Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 115
discussion of Example 3.5, (i), is an isomorphism of topological monoids [both of
which are, in fact, isomorphic to R ≥0 ]; (f) the collection of data in the above display
is isomorphic to the collection of data F ⊩ mod
of Example 3.5, (ii). A morphism of
F ⊩ -prime-strips is defined to be an isomorphism between collections of data as
discussed above. Following the conventions of §0, one thus has notions of capsules
of F ⊩ -prime-strips and morphisms of capsules of F ⊩ -prime-strips.
Remark 5.2.1.
(i) Note that it follows immediately from Definitions 4.1, (i), (iii); 5.2, (i),
(ii); Examples 3.2, (vi), (c), (d); 3.3, (iii), (b), (c), that there exists a functorial
algorithm for constructing D- (respectively, D ⊢ -) prime-strips from F- (respectively,
F ⊢ -) prime-strips.
(ii) In a similar vein, it follows immediately from Definition 5.2, (i), (ii); Examples
3.2, (vi), (f); 3.3, (iii), (e); 3.4, (i), (ii), that there exists a functorial algorithm
for constructing from an F-prime-strip ‡ F = { ‡ F v } v∈V an F ⊢ -prime-strip ‡ F ⊢
‡ F ↦→ ‡ F ⊢ = { ‡ F ⊢ v } v∈V
— which we shall refer to as the mono-analyticization of ‡ F. Next, let us recall from
the discussion of Example 3.5, (i), the relatively simple structure of the category
“Cmod ⊩ ”, i.e., which may be summarized, roughly speaking, as a collection, indexed
by V, of copies of the topological monoid R ≥0 , which are related to one another by
a “product formula”. In particular, we conclude that one may also construct from
the F-prime-strip ‡ F, via a functorial algorithm [cf. the constructions of Example
3.5, (i), (ii)], a collection of data
‡ F ↦→ ‡ F ⊩ def
= ( ‡ C ⊩ , Prime( ‡ C ⊩ ) ∼ → V, ‡ F ⊢ , { ‡ ρ v } v∈V )
— i.e., consisting of a category [which is, in fact, equipped with a Frobenioid
structure], a bijection, the F ⊢ -prime-strip ‡ F ⊢ , and an isomorphism of topological
monoids associated to ‡ C ⊩ and ‡ F ⊢ , respectively, at each v ∈ V — which is isomorphic
to the collection of data F ⊩ mod
of Example 3.5, (ii), i.e., which forms an
F ⊩ -prime-strip [cf. Definition 5.2, (iv)].
Remark 5.2.2. Thus, from the point of view of the discussion of Remark 5.1.3,
F-prime-strips are Kummer-ready [i.e., at v ∈ V non — cf. the theory of [FrdII], §2],
whereas F ⊢ -prime-strips are Kummer-blind.
Corollary 5.3. (Isomorphisms of Global Frobenioids, Frobenioid-Prime-
Strips, and Tempered Frobenioids) Relative to a fixed collection of initial
Θ-data:
(i) For i =1, 2, let i F ⊛ (respectively, i F ⊚ )beacategory which is equivalent
to the category † F ⊛ (respectively, † F ⊚ ) of Example 5.1, (iii). Thus, i F ⊛
(respectively, i F ⊚ ) is equipped with a natural Frobenioid structure [cf. [FrdI],
Corollary 4.11; [FrdI], Theorem 6.4, (i); Remark 3.1.5 of the present paper]. Write