Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

114 SHINICHI MOCHIZUKI
these observations concerning † F ⊚ , † F ⊛ , M ⊚ ( † D ⊚ ), † F ⊚ mod ,andM⊚ mod († D ⊚ )by
saying that † F ⊚ , † F ⊛ ,andM ⊚ ( † D ⊚ )areKummer-ready, whereas † F ⊚ mod and
M ⊚ mod († D ⊚ )areKummer-blind. In particular, the various copies of [and products
of copies of] † F ⊚ mod — i.e., † F ⊚ j , † F ⊚ 〈J〉 , † F ⊚ J
— considered in Example 5.1, (vii),
are also Kummer-blind.
Definition 5.2.
(i) We define a holomorphic Frobenioid-prime-strip, orF-prime-strip, [relative
to the given initial Θ-data] to be a collection of data
‡ F = { ‡ F v } v∈V
that satisfies the following conditions: (a) if v ∈ V non ,then ‡ F v is a category ‡ C v
which admits an equivalence of categories ‡ ∼
C v →Cv [where C v is as in Examples
3.2, (iii); 3.3, (i)]; (b) if v ∈ V arc ,then ‡ F v =( ‡ C v , ‡ D v , ‡ κ v )isacollection of data
consisting of a category, an Aut-holomorphic orbispace, and a Kummer structure
such that there exists an isomorphism of collections of data ‡ ∼
F v →Fv =(C v , D v ,κ v )
[where F v
is as in Example 3.4, (i)].
(ii) We define a mono-analytic Frobenioid-prime-strip, orF ⊢ -prime-strip, [relative
to the given initial Θ-data] to be a collection of data
‡ F ⊢ = { ‡ F ⊢ v } v∈V
that satisfies the following conditions: (a) if v ∈ V non ,then ‡ Fv
⊢ is a split Frobenioid,
whose underlying Frobenioid we denote by ‡ Cv ⊢ , which admits an isomorphism
∼
→Fv ⊢ [where Fv ⊢ is as in Examples 3.2, (v); 3.3, (i)]; (b) if v ∈ V arc ,then
‡ F ⊢ v
‡ Fv ⊢ is a triple of data, consisting of a Frobenioid ‡ Cv ⊢ , an object of TM ⊢ ,anda
splitting of the Frobenioid, such that there exists an isomorphism of collections of
data ‡ Fv
⊢ ∼
→Fv ⊢ [where Fv ⊢ is as in Example 3.4, (ii)].
(iii) A morphism of F- (respectively, F ⊢ -) prime-strips is defined to be a collection
of isomorphisms, indexed by V, between the various constituent objects of
the prime-strips. Following the conventions of §0, one thus has notions of capsules
of F- (respectively, F ⊢ -) and morphisms of capsules of F- (respectively, F ⊢ -)
prime-strips.
(iv) We define a globally realified mono-analytic Frobenioid-prime-strip, orF ⊩ -
prime-strip, [relative to the given initial Θ-data] to be a collection of data
‡ F ⊩ = ( ‡ C ⊩ , Prime( ‡ C ⊩ ) ∼ → V, ‡ F ⊢ , { ‡ ρ v } v∈V )
that satisfies the following conditions: (a) ‡ C ⊩ is a category [which is, in fact,
equipped with a Frobenioid structure] that is isomorphic to the category Cmod ⊩ of
Example 3.5, (i); (b) “Prime(−)” is defined as in the discussion of Example 3.5,
(i); (c) Prime( ‡ C ⊩ ) → ∼ V is a bijection of sets; (d) ‡ F ⊢ = { ‡ Fv ⊢ } v∈V is an F ⊢ -primestrip;
(e) ‡ ∼
ρ v :Φ‡ C ⊩ ,v → Φ rlf
‡ Cv
⊢,where“Φ‡ C ,v” and“Φ rlf
⊩ ‡ C
” are defined as in the
v ⊢