Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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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)]. ○
INTER-UNIVERSAL TEICHMÜLLER THEORY I 145 Remark 6.12.1. By applying Corollary 6.12, a similar remark to Remark 5.6.1 may be made concerning the Θ ± -bridges, Θ ell -bridges, andΘ ±ell -Hodge theaters studied in the present §6. We leave the routine details to the reader. Remark 6.12.2. Relative to a fixed collection of initial Θ-data: (i) Suppose that ( † F T → † F ≻ )isaΘ ± -bridge; write( † D T → † D ≻ )for the associated D-Θ ± -bridge [cf. Definition 6.11, (i)]. Then Proposition 6.7 gives a functorial algorithm for constructing a D-Θ-bridge ( † D T → † D > )fromthis D-Θ ± -bridge ( † D T → † D ≻ ). Suppose that this D-Θ-bridge ( † D T → † D > ) arises as the D-Θ-bridge associated to a Θ-bridge ( ‡ F J → ‡ F > ‡ HT Θ )[so J = T — cf. Definition 5.5, (ii)]. Then since the portion “ ‡ F J → ‡ F > ”ofthis Θ-bridge is completely determined [cf. Definition 5.5, (ii), (d)] by the associated D-Θ-bridge, one verifies immediately that one may regard this portion “ ‡ F J → ‡ F > ” of the Θ-bridge as having been constructed via a functorial algorithm similar to the functorial algorithm of Proposition 6.7 [cf. also Definition 5.5, (ii), (d); the discussion of Remark 5.3.1] from the Θ ± -bridge ( † F T → † F ≻ ). Since, moreover, isomorphisms between Θ-bridges are in natural bijective correspondence with isomorphisms between the associated D-Θ-bridges [cf. Corollary 5.6, (ii)], it thus follows immediately [cf. Corollary 5.3, (ii)] that isomorphisms between Θ-bridges are in natural bijective correspondence with isomorphisms between the portions of Θ-bridges [i.e., “ ‡ F J → ‡ F > ”] considered above. Thus, in summary, if ( ‡ F J → ‡ F > ‡ HT Θ )isaΘ-bridge for which the portion “ ‡ F J → ‡ F > ” is obtained via the functorial algorithm discussed above from the Θ ± -bridge ( † F T → † F ≻ ), then, for simplicity, we shall describe this state of affairs by saying that the Θ-bridge ( ‡ F J → ‡ F > ‡ HT Θ )isglued to the Θ ± -bridge ( † F T → † F ≻ ) via the functorial algorithm of Proposition 6.7. A similar [but easier!] construction may be given for D-Θ-bridges and D-Θ ± - bridges. We leave the routine details of giving a more explicit description [say, in the style of the statement of Proposition 6.7] of such functorial algorithms to the reader. (ii) Now observe that by gluing aΘ ±ell -Hodge theater [cf. Definition 6.11, (iii)] to a ΘNF- Hodge theater [cf. Definition 5.5, (iii)] along the respective associated Θ ± -andΘ-bridges via the functorial algorithm of Proposition 6.7 [cf. (i)], one obtains the notion of a “Θ ±ell NF-Hodge-theater” — cf. Definition 6.13, (i), below. Here, we note that by Proposition 4.8, (ii); Corollary 5.6, (ii), the gluing isomorphism that occurs in such a gluing operation
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completely determined by † D. INT
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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