So, let’s see. There’s a lot in what John was telling us about the categorified Heisenberg algebra that I’d like to slowly sort through, some of it tantalizingly related to stuff in the neighborhood of what I was just discussing.

To each span of groupoids (here we’ll always mean locally finite groupoids, like $\mathsf{S} = \mathrm{core}(\mathsf{FinSet})$) we can associate a bimodule = profunctor. Here spans of groupoids are composed not by taking ordinary pullbacks, but by taking homotopy pullbacks, which are basically isocomma category constructions. (This would hold also for spans between spans, I reckon, if and when we want to use those.)

To every span (of groupoids) $X \overset{f}{\leftarrow} U \overset{g}{\to} Y$ there is a corresponding bimodule from $X$ to $Y$, by forming the bimodule composite of $X(f-, -): X \nrightarrow U$ followed by $Y(-, g-): U \nrightarrow Y$. As I mentioned a few comments back in this thread, this is a right adjoint $f^\ast$ followed by a left adjoint $g_\ast$. The moral is that we are describing a 2-functor

$\mathrm{Span} \to \mathrm{Prof}(\mathsf{Gpd})$

from on the one hand groupoids, spans, and (what I shall call) maps between spans, to on the other hand groupoids, profunctors = bimodules between groupoids, and transformations between profunctors. (This latter is quite a nice example of a cartesian bicategory, but maybe put a pin through that for the moment.)

As part of this, it’s illuminating to consider on the one hand that every span as above is a composite of a right adjoint span followed by a left adjoint span,

$(X \overset{f}{\leftarrow} U \overset{1_U}{\to} U)\; ; \; (U \overset{1_U}{\leftarrow} U \overset{g}{\to} Y).$

This breaks down into a series of exercises, one of which is to check that the span composite of these two, as computed using the homotopy pullback, really does give back the original span. Actually, I think one needs to be a little bit careful in saying this. Literally speaking, there’s an *equivalence*, not an isomorphism, from $U$ to the apex of the homotopy pullback of $1_U$ with itself. I think maybe what we need to do here is consider that if $\mathrm{Span}$ is to be considered a bicategory (or eventually a symmetric monoidal bicategory, but let’s not jump too far ahead at the moment) then the 2-cells should not be literally maps between spans, but instead isomorphism classes of maps. Of course this is more or less the familiar idea of taking the “homotopy category” of a 2-category.

Another exercise to check is the basic adjunction

$(U \overset{1_U}{\leftarrow} U \overset{f}{\to} X) \dashv (X \overset{f}{\leftarrow} U \overset{1_U}{\to} U).$

Thus, consider a profunctor $\Phi: G \to \mathsf{Set}^{H^{op}}$. There may be more than one “$H$-parametrized” Grothendieck construction attached to such a profunctor, but the one I want to consider is where you take a comma category of the Yoneda embedding $y_H: H \to \mathsf{Set}^{H^{op}}$ toward $\Phi$. In other words, form the category whose objects are triples $(g \in G, h \in H, y(h) \to \Phi(g))$ and where morphisms are pairs $g \to g', h \to h'$ that behave in the expected way for a comma category.

So in fact we get out of this a span of groupoids

$G \overset{p}{\leftarrow} (y_H \downarrow \Phi) \overset{q}{\to} H$

where $p, q$ are the two projection functors out of the comma category, and the original profunctor $\Phi$ is isomorphic to the bimodule composite of $p^\ast$ followed by $q_\ast$, where $p^\ast$ denotes the right adjoint bimodule/profunctor $G(p-, -)$, and $q_\ast$ denotes the left adjoint bimodule $H(-, q-)$.

So just as in $\mathrm{Span}$, every 1-cell $\Phi$ in $\mathrm{Prof}(\mathsf{Gpd})$ can be considered canonically isomorphic to a composite of a right adjoint followed by a left adjoint.

Now, suppose instead that we have the other order, where a left adjoint bimodule $H(-, f-): G \to H$ is followed by a right adjoint bimodule $H(g-, -)$, where $f, g$ are functors

$G \overset{f}{\to} H \overset{g}{\leftarrow} K.$

A Yoneda-based calculation is that the bimodule composite $G \to K$ of these two is $H(g-, f-)$ — that should come as no surprise! But what happens when we compute its Grothendieck construction?

Answer: it’s equivalent to the homotopy pullback, i.e., the isocomma category $g \downarrow f$!

Anyway, this is what Joe and I were discussing a while back, as part of my general take on this Mackey business and what it has to do with Young symmetrizers. I also think it might be some good infrastructure for following Morton-Vicary.

I’ll stop here for the moment on that note. I haven’t even gotten to the categorified Heisenberg commutation relation which was the leitmotif of yesterday’s conversation!

## Re: Spans and the Categorified Heisenberg Algebra

Just a quick note that the link to your slides seems to be broken.