Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 121
to be a collection of poly-morphisms { e F → † F ⊚ } e∈E (respectively, { e F → † F} e∈E )
[cf. Definition 4.1, (vi)]. Thus, a poly-morphism { e F} e∈E → † F ⊚ (respectively,
{ e F} e∈E → † F) may be thought of as “lying over” an induced poly-morphism
{ e D} e∈E → † D ⊚ (respectively, { e D} e∈E → † D) [cf. Definition 4.1, (vi)].
(v) We continue to use the notation of (iv). Now observe that by Corollary
5.3, (i), (ii), there exists a unique poly-morphism
that lies over † φ NF
.
† ψ NF
: † F J → † F ⊚
(vi) We continue to use the notation of (v). Now observe that it follows from the
definition of † F ⊚ mod in terms of terminal objects of † D ⊛ [cf. Example 5.1, (iii)] that
any poly-morphism † F 〈J〉 → † F ⊚ [cf. the notation of (i)] induces, via “restriction”
[in the evident sense], an isomorphism class of functors [cf. the notation of Example
5.1, (vii)]
( † F ⊚ ⊆ † F ⊛ ⊇)
† F ⊚ ∼
mod
→ † F ⊚ 〈J〉
→ † F v〈J〉
for each v 〈J〉 ∈ V 〈J〉 which is independent of the choice of the poly-morphism
† F 〈J〉 → † F ⊚ [i.e., among its F l -conjugates]. That is to say, the fact that † F ⊚ mod
is defined in terms of terminal objects of † D ⊛ [cf. also the definition of F mod given
in Definition 3.1, (b)!] implies that this particular isomorphism class of functors is
immune to [i.e., fixed by] the various indeterminacies that appear in the definition
of the poly-morphism † F 〈J〉 → † F ⊚ ,aswellastothechoiceof † F 〈J〉 → † F ⊚ . Let
us write
( † F ⊚ ⊆ † F ⊛ ⊇)
† F ⊚ ∼
mod
→ † F ⊚ 〈J〉
→ † F 〈J〉
for the collection of isomorphism classes of restriction functors just defined, as v 〈J〉
ranges over the elements of V 〈J〉 . In a similar vein, we also obtain collections of
natural isomorphism classes of restriction functors
† F ⊚ J
→ † F J ;
† F ⊚ j → † F j
for j ∈ J. Finally, just as in Example 5.1, (vii), we obtain natural realifications
† F ⊚R
〈J〉
→ † F R 〈J〉 ; † F ⊚R
J
→ † F R J;
† F ⊚R
j
→ † F R j
of the various F-prime-strips and isomorphism classes of restriction functors discussed
so far.
(vii) We shall refer to as “pivotal distributions” the objects constructed in (vi)
† F ⊚ pvt → † F pvt ;
in the case j = 1 — cf. Fig. 5.2.
† F ⊚R
pvt
→ † F R pvt
Remark 5.4.1. The constructions of Example 5.4, (i), (ii) (respectively, Example
5.4, (iii), (iv), (v), (vi)) manifestly only require the D-Θ-bridge portion † φ Θ