Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

144 SHINICHI MOCHIZUKI
—where † D ⊚± is a category equivalent to D ⊚± ; T is an F ± l -torsor; † F T = { † F t } t∈T
is a capsule of F-prime-strips, indexed by [the underlying set of] T —tobea
D-Θ ell -bridge † φ Θell
± : † D T → † D ⊚± —wherewewrite † D T for the capsule of D-
prime-strips associated to † F T [cf. Remark 5.2.1, (i)]. We define a(n) [iso]morphism
of Θ ell -bridges
( † † ψ Θell
±
F T −→ † D ⊚± ) → ( ‡ F T ′
to be a pair of poly-isomorphisms
‡ ψ Θell
±
−→ ‡ D ⊚± )
† F T
∼
→ ‡ F T ′;
† D ⊚± ∼
→ ‡ D ⊚±
that determines a morphism between the associated D-Θ ell -bridges † φ Θell
± , ‡ φ Θell
± .
There is an evident notion of composition of morphisms of D-Θ ell -bridges.
(iii) We define a Θ ±ell -Hodge theater [relative to the given initial Θ-data] to be
a collection of data
† HT Θ±ell =( † † ψ Θ±
±
F ≻ ←− † † ψ Θell
±
F T −→ † D ⊚± )
— where the data † ψ± Θ± : † F T → † F ≻ forms a Θ ± -bridge; the data † ψ Θell : † F T →
† D ⊚± forms a Θ ell -bridge — such that the associated data { † φ Θ±
± , † φ Θell
± } [cf. (i),
(ii)] forms a D-Θ ±ell -Hodge theater. A(n) [iso]morphism of Θ ±ell -Hodge theaters
is defined to be a pair of morphisms between the respective associated Θ ± -and
Θ ell -bridges that are compatible with one another in the sense that they induce the
same poly-isomorphism between the respective capsules of F-prime-strips. There
is an evident notion of composition of morphisms of Θ ±ell -Hodge theaters.
Corollary 6.12. (Isomorphisms of Θ ± -Bridges, Θ ell -Bridges, and Θ ±ell -
Hodge Theaters) Relative to a fixed collection of initial Θ-data:
(i) The natural functorially induced map from the set of isomorphisms between
two Θ ± -bridges (respectively, two Θ ell -bridges; two Θ ±ell -Hodge theaters)
to the set of isomorphisms between the respective associated D-Θ ± -bridges
(respectively, associated D-Θ ell -bridges; associated D-Θ ±ell -Hodge theaters)
is bijective.
(ii) Given a Θ ± -bridge and a Θ ell -bridge, the set of capsule-+-full poly-isomorphisms
between the respective capsules of F-prime-strips which allow one to glue
the given Θ ± -andΘ ell -bridges together to form a Θ ±ell -Hodge theater forms a
torsor over the group
F ⋊±
l
×
({±1} V)
[cf. Proposition 6.6, (iv)]. Moreover, the first factor may be thought of as corresponding
to the induced isomorphisms of F ± l
-torsors between the index sets of the
capsules involved.
Proof. Assertions (i), (ii) follow immediately from Definition 6.11; Corollary 5.3,
(ii) [cf. also Proposition 6.6, (iv), in the case of assertion (ii)]. ○