Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

completely determined by † D.
INTER-UNIVERSAL TEICHMÜLLER THEORY I 81
(ii) Suppose that we are in the situation of (i). Then observe that by applying
the group-theoretic algorithm of [AbsTopI], Lemma 4.5, to the topological
group π 1 ( † D v )whenv ∈ V non , or by considering π 0 (−) of a cofinal collection of
“neighborhoods of infinity” [i.e., complements of compact subsets] of the underlying
topological space of † D v when v ∈ V arc , it makes sense to speak of the set of cusps
of † D v ; a similar observation applies to † D v ,forv ∈ V. If v ∈ V, then we define
a label class of cusps of † D v to be the set of cusps of † D v that lie over a single
“nonzero cusp” [i.e., a cusp that arises from a nonzero element of the quotient “Q”
that appears in the definition of a “hyperbolic orbicurve of type (1,l-tors) ± ”given
in [EtTh], Definition 2.1] of † D v ;write
LabCusp( † D v )
for the set of label classes of cusps of † D v . Thus, for each v ∈ V, LabCusp( † D v )
admits a natural F l
-torsor structure [cf. [EtTh], Definition 2.1]. Moreover, [for
any v ∈ V!] one may construct, solely from † D v ,acanonical element
† η v
∈ LabCusp( † D v )
determined by “ɛ v ” [cf. the notation of Definition 3.1, (f)]. [Indeed, this follows
from [EtTh], Corollary 2.9, for v ∈ V bad , from Corollary 1.2 for v ∈ V good ⋂ V non ,
and from the evident translation into the “language of Aut-holomorphic orbispaces”
of Corollary 1.2 for v ∈ V arc .]
(iii) We define a mono-analytic base-prime-strip, orD ⊢ -prime-strip, [relative
to the given initial Θ-data] to be a collection of data
† D ⊢ = { † D ⊢ v } v∈V
that satisfies the following conditions: (a) if v ∈ V non ,then † Dv ⊢ is a category which
admits an equivalence of categories † Dv
⊢ ∼
→Dv ⊢ [where Dv ⊢ is as in Examples 3.2,
(i); 3.3, (i)]; (b) if v ∈ V arc ,then † Dv
⊢ is an object of the category TM ⊢ [so, if Dv
⊢
is as in Example 3.4, (ii), then there exists an isomorphism † Dv
⊢ ∼
→Dv ⊢ in TM ⊢ ].
(iv) A morphism of D- (respectively, D ⊢ -) prime-strips is defined to be a collection
of morphisms, indexed by V, between the various constituent objects of the
prime-strips. Following the conventions of §0, one thus has a notion of capsules of
D- (respectively, D ⊢ -) and morphisms of capsules of D- (respectively, D ⊢ -) primestrips.
Note that to any D-prime-strip † D, one may associate, in a natural way,
a D ⊢ -prime-strip † D ⊢ — which we shall refer to as the mono-analyticization of
† D — by considering appropriate subcategories at the nonarchimedean primes [cf.
Examples 3.2, (i), (vi); 3.3, (i), (iii)], or by applying the construction of Example
3.4, (ii), at the archimedean primes.
(v) Write
D ⊚ def
= B(C K ) 0