I think I can make a guess as to how these approaches are connected. I’m going to describe how I *think* it should all work out, but I’ve by no means worked through all the details, so let me know where I get it wrong.

An important category in JM’s paper is the 2-category SGpd of spans of groupoids, which is defined with

objects as groupoids;

morphisms as spans of groupoids, i.e. triples $(F,F_A,F_B):A\rightarrow B$, where $F$ is a groupoid, and $F_A:F\rightarrow A$ and $F_B:F\rightarrow B$ are functors;

2-morphisms as maps of spans of groupoids, i.e. a 2-morphism $m:(F,F_A,F_B)\Rightarrow(G,G_A,G_B)$ is a functor $m:F\rightarrow G$ between groupoids $F$ and $G$, such that $G_B \circ m=F_B$ and $G_A \circ m=F_A$.

I’m going to concentrate on the 1-categorical structure of SGpd, ignoring the 2-morphisms. They’re by no means unimportant, but JV’s paper is entirely 1-categorical, and we’ll be able to do without them. I would love to know how they enter the scene!

The only thing that’s not obvious about the category SGpd — for now, just considered as a 1-category — is what to do about composition of morphisms. We do this using pullbacks. Given $(F,F_A,F_B):A\rightarrow B$ and $(G,G_B,G_C):B\rightarrow C$, then the pullback of $F_B$ and $G_B$ gives a pair of functors $P_F:P\rightarrow F$ and $P_G:P \rightarrow G$; we then define $(G,G_B,G_C) \circ (F,F_A,F_B) = (P,F_A \circ P_F, G_C \circ P_G)$.

So, what properties does SGpd have? First of all, it’s a $\dagger$-category: given any $(F,F_A,F_B):A\rightarrow B$, we can easily obtain $(F,F_A,F_B) ^\dagger = (F,F_B,F_A):B \rightarrow A$. It also has products and coproducts, which coincide to give $\dagger$-biproducts; for groupoids $A$ and $B$, their biproduct is given by $A+B$, the coproduct in Gpd. Weirdly, the product structure in Gpd has not survived to SGpd, so we explicitly retain it by putting a symmetric monoidal structure on SGpd: the tensor product of two morphisms $(F,F_A,F_B) \otimes (G,G_C,G_D)$ is given by the product of the defining functors in Gpd, i.e., $(F \times G, F_A \times G_C, F_B \times G_D)$.

So, it seems to me that SGpd is a symmetric monoidal $\dagger$-category with $\dagger$-biproducts. But this is exactly what JV’s paper requires to construct a model of the harmonic oscillator!

For every groupoid $G$, we have to construct a Fock space groupoid $F(G)$ over it. (F will stand for ‘Fock space’ from now on.) Among other things, we need all of these Fock space groupoids to carry a canonical commutative monoid structure. So, defining 1 to be the trivial groupoid with only one morphism, what could $F(1)$ be? It surely has to be FinSet${}_0$! A naive answer would be the groupoid $1+1+1+...$, but my guess is that would be associated to the free *non*commutative monoid over 1. So, does this work? Does $\mathrm{FinSet}_0$ deserve to be called the ‘free commutative monoid groupoid over 1’?

So, what’s a state of $F(1)$? It’s a morphism $(\Phi, \Phi_1, \Phi_{F(1)}) : 1 \rightarrow F(1)$, where 1 is the monoidal unit object. Since $1$ is the terminal object in Gpd, $\Phi_1$ is unique, and so specifying this state amounts to choosing a functor $\Phi_{F(1)} : \Phi \rightarrow F(1) \simeq \mathrm{FinSet}_0$ — in other words, a stuff type! Raising and lowering operators should emerge using JV’s prescription, and should turn out to be just the same as the raising and lowering stuff operators defined in JM’s paper.

As far as this goes, though, the fact that we started with the category of groupoids doesn’t seem important. If groupoids are more important than any other sort of algebraic structure, then there must be some special reason to use them. Something to do with groupoid cardinality, perhaps? What would happen if we used the category of spans of categories, rather than spans of groupoids?

## Re: Categorifying Quantum Mechanics

I think I can make a guess as to how these approaches are connected. I’m going to describe how I

thinkit should all work out, but I’ve by no means worked through all the details, so let me know where I get it wrong.An important category in JM’s paper is the 2-category SGpd of spans of groupoids, which is defined with

I’m going to concentrate on the 1-categorical structure of SGpd, ignoring the 2-morphisms. They’re by no means unimportant, but JV’s paper is entirely 1-categorical, and we’ll be able to do without them. I would love to know how they enter the scene!

The only thing that’s not obvious about the category SGpd — for now, just considered as a 1-category — is what to do about composition of morphisms. We do this using pullbacks. Given $(F,F_A,F_B):A\rightarrow B$ and $(G,G_B,G_C):B\rightarrow C$, then the pullback of $F_B$ and $G_B$ gives a pair of functors $P_F:P\rightarrow F$ and $P_G:P \rightarrow G$; we then define $(G,G_B,G_C) \circ (F,F_A,F_B) = (P,F_A \circ P_F, G_C \circ P_G)$.

So, what properties does SGpd have? First of all, it’s a $\dagger$-category: given any $(F,F_A,F_B):A\rightarrow B$, we can easily obtain $(F,F_A,F_B) ^\dagger = (F,F_B,F_A):B \rightarrow A$. It also has products and coproducts, which coincide to give $\dagger$-biproducts; for groupoids $A$ and $B$, their biproduct is given by $A+B$, the coproduct in Gpd. Weirdly, the product structure in Gpd has not survived to SGpd, so we explicitly retain it by putting a symmetric monoidal structure on SGpd: the tensor product of two morphisms $(F,F_A,F_B) \otimes (G,G_C,G_D)$ is given by the product of the defining functors in Gpd, i.e., $(F \times G, F_A \times G_C, F_B \times G_D)$.

So, it seems to me that SGpd is a symmetric monoidal $\dagger$-category with $\dagger$-biproducts. But this is exactly what JV’s paper requires to construct a model of the harmonic oscillator!

For every groupoid $G$, we have to construct a Fock space groupoid $F(G)$ over it. (F will stand for ‘Fock space’ from now on.) Among other things, we need all of these Fock space groupoids to carry a canonical commutative monoid structure. So, defining 1 to be the trivial groupoid with only one morphism, what could $F(1)$ be? It surely has to be FinSet${}_0$! A naive answer would be the groupoid $1+1+1+...$, but my guess is that would be associated to the free

noncommutative monoid over 1. So, does this work? Does $\mathrm{FinSet}_0$ deserve to be called the ‘free commutative monoid groupoid over 1’?So, what’s a state of $F(1)$? It’s a morphism $(\Phi, \Phi_1, \Phi_{F(1)}) : 1 \rightarrow F(1)$, where 1 is the monoidal unit object. Since $1$ is the terminal object in Gpd, $\Phi_1$ is unique, and so specifying this state amounts to choosing a functor $\Phi_{F(1)} : \Phi \rightarrow F(1) \simeq \mathrm{FinSet}_0$ — in other words, a stuff type! Raising and lowering operators should emerge using JV’s prescription, and should turn out to be just the same as the raising and lowering stuff operators defined in JM’s paper.

As far as this goes, though, the fact that we started with the category of groupoids doesn’t seem important. If groupoids are more important than any other sort of algebraic structure, then there must be some special reason to use them. Something to do with groupoid cardinality, perhaps? What would happen if we used the category of spans of categories, rather than spans of groupoids?