Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

134 SHINICHI MOCHIZUKI
∼
— where we require that the bijection of index sets F l → T induced by the second
isomorphism determine an isomorphism of F ± l
-torsors — conjugation by which
maps φ Θell
± ↦→ † φ Θell
± . We define a(n) [iso]morphism of D-Θ ell -bridges
( † † φ Θell
±
D T −→ † D ⊚± ) → ( ‡ D T
′
to be a pair of poly-morphisms
‡ φ Θell
±
−→ ‡ D ⊚± )
† D T
∼
→ ‡ D T ′;
† D ⊚± ∼
→ ‡ D ⊚±
—where † ∼
D T → ‡ D T
′ is a capsule-+-full poly-isomorphism whose induced morphism
on index sets T → ∼ T ′ is an isomorphism of F ± l -torsors; † D ⊚± → ‡ D ⊚± is a
poly-morphism which is an Aut csp ( † D ⊚± )- [or, equivalently, Aut csp ( ‡ D ⊚± )-] orbit
of isomorphisms — which are compatible with † φ Θell
± , ‡ φ Θell
± . There is an evident
notion of composition of morphisms of D-Θ ell -bridges.
(iii) We define a base-Θ ±ell -Hodge theater, orD-Θ ±ell -Hodge theater, [relative
to the given initial Θ-data] to be a collection of data
† HT D-Θ±ell =( † † φ Θ±
±
D ≻ ←− † † φ Θell
±
D T −→ † D ⊚± )
—whereT is a F ± l -group; † φ Θ±
± is a D-Θ ± -bridge; † φ Θell
± is a D-Θ ell -bridge [relative
to the F ± l -torsor structure determined by the F± l
-group structure on T ]—such
that there exist isomorphisms
D ≻
∼
→ † D ≻ ; D ±
∼
→ † D T ; D ⊚± ∼
→ † D ⊚±
conjugation by which maps φ Θ±
± ↦→ † φ Θ±
± , φ Θell
± ↦→ † φ Θell
± . A(n) [iso]morphism of
D-Θ ±ell -Hodge theaters is defined to be a pair of morphisms between the respective
associated D-Θ ± -andD-Θ ell -bridges that are compatible with one another in the
sense that they induce the same poly-isomorphism between the respective capsules
of D-prime-strips. There is an evident notion of composition of morphisms of D-
Θ ±ell -Hodge theaters.
The following additive analogue of Proposition 4.7 follows immediately from
the various definitions involved. Put another way, the content of Proposition 6.5
below may be thought of as a sort of “intrinsic version” of the constructions carried
out in Examples 6.2, 6.3.
Proposition 6.5.
Bridges) Let
(Transport of ±-Label Classes of Cusps via Base-
† HT D-Θ±ell =( † † φ Θ±
±
D ≻ ←− † † φ Θell
±
D T −→ † D ⊚± )
be a D-Θ ±ell -Hodge theater [relative to the given initial Θ-data]. Then:
(i) For each v ∈ V, t ∈ T ,theD-Θ ell -bridge † φ Θell
± induces a [single, welldefined!]
bijection of sets of ±-label classes of cusps
† ζ Θell
v t
: LabCusp ± ( † D vt ) ∼ → LabCusp ± ( † D ⊚± )