Morton on 2Vector Spaces and Groupoids
Posted by John Baez
My student Jeffrey Morton has come out with a paper based on his thesis:

Jeffrey Morton, 2vector spaces and groupoids.
Abstract: This paper describes a relationship between essentially finite groupoids and 2vector spaces. In particular, we show to construct 2vector spaces of Vectvalued presheaves on such groupoids. We define 2linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation — a weak functor — from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail. It has applications in constructing quantum field theories, among others.
Jeffrey Morton’s paper builds a 2functor
$\Lambda : 2Span(Gpd) \to 2Vect$
Here 2Span(Gpd) is the 2category of:
 finite groupoids,
 spans of finite groupoids, and
 (equivalence classes of) spans of spans of finite groupoids
while 2Vect is the 2category of:
 Kapranov–Voevodsky 2vector spaces,
 2linear maps between such 2vector spaces, and
 natural transformations between linear maps between such 2vector spaces.
Let me explain some of this stuff.
A (Kapranov–Voevodsky) 2vector space is a category equivalent to $Vect^n$ for some finite $n$, where $Vect$ is the category of finitedimensional vector spaces. A 2linear map between 2vector spaces is a functor that’s linear on homsets and preserves direct sums. More concretely, we can think of any linear map like this:
$f : Vect^n \to Vect^m$
as an $m \times n$ matrix of finitedimensional vector spaces. So, we’re doing categorified quantum mechanics: matrix mechanics with vector spaces replacing complex numbers! This is an old idea, promoted by Louis Crane.
From a finite groupoid $X$, Morton constructs the 2vector space
$\Lambda(X) = Vect^X$
whose objects are functors $\psi : X \to Vect$, and whose morphisms are natural transformations between these. Mathematicians should think of $\psi: X \to Vect$ as a representation of $X$, since that’s all it is when $X$ is a group. Physicists should think of $\psi : X \to Vect$ as a categorified ‘wavefunction’, since in quantum mechanics a wavefunction is a function $\psi : X \to \mathbb{C}$ where $X$ is a mere set.
From a span of finite groupoids:
$X \leftarrow S \rightarrow Y$
Morton constructs a 2linear map:
$\Lambda(S) : Vect^X \to Vect^Y$
This map takes any functor $\psi: X \to Vect$, pushes it forward from $X$ to $S$, and then pulls it back from $S$ to $Y$. This is one of the ‘pushpull’ constructions we see so often on this blog.
Finally, from a span of spans of groupoids — it’s tough for me to draw such a thing beautifully here, so look at the picture at the beginning of Section 5 of the paper — Morton constructs a natural transformation between linear maps between 2vector spaces. The interesting thing about this step of the construction is that it makes essential use of groupoid cardinality!
What’s the point of all this business? One point is that it lets Morton construct the Dijkgraaf–Witten model as an extended topological quantum field theory. For that, see his thesis and also the paper where he constructs a weak 2category nCob_{2} consisting of:
 compact $(n2)$manifolds,
 cobordisms between such manifolds,
 (equivalence classes) of cobordisms between cobordisms between such manifolds.
Given any finite group $G$, Morton gets a weak 2functor
$Z : n Cob_2 \to 2 Vect$
This is the untwisted Dijkgraaf–Witten model, viewed as an extended TQFT. He builds $Z$ in two stages. First he constructs a weak 2functor
$Z_0 : n Cob_2 \to 2Span(Gpd)$
Then he composes this with the weak 2functor I just described:
$\Lambda : 2 Span(Gpd) \to 2 Vect$
So, $\Lambda$ — short for linearization — is the algebraic essence of the Dijkgraaf–Witten model!
Cobordisms
Thanks for commenting on that. I should point out that the previous paper you linked to has a new version, since I decided to clarify part of it and it grew into a twopart story. The old version contains the cobordism stuff.
I should also point out the thing which this paper corrects that was flawed in my thesis, in that the construction of the natural transformation for a span of spans wasn’t properly defined before. There is some essential use of the groupoid cardinality, which was there before, but also some representation theory which I hadn’t dealt with properly. Specifically, it involves the representations of the various automorphism groups of the objects  it’s what degroupoidification into a vector space discards.