The deck weight character (a lemma of an earlier draft) #
An earlier draft of the manuscript (lem:deck-weight-character, not in the current
revision): "Define the character of a
closed walk in K to be the element of L = Hom_ℤ(H_ℤ, ℤ) sending an integer ray weight h
to the weight of the walk under a mesh-weight lift of h; by Lemma 7.4 this does not depend on
the lift. A closed walk whose lift joins X̃ to gX̃ has character −θ_g."
Formalization. The deck group is the fundamental group of the coned complex at the basepoint
(Deck), so a closed walk is a deck element g and its weight under a mesh weight W is the
transport potW of the edge weights along g (MeshWt.walkWt). The deck class θ_g is
ζ(g o₀) − ζ(o₀) for the object values ζ of the universal ray weight on the lifted projective
category (Proposition 7.2(2)), defined for g in the stabilizer H of the base component
(the deviation of Proposition 7.2(1)). Deviation: with the orientation conventions of the
formalization (arrows of A₀ go along paths, rv is the weight of the path) the identity reads
walkWt g = h(θ_g) (walkWt_eq), the sign of the source coming from its opposite convention
for object values (−b_x at P_x).
The weight of a closed walk under a mesh weight: its transport along the deck element.
Instances For
The object value at a lifted vertex is the weight of its sheet.
The deck weight character (an earlier draft's lemma): the weight of a closed walk g in the stabilizer of the base component
under any mesh-weight lift W of an integer ray weight h is h(θ_g), where
θ_g = ζ(g o₀) − ζ(o₀) is the deck class of Proposition 7.2(1).
The character does not depend on the mesh-weight lift (weights-and-parameters.tex).