Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

102 SHINICHI MOCHIZUKI
(ii) Since the F l
-symmetry that appears in Proposition 4.9, (i), is transitive,
it follows that one may use this action to perform comparisons as discussed in (i).
This prompts the question:
What is the difference between this F l -symmetry and the S l -symmetry
of the output data of the functors of Proposition 4.9, (ii), (iii)?
Inaword,restrictingtotheF l
-symmetry of Proposition 4.9, (i), amounts to the
imposition of a “cyclic structure” on the index set J [i.e., a structure of F l -torsor
on J]. Thus, relative to the issue of comparability raised in (i), this F l -symmetry
allows comparison between — i.e., involves isomorphisms between the non-labeled
D-prime-strips corresponding to — distinct members of this index set J, without
disturbing the cyclic structure on J. This cyclic structure may be thought of as
a sort of combinatorial manifestation of the link to the global object † D ⊚ that
appears in a D-NF-bridge. On the other hand,
in order to compare these D-prime-strips indexed by J “in the absolute”
to D-prime-strips that have nothing to do with J, it is necessary
the “forget the cyclic structure on J”.
This is precisely what is achieved by considering the functors of Proposition 4.9,
(ii), (iii), i.e., by working with the “full S l -symmetry”.
Remark 4.9.2.
(i) The various elements of the index set of the capsule of D-prime-strips of a
D-NF-bridge are synchronized in their correspondence with the labels “1, 2,... ,l ”,
in the sense that this correspondence is completely determined up to composition
withtheactionofanelementofF l
. In particular, this correspondence is always
bijective.
One may regard this phenomenon of synchronization, orcohesion, as
an important consequence of the fact that the number field in question is
represented in the D-NF-bridge via a single copy [i.e., as opposed to a
capsule whose index set is of cardinality ≥ 2] of D ⊚ .
Indeed, consider a situation in which each D-prime-strip in the capsule † D J is
equipped with its own “independent globalization”, i.e., copy of D ⊚ ,towhichit
is related by a copy of “φ NF
j ”, which [in order not to invalidate the comparability
of distinct labels — cf. Remark 4.9.1, (i)] is regarded as being known only up to
composition with the action of an element of F l
. Then if one thinks of the [manifestly
mutually disjoint — cf. Definition 3.1, (f); Example 4.3, (i)] F l
-translates of
V ±un ⋂ V(K) bad [whose union is equal to V Bor ⋂ V(K) bad ]asbeinglabeled by the
elements of F l ,theneach D-prime-strip in the capsule † D J — i.e., each “•” in Fig.
4.5 below — is subject, as depicted in Fig. 4.5, to an independent indeterminacy
concerning the label ∈ F l
to which it is associated. In particular, the set of all
possibilities for each association includes correspondences between the index set J
of the capsule † D J and the set of labels F l
which fail to be bijective. Moreover,