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February 20, 2024

Spans and the Categorified Heisenberg Algebra

Posted by John Baez

I’m giving this talk at the category theory seminar at U. C. Riverside, as a kind of followup to one by Peter Samuelson on the same subject. My talk will not be recorded, but here are the slides:

Abstract. Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from ‘spans’, where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation. More recently, Khovanov introduced a ‘categorified’ Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The meaning of these new relations was initially rather mysterious, at least to me. However, Jeffery Morton and Jamie Vicary have shown that these, too, have a nice interpretation in terms of spans.

I feel like reviving interest in Morton and Vicary’s approach to Khovanov’s categorified Heisenberg algebra, because they shed a lot of light on its combinatorial underpinnings—which may in turn may shed new light on quantum physics, but only if someone works to dig deeper!

Some of Morton and Vicary’s work remains conjectural, namely that there’s a bicategory Span(FinGpd)\mathbf{Span}(\mathbf{FinGpd}) of

  • locally finite groupoids
  • spans of such
  • (equivalence classes of) span of spans of such

and a 2-functor from this to the 2-category 2Vect\mathbf{2Vect} of

  • 2-vector spaces
  • exact \mathbb{C}-linear functors
  • natural transformations

which preserves direct sums of morphisms and also sums of 2-morphisms. This relates two popular approaches to categorifed linear algebra.

Rune Haugesang has studied higher categories of iterated spans, and his technology could perhaps to be used to give an elegant construction of the categorified Heisenberg algebra following the ideas in Morton and Vicary’s work. But I’m not aware of anyone actually having done this! So if someone has gone further with Morton and Vicary’s ideas, please let me know. And if nobody has… give it a try, I think it will be worthwhile!

Posted at February 20, 2024 10:51 PM UTC

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Re: Spans and the Categorified Heisenberg Algebra

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

Posted by: Evan Patterson on February 21, 2024 12:31 AM | Permalink | Reply to this

Re: Spans and the Categorified Heisenberg Algebra


Posted by: John Baez on February 21, 2024 4:59 AM | Permalink | Reply to this

Re: Spans and the Categorified Heisenberg Algebra

I gave a warmup version of this talk to Todd Trimble and Joe Moeller, since we’ve been working on our second paper on the 2-vector space of Schur functors, which is closely related to Khovanov’s work. Let me summarize:

The 2-vector space Schur\mathsf{Schur} is the category of ‘finite representations’ of the groupoid S\mathsf{S} of finite sets. A finite representation of a groupoid G\mathsf{G} is a functor

F:GVect F: \mathsf{G} \to \mathsf{Vect}

that send every object to a finite-dimensional vector space, and all but finitely many isomorphism classes of objects to zero-dimensional vector spaces.

Indeed, the 2-functor mentioned in my post — the 2-functor

FinRep:FinSpan(FinGpd)2Vect \mathrm{FinRep}: \mathbf{FinSpan}(\mathbf{FinGpd}) \to \mathbf{2Vect}

that sends

  • locally finite groupoids (that is, groupoids with finite homsets)
  • spans of such
  • (equivalence classes of) span of spans of such


  • 2-vector spaces
  • exact \mathbb{C}-linear functors
  • natural transformations

starts precisely by sending each locally finite groupoid to its category of finite representations! Then comes the fun: how a span gets sent to an exact functor by a push-pull construction.

A map gives a span. So, the functor

+1:SS - + 1 : \mathsf{S} \to \mathsf{S}

gives a span of locally finite groupoids, and thus a exact \mathbb{C}-linear functor

A:SchurSchur \mathbf{A}: \mathsf{Schur} \to \mathsf{Schur}

This is called the annihilation operator. But we can also turn this span around and get another exact \mathbb{C}-linear functor

A :SchurSchur \mathbf{A}^\dagger: \mathsf{Schur} \to \mathsf{Schur}

called the creation operator. These two functors are both left and right adjoint to each other, by general abstract nonsense. Those adjunctions give a bunch of natural transformations — two units and two counits. And Khovanov discovered a bunch of equations these obey!

But before getting to Khovanov’s equations, there are a lot of other things to say. And Todd has given me permission to quote some of his thoughts, so I’ll do that in the next comment.

Posted by: John Baez on February 22, 2024 11:54 PM | Permalink | Reply to this

Re: Spans and the Categorified Heisenberg Algebra

Todd wrote:

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 S=core(FinSet)\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) XfUgYX \overset{f}{\leftarrow} U \overset{g}{\to} Y there is a corresponding bimodule from XX to YY, by forming the bimodule composite of X(f,):XUX(f-, -): X \nrightarrow U followed by Y(,g):UYY(-, g-): U \nrightarrow Y. As I mentioned a few comments back in this thread, this is a right adjoint f *f^\ast followed by a left adjoint g *g_\ast. The moral is that we are describing a 2-functor

SpanProf(Gpd)\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,

(XfU1 UU);(U1 UUgY).(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 UU to the apex of the homotopy pullback of 1 U1_U with itself. I think maybe what we need to do here is consider that if Span\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

(U1 UUfX)(XfU1 UU).(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 Φ:GSet H op\Phi: G \to \mathsf{Set}^{H^{op}}. There may be more than one “HH-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:HSet H opy_H: H \to \mathsf{Set}^{H^{op}} toward Φ\Phi. In other words, form the category whose objects are triples (gG,hH,y(h)Φ(g))(g \in G, h \in H, y(h) \to \Phi(g)) and where morphisms are pairs gg,hhg \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

Gp(y HΦ)qHG \overset{p}{\leftarrow} (y_H \downarrow \Phi) \overset{q}{\to} H

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

So just as in Span\mathrm{Span}, every 1-cell Φ\Phi in Prof(Gpd)\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):GHH(-, f-): G \to H is followed by a right adjoint bimodule H(g,)H(g-, -), where f,gf, g are functors

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

A Yoneda-based calculation is that the bimodule composite GKG \to K of these two is H(g,f)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 gfg \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!

Posted by: John Baez on February 23, 2024 12:20 AM | Permalink | Reply to this

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