# Provide a citation for the “spine lemma”?

I'm looking for a citable reference for the following (perhaps folkloric?) result on topological field theories.

(There are obviously generalizations to other dimensions; I'm happy with just the 2-dimensional case described below.)

Let $Y$ be a 2-manifold, $c$ be a finite set of oriented points in $\partial Y$. A spine for $(Y; c)$ is a oriented 1-complex $G\subset Y$ such that $G \cap \partial Y = c$ and $Y$ deformation retracts onto $G$.

If $C$ is a semisimple pivotal category (or something similar enough for the following to make sense), and now $c$ is a finite set of oriented points in $\partial Y$ labelled by simple objects of $C$, then a labeling of a spine $G$ is a labeling of the edges of $G$ by simple objects of $C$ (compatible with $c$) and labeling of the vertices of $G$ by (appropriate) morphisms of $C$.

Spine Lemma Suppose $(Y, c)$ has a spine (exactly when each connected component has non-empty boundary). The Turaev-Viro vector space for $(Y,c)$ has a basis the set of labellings of any fixed spine $G$, where the vertex labels should be drawn from a basis for the relevant morphism space.

Can anyone point to a place in the literature this (or a generalization) has been proved?

(Sketch: take an arbitrary element of the Turaev-Viro vector space, compress it onto the spine, using basic relations. In the other direction, labellings are linearly independent, using a nondegenerate pairing on each morphism space.)

• Balsam-Kirillov and Turaev-Virelizier both have rigorous generalizations of TV beyond triangulations, but I'm not sure if either has exactly what you're looking for. – Noah Snyder Sep 18 '13 at 14:31
• Thanks Noah! I'm not finding this there. For the sake of linking, here's Balsam-Kirillov I, II, and III and Turaev-Virelizier. – Scott Morrison Sep 19 '13 at 1:40
• I suspect the place to look is in the skein module literature of the late 1980's and early 1990's. – Kevin Walker Sep 19 '13 at 2:00