Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

130 SHINICHI MOCHIZUKI
(vi) Let
† D ⊚±
be any category isomorphic to D ⊚± . Then just as in the discussion of (iii) in the
case of “v ∈ V good ⋂ V non ”, it makes sense [cf. [AbsTopI], Lemma 4.5] to speak
of the set of cusps of † D ⊚± ,aswellastheset of ±-label classes of cusps
LabCusp ± ( † D ⊚± )
— which, in this case, may be identified with the set of cusps of † D ⊚± .
(vii) Recall from [AbsTopIII], Theorem 1.9 [cf. Remark 3.1.2] that [just as in
the case of D ⊚ — cf. the discussion of Definition 4.1, (v)] there exists a grouptheoretic
algorithm for reconstructing, from π 1 (D ⊚± )[cf.§0], the algebraic closure
“F ” of the base field “K”, hence also the set of valuations “V(F )” from D ⊚± [e.g.,
as a collection of topologies on F — cf., e.g., [AbsTopIII], Corollary 2.8]. Moreover,
for w ∈ V(K) arc , let us recall [cf. Remark 3.1.2; [AbsTopIII], Corollaries 2.8, 2.9]
that one may reconstruct group-theoretically, fromπ 1 (D ⊚± ), the Aut-holomorphic
orbispace X w associated to X w .Let † D ⊚± be as in (vi). Then let us write
V( † D ⊚± )
for the set of valuations [i.e., “V(F )”], equipped with its natural π 1 ( † D ⊚± )-action,
V( † D ⊚± )
def
= V( † D ⊚± )/π 1 ( † D ⊚± )
for the quotient of V( † D ⊚± )byπ 1 ( † D ⊚± ) [i.e., “V(K)”], and, for w ∈ V( † D ⊚± ) arc ,
X( † D ⊚± ,w)
[i.e., “X w ” — cf. the discussion of [AbsTopIII], Definition 5.1, (ii)] for the Autholomorphic
orbispace obtained by applying these group-theoretic reconstruction
algorithms to π 1 ( † D ⊚± ). Now if U is an arbitrary Aut-holomorphic orbispace, then
let us define a morphism
U → † D ⊚±
to be a morphism of Aut-holomorphic orbispaces [cf. [AbsTopIII], Definition 2.1,
(ii)] U → X( † D ⊚± ,w)forsomew ∈ V( † D ⊚± ) arc . Thus, it makes sense to speak of
the pre-composite (respectively, post-composite) of such a morphism U → † D ⊚±
with a morphism of Aut-holomorphic orbispaces (respectively, with an isomorphism
[cf. §0] † D ⊚± ∼ → ‡ D ⊚± [i.e., where ‡ D ⊚± is a category equivalent to D ⊚± ]).
Remark 6.1.1. In fact, in the notation of Example 4.3, (i); Definition 6.1, (v), it
is not difficult to verify that V ± = V ±un (⊆ V(K)).
Example 6.2.
Model Base-Θ ± -Bridges.
(i) In the following, let us think of F l as a F ± l
-group [relative to the evident
F ± l -group structure]. Let D ≻ = {D ≻,v } v∈V ;
D t = {D vt } v∈V