The n-Category Café

I really do NOT want to distract from the project, which fascinates me: “categorify the theory of quantum groups.”

But I am moved to say that you may have answered questions literally in the Voir Dire (questions to potential jurors). It is more fun if you first consider the question/answer system metalinguistically, and generate word strings which increase your probability of actually being on a Jury.

There was a murder trial that looked entertaining, so I resolved th give the minimal answers that were technically true (I was under oath) but most likely to lead judge and attorneys away from seeing where I really was in mental state space.

Lawyer: “Do you have a college degree?”

JVP: “Yes.” [thinking many, many, you specialist fool]

Lawyer: “what degree?” [falsely assuming one and only one]

JVP: “English.” [not mentioning that it was a B.S. from Caltech simultaneously with a B.S. in Math (advanced Logic), followed by an M.S. in Computers (Cybernetics and AI), and then…]

Lawyer: “Why?”

JVP: “I like to read.”

So I slip through the net intended to catch and throw back into the pool those actually intelligent, and capable of critical thought.

I become Jury Foreman. The trial takes a day. The jury argues for 4 more days, and keeps sending questions to the judge. I hang the jury on Murder (the burden of proof was lacking, as was motive, and a murder weapon). I lead them to acquit on Attempted Murder (as they failed to prove that the alleged victim of that even existed, let alone was in California at the time of the manifestly fatal shooting of the alleged murder victim, let alone at the scene of the crime).

A man walked free (or, I think, back to his cell where he was serving time for something else, as the outfit and handcuffs and leg shackles suggested) – a very scary man whom I suspected of being a stone cold killer – and he gave me a huge smile and thumbs up. Because the State had failed their duty, and I followed mine.

I had an extensive discussion afterwards with both Prosecution and Defense attorneys.

They: “How did you get the Jury to ask such tricky questions about fingerprints, testimony of plea-bargained crack dealer, temperature of muzzle in firing those particular bullets, and so forth?

JVP: You were sloppy in Voir Dire. I have other degrees, and my English degree led me to be an Active Member of Mystery Writers of America.

They: “A damned Mystery Writer. We’ll never let THAT happen again. As it turns out, Prosecution lost a Motion in Limine, so the jury never got to hear that this was a crack deal gone bad, in which Defendant shot a dealer from a rival gang WHO HAD PREVIOUSLY SHOT DEFENDANT’S BROTHER.”

JVP: “I strongly urge you to try a second time to convict that guy. But this time, get your ducks in a row first. The Jury found the narrative too thin. They wanted to convict the guy, whom they agreed LOOKED like a killer. But you need to provide a REASON for his actions, and a better presentation of the circumstantial evidence which, better packages, could and should lead to conviction.”

They: “You didn’t ever go to Law School, or did you…?”

JVP: “No,” [not knowing that my son would, at age eighteen, be in USC Law School]. “Math and Physics are hard. Law is easy. So do your homework, and try again.”

Again, sorry for the digression, but Jury Duty can be a spectacular way to go beyond naive philosophical and sociological theory and grapple with the multidimensional writhing knotted paradoxical glory of The Human Condition.

Now, back to Quantum Groups – which, naively speaking, are not quite Groups as the name suggests….

The witness may now step down.

I want to ask the following question somewhere in public. It is closely related to the content of the Geometric Representation Theory Seminar, if not to this particular session.

Question. For $G$ some $n$-group, $n \in \mathbb{N}$, and $\mathbf{B} G$ the corresponding 1-object $n$-groupoid; how do we characterize $n$-functors $$p : C \to \mathbf{B}G$$ which come from $C$ being the action $n$-groupoid of $G$ acting on something?

For $n=1$ we know we demand the functor $p$ to be faithful. What’s the corresponding condition for higher $n$?

From Lie $\infty$-algebraic considerations # I have the suspicion that a particularly elegant answer to this should be possible using $\mathbb{Z}$-groupoids: I expect that $\mathbb{Z}$-groupoids $C$ with morphisms $$p : C \to \mathbf{B}G$$ which are injective on positive degree Hom-spaces can be nicely related to actions of $G$ on $k$-things, the $k$-thing sitting in degrees $0, -1, \cdots, -k$ in $C$.

Notice that for the ordinary action of a group on a set (a 0-thing), this means $C$ has that thing in degree 0 and the group faithfully in degree 1, which is indeed the case.

I know from playing some Lie $\infty$-algebraic tricks that something like this is true.

I expect it is too much to hope that the answer to the above question is already known (but am prepared for pleasant surprises). But since I think it is a good and important question, somebody should think about it.

And clearly, this will become relevant to the issue Geometric Representation Theory as soon as somebody begins trying to groupoidify groupoidification, in order to describe 2-vector spaces combinatorially.

So it’s of relevance here. And in some other contexts, too, I think.

Maybe some of you would enjoy this puzzle. (Others may already know the answer; if so don’t spoil it.)

Take the $A_3$ quiver:

$$\bullet \longrightarrow \bullet \longrightarrow \bullet$$

A representation of this quiver is a pathetically simple thing, just a gadget like this:

$$V_1 \stackrel{f_1}{\longrightarrow} V_2 \stackrel{f_2}{\longrightarrow} V_3$$

consisting of 3 vector spaces and 2 linear maps.

There’s an obvious notion of ‘direct sum’ for such gadgets: the direct sum of

$$V_1 \stackrel{f_1}{\longrightarrow} V_2 \stackrel{f_2}{\longrightarrow} V_3$$

and

$$W_1 \stackrel{g_1}{\longrightarrow} W_2 \stackrel{g_2}{\longrightarrow} W_3$$

is

$$V_1 \oplus W_1 \stackrel{f_1 \oplus g_1}{\longrightarrow} V_2 \oplus W_2 \stackrel{f_2 \oplus g_2}{\longrightarrow} V_3 \oplus W_3$$

Puzzle: How many such gadgets are there that aren’t direct sums of other such gadgets, except in a trivial way?

These are called indecomposable representations of the $A_3$ quiver.

Just a comment about finding the indecomposable representations - one can do this quite easily for Dynkin quivers of types ADE (and for many others too) using the Auslander-Reiten theory (refer: Elements of representation theory of associative algebras by Simson, Assem and Skowronski).