Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

82 SHINICHI MOCHIZUKI
[cf. §0]. Then recall from [AbsTopIII], Theorem 1.9 [cf. Remark 3.1.2] that there
exists a group-theoretic 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 )” [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 C w associated to C w .Let † D ⊚ be a category equivalent to D ⊚ .Thenlet
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 ,
C( † D ⊚ ,w)
[i.e., “C 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 → C( † 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 ⊚ ]). Finally, just as in
the discussion of (ii) in the case of “v ∈ V good ⋂ V non ”, we may apply [AbsTopI],
Lemma 4.5, to conclude that it makes sense to speak of the set of cusps of † D ⊚ ,as
well as the set of label classes of cusps
LabCusp( † D ⊚ )
of † D ⊚ , which admits a natural F l
-torsor structure.
(vi) Let † D ⊚ be a category equivalent to D ⊚ , † D = { † D v } v∈V a D-primestrip.
If v ∈ V, then we define a poly-morphism † D v → † D ⊚ to be a collection of
morphisms † D v → † D ⊚ [cf. §0 whenv ∈ V non ;(v)whenv ∈ V arc ]. We define a
poly-morphism
† D → † D ⊚
to be a collection of poly-morphisms { † D v → † D ⊚ } v∈V . Finally, if { e D} e∈E is a
capsule of D-prime-strips, then we define a poly-morphism
{ e D} e∈E → † D ⊚ (respectively, { e D} e∈E → † D)
to be a collection of poly-morphisms { e D → † D ⊚ } e∈E (respectively, { e D → † D} e∈E ).