The n-Category Café

Hi Jamie,

Thank you for reading my paper.

The big theorem of this topic (Gabriel’s Theorem) says that the representation category of a quiver is of finite type if and only if the quiver is simply laced. I should definitely be more specific about why I can take the cardinality of this set and expect it to be finite (i.e. that I am using isomorphism classes).

As far as generalizing to non-simply laced quivers, It would take a different approach since the important sets here would not be finite. This would definitely be another direction to go. If you have every studied Lie theory, you will remember that the representation theory of lie algebras that are not simple becomes very difficult, and in many instances not even understood yet.

As for the two different forms of P, I have been going back and forth on this one, and may end up at a cardinality notion like you suggested in the end. There are some notational conflicts in the groupoidification program that I was trying to avoid.

Let’s not have a heated debate! Everyone should simply agree that there’s no universally-agreed orientation.

Some people have their inputs at the bottom and work upwards; others do the opposite. Sometimes I have my inputs on the left and work rightwards. I’m sure there are situations in which I’d want to have the inputs on the right and work leftwards.

Different people do different things at different times, and that’s all there is to it.

Hi “other” Chris,

Thank you for the catches on typos.

I think subconsciously I use “problematic” because I think anything that is not compatible is a problem. I may re-word it to sound less like an opinion.

As my wife the sociologist would say, diagram direction is just the oppressive right-handed society trying to keep us poor left-handed folk down. I mean they get to be called “right”-handed like there’s something “wrong” with the other hand. :)

In all seriousness, I will remove the reference to “standard” practice here, as there seems to be a difference of opinions.

Chris wrote:

A small comment/question: You say a few times in the introduction that the incompatibility between the multiplication and comultiplication is a “problematic feature”. Why should I think of this as a problem, exactly?

A bialgebra is a beautiful thing: it’s a monoid in the category of comonoids in $Vect$ — or equivalently, a comonoid in the category of monoids in $Vect$ Thanks to this elegant definition, the power of category theory kicks in, and we get all sorts of wonderful results. For starters, the category of representations of any bialgebra is a monoidal category.

But with the usual approach to Hall algebra, instead of a bialgebra we get something that’s sort of close to a bialgebra, but where the compatibility condition fails, due the annoying intrusion of a mysterious ‘fudge factor’. People call this gadget a ‘twisted bialgebra’, but that’s just jargon to summarize a mystery: this gadget doesn’t fit into any clear pattern.

At least that’s what people usually say! But Christopher has shown that in fact this gadget is a bialgebra object — only not in $Vect$, but in some other braided monoidal category!

More precisely, he takes the Grothendieck group of the category of representations of a simply-laced Dynkin quiver, say $K$, and puts an interesting braiding on the category of $K$-graded vector spaces, say $Vect^K$.

Then Christopher shows the Hall algebra is a bialgebra object in $Vect^K$. In other words, it’s a monoid object in the category of comonoid objects in $Vect^K$ — or equivalently, a comonoid object in the category of monoid objects in $Vect^K$.

So, the power of category theory kicks in again! Now we can easily do all sorts of wonderful things with the Hall algebra — things we’d normally do with a bialgebra, but now with $Vect^K$ taking the place of $Vect$.

For example, now we instantly know that the Hall algebra has a monoidal category of representations in $Vect^K$. Back when it was a mere ‘twisted bialgebra’, this would be utterly nonobvious: even if you were enough of a genius to guess it, you’d have to check it by playing around with ‘fudge factors’ and discovering ‘surprising cancellations’. But now it’s an automatic consequence of general facts.

Of course, some people enjoy all sorts of ‘twisted’, ‘deformed’, or ‘warped’ mathematical gadgets. Some people enjoy complexity for its own sake, because it offers scope for cleverness. But simplicity is better, because it lets you do good things without cleverness. So it’s always best when you can take a ‘twisted’ gadget and reinterpret it as a plain old fully functional gadget in another category.

Hi Theo,

The point of the paper is the bialgebra compatibility condition. The existence of an antipode was simply a nice addition point. You are correct that the definition of an antipode does not require information about the braiding, but I have never thought about antipodes independently. I have always thought of a Hopf algebra as a bialgebra with an antipode, and so this was what I meant in the last sentence.

We can certainly define two kinds of bialgebra in a braided monoidal category, depending on which string crosses over which in this picture:

Alternatively, we can pick our favorite way, use that to define our favorite kind of bialgebra, and note that any braided monoidal category has a kind of ‘opposite’, where we use this new braiding:

$$B^{opp}_{X,Y} = B^{-1}_{Y,X}$$

Then the other kind of bialgebra can be thought of as our favorite kind of bialgebra, but in the ‘opposite’ braided monoidal category.

Of course the terminology gets a bit confusing here. We can take the opposite of a category by reversing the arrows, we can take the ‘opposite’ of a monoidal category by reversing the tensor product:

$$X \otimes_{opp} Y = Y \otimes X$$

and we can take the opposite of a braided monoidal category by reversing the braiding. We can even combine these different options, for a total of $2^3$ choices. That’s because a braided monoidal category is a 3-category, so its morphisms can be drawn as 3-dimensional globes, which have a reflection symmetry group of size $2^3$.

But when you said “multiple” distinct notions of bialgebra, did you mean more than two?

I used to know whether you could take a monoid $X$ in a braided monoidal category, with multiplication

$$m: X \otimes X \to X \, ,$$

and define a new monoid with a “$n$-tuply twisted multiplication”:

$$m_{new} = m \circ B_{X,X}^n$$

The trick is checking the associative law. I forget the answer.

I don’t think I ever had the brains to check whether a braided monoidal category gives an infinite series of braided monoidal categories where we braid things around a bunch of times, e.g.:

$$B^{new}_{X,Y} = B_{X,Y} B_{Y,X} B_{X,Y}$$

If the answer to this question were “yes”, then the answer to the previous question would also be “yes”. Furthermore, we would get an infinite series of interesting concepts of bialgebra in a given monoidal category.

I’m too busy to draw the string diagrams needed to check this! But if I had to go by gut instinct, I’d guess the answer is “no”.

It’s true. Christopher should cite Majid’s work when introducing the concept of a Hopf algebra in a braided monoidal category.

I also told Christopher to read and cite Joyal and Street’s paper where they get a braiding on the category of $A$-graded vector spaces from a bilinear form on the abelian group $A$. It’s one of these, or perhaps both:

• A. Joyal and R. Street, Braided monoidal categories, Macquarie Math Reports 860081 (1986). Available at http://www.maths.mq.edu.au/~street/JS1.pdf.
• A. Joyal and R. Street, Braided tensor categories, Adv. Math. 102 (1993) 20-78.

Hmm, yes, it’s Theorem 12 together with Proposition 13 on page 44 of the first, unpublished, paper. Better to cite the second one if it’s also in there — I don’t have it on me right now.

As usual, they prove this result in such great generality that it’s a bit hard to understand at first. But it’s not hard to extract the desired fact from what they prove.

An interesting fact is that Theorem 12 goes back to the work of Eilenberg and Mac Lane on the cohomology of groups. All this stuff can now be understood in terms of Postnikov towers for $n$-groupoids.

Carl - I’m coming to this thread late, but your paper on path integrals via mutually unbiased bases actually looks decent to me. I’m curious, did you end up at a grad school?