Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

66 SHINICHI MOCHIZUKI
for □ ∈{X v , C v , X v , C v , −→v X , −→v C },wehaveacomplex archimedean topological field
[i.e., a “CAF” — cf. §0]
A □
[cf. [AbsTopIII], Definition 4.1, (i)] which may be algorithmically constructed from
def
□; writeA □ = A □ \{0}. Next, let us write
and
D v
def
= X −→v
C v
for the archimedean Frobenioid as in [FrdII], Example 3.3, (ii) [i.e., “C” ofloc. cit.],
wherewetakethebase category [i.e., “D” ofloc. cit.] to be the one-morphism
category determined by Spec(K v ). Thus, the linear morphisms among the pseudoterminal
objects of C determine unique isomorphisms [cf. [FrdI], Definition 1.3,
(iii), (c)] among the respective topological monoids “O ⊲ (−)”—wherewerecall
[cf. [FrdI], Theorem 3.4, (iii); [FrdII], Theorem 3.6, (i), (vii)] that these topological
monoids may be reconstructed category-theoretically from C. In particular, it makes
sense to write “O ⊲ (C v )”, “O × (C v ) ⊆O ⊲ (C v )”. Moreover, we observe that, by
construction, there is a natural isomorphism
O ⊲ (C v ) →O ∼ K ⊲ v
of topological monoids. Thus, one may also think of C v as a “Frobenioid-theoretic
representation” of the topological monoid OK ⊲ v
[cf. [AbsTopIII], Remark 4.1.1].
∼
Observe that there is a natural topological isomorphism K v → ADv ,whichmaybe
restricted to OK ⊲ v
to obtain an inclusion of topological monoids
κ v : O ⊲ (C v ) ↩→ A Dv
— which we shall refer to as the Kummer structure on C v [cf. Remark 3.4.2 below].
Write
def
F v
=(C v , D v ,κ v )
[cf. Example 3.2, (i); Example 3.3, (i)].
(ii) Next, recall the category TM ⊢ of “split topological monoids” of [AbsTopIII],
Definition 5.6, (i) — i.e., the category whose objects (C, −→ C ) consist of a topological
monoid C isomorphic to OC
⊲ and a topological submonoid −→ C ⊆ C [necessarily
isomorphic to R ≥0 ] such that the natural inclusions C × ↩→ C [where C × ,
which is necessarily isomorphic to S 1 , denotes the topological submonoid of invertible
elements], −→ C ↩→ C determine an isomorphism C × × −→ C → ∼ C of topological
monoids, and whose morphisms (C 1 , −→ C 1 ) → (C 2 , −→ C 2 ) are isomorphisms of topological
monoids C 1 → C2 that induce isomorphisms −→ ∼ −→
∼
C 1 → C 2 . Note that the
CAF’s K v , A Dv determine, in a natural way, objects of TM ⊢ .Write
τ ⊢ v