The n-Category Café

Sorry, “Shavarevich” was a typo for the famous algebraic geometer Shafarevich. I’ll fix that now.

Glad you’re enjoying it, and thanks for the suggestions/corrections!

I guess a lower-case $v$ would be nicer, so we could write $[\mathrm{Sv}]$ as the result of applying the span of groupoids $S$ to $v$, and get

$$[\mathrm{Sv}]=[S][v]$$

It’s also true that what really matters is the morphism $p:v\to X$, not $v$ itself — but this notation gets a bit tricky when we apply it to a span, or matrix, which has one object and two morphisms.

Probably Jim Dolan’s approach is best: we should think of each morphism as an ‘index’, in a categorified version of Penrose’s abstract index notation. Then a groupoid over $X$:

$$i:v\to X$$

can be written as a vector with one index:

$$v_i$$

Similarly, a span of groupoids:

$$i:S\to X,j:S\to Y$$

can be written as a matrix with two indices:

$$S_{ij}$$

And so on for higher-rank tensors, as David Corfield pointed out.

Summing over repeated indices is then our notation for taking weak pullbacks! And we don’t need to distinguish between upper and lower indices, due to the sneaky properties of groupoidification.

More abstractly, we can think of $v$ as an object in $[\mathrm{groupoids}\mathrm{over}X]$ and drop the index. Similarly, we can think of $S$ as an object in $[\mathrm{groupoids}\mathrm{over}X\times Y]$ and drop the indices.

So, we have both the usual mathematicians’ notation and the usual physicists’ notation available to us! Everything will look quite ordinary, but it’s all been groupoidified!

In my 2004 course notes I mention not only the fact you cite — that the groupoid of $n$-colored finite sets has cardinality $e^n$ — but also that the groupoid of $\frac{1}{2}$-colored finite sets has cardinality $e^{1/2}$, and that the groupoid of finite-set-colored finite sets has cardinality $e^e\approx 15.154$.

These are cute examples of a more general theory, explained in those notes.

So Lie $n$-groupoids come in symplectic flavour.

While I feel quite unsure about symplectic groupoids, I do understand symplectic Lie $n$-algebroids. These are just “NQP”-manifolds, i.e. graded manifolds with a nilpotent odd derivative and with a compatible graded non-degenerate Poisson structure.

We know that without the P, these NQ manifolds are nothing but Lie $n$-algebroids with weak Jacobi identity (partly because we really know, partly by definition).

I am still hoping that adding the “P” to these is nothing but also weakening the skew symmetry.

But maybe I should explain in more detail why I think so:

recall why it is that a Courant algebroid comes from a symplectic structure on a Lie algebroid:

consider the simple example where we live over a point, have just an ordinary Lie algebra $g$ in degree 1, and $ℝ$-woth of 2-ismorphisms on each such 1-morphism $${\wedge }^{•}(s{g}^{*}\oplus \mathrm{ss}{ℝ}^{*}).$$

Let $\{t^a\}$ be a basis of $s{g}^{*}$. Then a symplectic structure on this graded manifold is $$\omega ={\omega }_{\mathrm{ab}}d{t}^a\wedge d{t}^b,$$ where $d{t}^a$ denotes the graded exterior differential of the graded “coordinate” $t^a$.

The point is that with $t^a$ odd, $d{t}^a$ is even and hence this $\omega$ defines a symmetric bilinear structure on $g$!

Also, $\omega$ is required to be compatible with the differential (the “Q”). This makes it precisely an invariant degree 2 polynomial on the Lie algebra.

This way we find that the Courant algebroid over the point comes from a Lie algebra together with a bilinear invariant form on it.

In fact, in general, for $g_{(n)}$ any Lie $n$-algebroid, the symplectic structure should be nothing but a closed bilinear element in $\mathrm{inn}(g_{(n)}{)}^{*}$, otherwise known as the odd tangent bundle.

So we have a construction precisely as for Chern Lie $(2n+1)$-algebras, only that there this bilinear piece is regarded as a higher coherence of the Jacobiator.

I expect we can just reinterpret this Jacobi failure as a skew failure. Notice that the skew-symmetrizator $$S:[x,y]\stackrel{\sim }{\to }-[y,x]$$ does send two objects to an isomorphism starting at them.

So, in the dual picture, this could come from a degree -2 map $S^{*}$ which maps the canonical basis $\{b\}$ of $ss{ℝ}^{*}$ to the “symplectic form” $$S^{*}:b\mapsto {\omega }_{\mathrm{ab}}d{t}^a\wedge d{t}^b.$$

Indeed, that symplectic form is usually required to be of degree -2, in the BV context.

Well, sorry if that sounds incoherent. If only I were not busy with something else, I would sit down and try to clear this up. But maybe with these hints somebody else can see the Fata Morgana that I am talking about

At some point won’t you need to find an analogue of phased sets for groupoids? Or has this been done already?

I was thinking that a $G$-phased groupoid is a groupoid over $\mathrm{INN}(G)$.

First of all, a discrete groupoid over $\mathrm{INN}(G)$ is a $G$-phased set. Then, if two objects in the groupoid are related by the action of $g\in G$, then their $G$-phase should differ by that $g$. That’s exactly what happens for groupoids over $\mathrm{INN}(G)$.

I am enjoying the following combination of things I have been thinking about in the context of tangent categories and $n$-curvature, and the Tale of Groupoidification:

we may start with a $G$-bundle with connection, presented by a functor $$\mathrm{tra}:P_1(X)\to \Sigma G.$$ But we then realize that we rather ought to be looking at its curvature $$\delta \mathrm{tra}:{\Pi }_2(X)\to \mathrm{Grp}\downarrow \Sigma G.$$ This sends each point $x$ in $X$ to the groupoid $\mathrm{INN}(G)$ regarded as a groupoid over $\Sigma G$. By the Tale, I may think of this as describing a connection on a (trivial) bundle whose fibers are like a certain vector space with a $G$-action.

(Think for instance about $G={\mathbb{Z}}_n$ as an approximation to $G=U(1)$).

What’s interesting now is that a section of $\delta \mathrm{tra}$ thought of as a morphism into $\delta \mathrm{tra}$ is

- over each point $x\in X$ a groupoid over $\mathrm{INN}(G)$, hence a “$G$-phase groupoid”.

Again, if I translate this to the Tale using $G={\mathbb{Z}}_n$ it seems to say that over each point the section is (the approximation to) a complex number!

So it looks like by combining the curvature technique with the tale, we automatically turn principal $U(1)$-bundles into line bundles, whose sections are, locally, complex numbers over each point.

David Corfield wrote:

At some point won’t you need to find an analogue of phased sets for groupoids? Or has this been done already?

Jeffrey Morton introduced phased groupoids under the name of ‘$\mathrm{U}(1)$–groupoids’ in his paper Categorified algebra and quantum mechanics — see Section 6.1.2, page 46.

Then you could have things like spans of phased groupoids as unitary operators.

Exactly!

As you know, Jim Dolan and I introduced a theory of ‘stuff types’ generalizing Joyal’s ‘species’ so we could categorify the quantum harmonic oscillator and the theory of Feynman diagrams. The big problem with our work was that it didn’t include the phases needed to describe the unitary time evolution of the free harmonic oscillator!

This is what Jeffrey did in his paper, using ‘$\mathrm{U}(1)$-stuff types’ and ‘$\mathrm{U}(1)$-stuff operators’.

Everyone should read this paper! It explains this business very gently, with lots of pictures. I’ll give an ultra-terse summary now, but only to tempt people into reading the real story.

Now: having decided that the term ‘$\mathrm{U}(1)$-set’ is a really lousy term for a set equipped with a map to $\mathrm{U}(1)$, and having switched to calling such a thing a phased set, let’s be consistent in adopting this new style of terminology.

So: we hereby define a phased groupoid to be a groupoid equipped with a functor to the groupoid with $\mathrm{U}(1)$ as its set of objects and only identity morphisms.

In other words: a phased groupoid is a groupoid such that each object is labelled by a phase (an element of $\mathrm{U}(1)$), and isomorphic objects are labelled by the same phase.

A phased set is then a phased groupoid with only identity morphisms.

A phased stuff type is a phased groupoid whose underlying groupoid is equipped with a functor to the groupoid of finite sets.

Jeffrey shows how every phased stuff type gives a (possibly non-normalizable) state of the quantum harmonic oscillator.

A phased stuff operator is a phased groupoid whose underlying groupoid G is part of a span:

                       G
                      / \
                     /   \
                    /     \
                   v       v
                  B         B

where B is the groupoid of finite sets.

Jeffrey shows how every phased stuff operator gives a (possibly unbounded) operator on the Hilbert space for the quantum harmonic oscillator.

And using this, he shows how to categorify the theory of Feynman diagrams as a tool for computing time evolution of the perturbed quantum harmonic oscillator!

I hope you see that this is all part of the Tale of Groupoidification…

So how does the counting up go? In the case of phased sets, we add up the phases.

Right — you’ll find this idea nicely explained in Feynman’s popular book QED. A set of possible ways for a quantum system to get from here to there is really a phased set. To compute the amplitude for it to get from here to there, we just compute the cardinality of the phased set — that is, add up the phases. The cardinality of a finite phased set can be any complex number.

But, this generalizes nicely to phased groupoids.

How do we add up the phases in, say, a $\mathrm{U}(1)$-phased groupoid?

I’m sure you can guess! Or, read the material in Jeff’s paper leading up to the answer in Section 6.2.1, page 50.