Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 123
This point of view may be thought of as a sort of abstract, Frobenioid-theoretic
analogue of the normalization of arithmetic degrees or heights in which the height
of a point is multiplied by the degree of the field extension when one executes a
change of the base field.
The notions defined in the following “Frobenioid-theoretic lifting” of Definition
4.6 will play a central role in the theory of the present series of papers.
Definition 5.5.
(i) We define an NF-bridge [relative to the given initial Θ-data] to be a collection
of data
( ‡ ‡ ψ NF
F J −→ ‡ F ⊚
‡ F ⊛ )
as follows:
(a) ‡ F J = { ‡ F j } j∈J is a capsule of F-prime-strips, indexed by a finite index
set J. Write ‡ D J = { ‡ D j } j∈J for the associated capsule of D-prime-strips
[cf. Remark 5.2.1, (i)].
(b) ‡ F ⊚ , ‡ F ⊛ are categories equivalent to the categories † F ⊚ , † F ⊛ , respectively,
of Example 5.1, (iii). Thus, each of ‡ F ⊚ , ‡ F ⊛ is equipped with
a natural Frobenioid structure [cf. [FrdI], Corollary 4.11; [FrdI], Theorem
6.4, (i); Remark 3.1.5 of the present paper]; write ‡ D ⊚ , ‡ D ⊛ for the
respective base categories of these Frobenioids.
(c) The arrow “” consists of the datum of a morphism ‡ D ⊚ → ‡ D ⊛
which is abstractly equivalent [cf. §0] to the natural morphism † D ⊚ →
† D ⊛ of Example 5.1, (i), together with the datum of an isomorphism
‡ F ⊚ → ∼ ‡ F ⊛ |‡ D⊚ [cf. the discussion of Example 5.1, (iii)].
(d) ‡ ψ
NF is a poly-morphism that lifts [uniquely! — cf. Corollary 5.3, (i), (ii)]
a poly-morphism ‡ φ NF
: ‡ D J → ‡ D ⊚ such that ‡ φ NF
forms a D-NF-bridge
[cf. Example 5.4, (v); Remark 5.4.1].
Thus, one verifies immediately that any NF-bridge as above determines an associated
D-NF-bridge ( ‡ φ NF
: ‡ D J → ‡ D ⊚ ). We define a(n) [iso]morphism of
NF-bridges
( 1 1 ψ NF
F J1 −→ 1 F ⊚
to be a collection of arrows
1 F ⊛ ) → ( 2 2 ψ NF
F J2 −→ 2 F ⊚
2 F ⊛ )
1 F J1
∼
→ 2 F J2 ;
1 F ⊚ ∼
→ 2 F ⊚ ;
1 F ⊛ ∼
→ 2 F ⊛
—where 1 F J1
∼
→ 2 F J2 is a capsule-full poly-isomorphism [cf. §0], hence induces a
poly-isomorphism 1 D J1
∼
→ 2 D J2 ; 1 F ⊚ ∼ → 2 F ⊚ is a poly-isomorphism which lifts
a poly-isomorphism 1 D ⊚ ∼ → 2 D ⊚ such that the pair of arrows 1 D J1
∼
→ 2 D J2 ,
1 D ⊚ ∼ → 2 D ⊚ forms a morphism between the associated D-NF-bridges; 1 F ⊛ ∼ → 2 F ⊛
is an isomorphism — which are compatible [in the evident sense] with the i ψ NF
[for