Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

94 SHINICHI MOCHIZUKI
indeterminacies in the specification of J and H means that one cannot specify the
inclusion ι : J↩→ H independently of the inclusion ζ : J↩→ H g−1 [i.e., arising from
J g ⊆ H]. One way to express this state of affairs is as follows. Write “
out
↩→ ”
for the outer homomorphism determined by an injective homomorphism between
groups. Then the collection of factorizations J
out
↩→ H out
↩→ G of the natural
“outer” inclusion J
out
↩→ G through some G-conjugate of H — i.e., put another
way,
the collection of outer homomorphisms
J
out
↩→
that are compatible with the “structure morphisms” J
H
out
↩→ G determined by the natural inclusions
H
out
↩→ G,
—iswell-defined, inafashionthatiscompatible with independent G-conjugacy
indeterminacies in the specification of J and H. That is to say, this collection
of outer homomorphisms amounts to the collection of inclusions J g 1
↩→ H g 2
,for
g 1 ,g 2 ∈ G. By contrast, to specify the inclusion ι : J ↩→ H [together with, say,
its G-conjugates {ι γ } γ∈G ] independently of the inclusion ζ : J↩→ H g−1 [and its G-
conjugates {ζ γ } γ∈G ] amounts to the imposition of a partial synchronization —
i.e., a partial deactivation — of the [a priori!] independent G-conjugacy indeterminacies
in the specification of J and H. Moreover, such a “partial deactivation”
can only be effected at the cost of introducing certain arbitrary choices into the
construction under consideration.
(ii) Relative to the factorizations considered in (i), we make the following
observation. Given a G-conjugate H ∗ of H and a subgroup I ⊆ H ∗ , the condition
on I that
(∗ ⊆ ) I be a G-conjugate of J
is a condition that is independent of the datum H ∗ , while the condition on I that
(∗ ∼= ) I be a G-conjugate of J such that (H ∗ ,I) ∼ = (H, J)
[where the “ ∼ =” denotes an isomorphism of pairs consisting of a group and a subgroup
— cf. the discussion of (i)] is a condition that depends, in an essential fashion,
on the datum H ∗ . Here, (∗ ⊆ ) is precisely the condition that one must impose when
one considers arbitrary factorizations as in (i), while (∗ ∼= ) is the condition that one
must impose when one wishes to restrict one’s attention to factorizations whose
first arrow gives rise to a pair isomorphic to the pair determined by ι. That is to
say, the dependence of (∗ ∼= ) on the datum H ∗ may be regarded as an explicit formulation
of the necessity for the “imposition of a partial synchronization” as discussed
in (i), while the corresponding independence, exhibited by (∗ ⊆ ), of the datum H ∗
may be regarded as an explicit formulation of the lack of such a necessity when one
considers arbitrary factorizations as in (i). Finally, we note that by reversing the
direction of the inclusion “⊆”, one may consider a subgroup of I ⊆ G that contains
agivenG-conjugate J ∗ of J, i.e., I ⊇ J ∗ ; then analogous observations may be made
concerning the condition (∗ ⊇ )onI that I be a G-conjugate of H.