
D-ΘNF-Hodge theaters
→
category of l -capsules
of D-prime-strips
and capsule-full polyisomorphisms
of l -capsules
† HT D-ΘNF ↦→ † D J
whose output data admits an S l -symmetry [where we write S l for the symmetric
group on l letters] which acts transitively on the index set [i.e., “J”] of this
output data. Thus, this functor may be thought of as an operation that consists of
forgetting the labels ∈ F l
of Proposition 4.7, (i). In particular, if one is only
given this output data † D J up to isomorphism, then there is a total of precisely l