Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 119
for the F-prime-strip tautologically associated to this Θ-Hodge theater [cf. the data
“{ † F v
} v∈V ” of Definition 3.6; Definition 5.2, (i)]. Thus, † D > may be identified with
the D-prime-strip associated [cf. Remark 5.2.1, (i)] to † F > .
Example 5.4.
Model Θ- and NF-Bridges.
(i) For j ∈ J, let
† F j = { † F vj } vj ∈V j
be an F-prime-strip whose associated D-prime-strip [cf. Remark 5.2.1, (i)] is equal
to † D j ,
† F 〈J〉 = { † F v〈J〉 } v〈J〉 ∈V 〈J〉
an F-prime-strip whose associated D-prime-strip we denote by † D 〈J〉 .Write
† F J
def
= ∏ j∈J
† F j
— where the “formal product ∏ ” is to be understood as denoting the capsule with
index set J for which the datum indexed by j is given by † F j .Thus, † F 〈J〉 may be
related to † F > ,inanatural fashion, via the full poly-isomorphism
† F 〈J〉
∼
→ † F >
and to † F J via the “diagonal arrow”
† F 〈J〉 → † F J = ∏ j∈J
† F j
— i.e., the arrow defined as the collection of data indexed by J for which the
datum indexed by j is given by the full poly-isomorphism † ∼
F 〈J〉 → † F j . Thus, we
think of † F j as a copy of † F > “situated on” the constituent labeled j of the capsule
† D J ; we think of † F 〈J〉 as a copy of † F > “situated in a diagonal fashion on” all the
constituents of the capsule † D J .
(ii) Note that in addition to thinking of † F > as being related to † F j [for j ∈ J]
via the full poly-isomorphism † ∼
F > → † F j , we may also regard † F > as being related
to † F j [for j ∈ J] via the poly-morphism
† ψ Θ j : † F j → † F >
uniquely determined by † φ Θ j
[i.e., as discussed in Remark 5.3.1]. Write
† ψ Θ : † F J → † F >
for the collection of arrows { † ψj Θ} j∈J
collection of arrows † φ Θ = { † φ Θ j } j∈J.
—whichwethinkofas“lying over” the
(iii) Next, let † F ⊛ , † F ⊚ be as in Example 5.1, (iii); δ ∈ LabCusp( † D ⊚ ). Thus,
[cf. the discussion of Example 4.3, (i)] there exists a unique Aut ɛ ( † D ⊚ )-orbit of