Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 59
(d) the category D v may be reconstructed category-theoretically either from
F v
[cf. [EtTh], Theorem 4.4; [EtTh], Proposition 5.1] or from C v [cf.
[FrdI], Theorem 3.4, (v); [FrdII], Theorem 1.2, (i); [FrdII], Example 1.3,
(i); [SemiAnbd], Example 3.10; [SemiAnbd], Remark 3.4.1].
Next, let us observe that by (b), (d), together with the discussion of (ii) concerning
the category-theoreticity of Θ v
, it follows [cf. Remark 3.2.1 below] that
(e) one may reconstruct the split Frobenioid Fv
Θ [up to the l·Z indeterminacy
in Θ v
discussed in (ii); cf. also Remark 3.2.3 below] category-theoretically
from F v
[cf. [FrdI], Theorem 3.4, (i), (v); [EtTh], Proposition 5.1].
Next, let us recall that the values of Θ v
may be computed by restricting the corresponding
Kummer class, i.e., the “étale theta function” [cf. [EtTh], Proposition
1.4, (iii); cf. the proof of [EtTh], Theorem 1.10, (ii); the proof of [EtTh], Theorem
5.7], which may be reconstructed category-theoretically from D v [cf. [EtTh], Corollary
2.8, (i)]. Thus, by applying the isomorphisms of cyclotomes of [AbsTopIII],
Corollary 1.10, (c); [AbsTopIII], Remark 3.2.1 [cf. also [AbsTopIII], Remark 3.1.1]
to these Kummer classes, one concludes from (a), (d) that
(f) one may reconstruct the split Frobenioid Fv
⊢ category-theoretically from
C v , hence also [cf. (iii)] from F v
[cf. Remark 3.2.1 below].
Remark 3.2.1.
(i) In [FrdI], [FrdII], and [EtTh] [cf. [EtTh], Remark 5.1.1], the phrase “reconstructed
category-theoretically” is interpreted as meaning “preserved by equivalences
of categories”. From the point of view of the theory of [AbsTopIII] — i.e., the discussion
of “mono-anabelian” versus “bi-anabelian” geometry [cf., [AbsTopIII], §I2,
(Q2)] — this sort of definition is “bi-anabelian” in nature. In fact, it is not difficult
to verify that the techniques of [FrdI], [FrdII], and [EtTh] all result in explicit reconstruction
algorithms, whoseinput data consists solely of the category structure
of the given category, of a “mono-anabelian” nature that do not require the use of
some fixed reference model that arises from scheme theory [cf. the discussion of
[AbsTopIII], §I4]. For more on the foundational aspects of such “mono-anabelian
reconstruction algorithms”, we refer to the discussion of [IUTchIV], Example 3.5.
(ii) One reason that we do not develop in detail here a “mono-anabelian approach
to the geometry of categories” along the lines of [AbsTopIII] is that, unlike
the case with the mono-anabelian theory of [AbsTopIII], which plays a quite essential
role in the theory of the present series of papers, much of the category-theoretic
reconstruction theory of [FrdI], [FrdII], and [EtTh] is not of essential importance
in the development of the theory of the present series of papers. That is to say, for
instance, instead of quoting results to the effect that the base categories or divisor
monoids of various Frobenioids may be reconstructed category-theoretically, one
could instead simply work with the data consisting of “the category constituted by