Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 143
spokes. Finally, [cf. the situation discussed in Corollary 3.9, (i)] this diagram satisfies
the important property of admitting arbitrary permutation symmetries
among the spokes [i.e., among the labels n ∈ Z of the D-Θ ±ell -Hodge-theaters].
(iv) The constructions of (i), (ii), (iii) are compatible, respectively, with
the constructions of Corollary 4.12, (i), (ii), (iii), relative to the functor [i.e.,
determined by the functorial algorithm] of Proposition 6.7, in the evident sense [cf.
the compatibility discussed in Proposition 6.9, (iii)].
Finally, we conclude with additive analogues of Definition 5.5, Corollary 5.6.
Definition 6.11.
(i) We define a Θ ± -bridge [relative to the given initial Θ-data] to be a polymorphism
† † ψ Θ±
±
F T −→
—where † F ≻ is a F-prime-strip; T is an F ± l -group; † F T = { † F t } t∈T is a capsule of
F-prime-strips, indexed by [the underlying set of] T — that lifts a D-Θ ± -bridge
† φ Θ±
± : † D T → † D ≻ [cf. Corollary 5.3, (ii)]. In this situation, we shall write
† F ≻
† F |T |
for the l ± -capsule obtained from the l-capsule † F T by forming the quotient |T | of
the index set T of this underlying capsule by the action of {±1} and identifying the
components of the capsule † F T indexed by the elements in the fibers of the quotient
T ↠ |T | via the constituent poly-morphisms of † ψ± Θ± = { † ψt Θ± } t∈Fl [so each consitutent
F-prime-strip of † F |T | is only well-defined up to a positive automorphism,
but this indeterminacy will not affect applications of this construction — cf. the
discussion of Definition 6.4, (i)]. Also, we shall write
† F T
for the l -capsule determined by the subset T def
= |T |\{0} of nonzero elements of
|T |. We define a(n) [iso]morphism of F-Θ ± -bridges
( † † ψ Θ±
±
F T −→ † F ≻ ) → ( ‡ F T ′
‡ ψ Θ±
±
−→ ‡ F ≻ )
to be a pair of poly-isomorphisms
† F T
∼
→ ‡ F T ′;
† F ≻
∼
→ ‡ F ≻
that lifts a morphism between the associated D-Θ ± -bridges † φ Θ±
± , ‡ φ Θ±
± .Thereis
an evident notion of composition of morphisms of F-Θ ± -bridges.
(ii) We define a Θ ell -bridge [relative to the given initial Θ-data]
† F T † ψ Θell
±
−→ † D ⊚±