The String Coffee Table

Remember, otimes is not an operation in the derived category. Instead, its left derived functor is. This is usually denoted by an otimes with an L on top of it. You have to take projective resolutions to define it. (See Gelfand and Manin or Weibel).

Unfortunately, I’m too doped up on cold medication right now to think about the answers to your questions.

Many thanks indeed for your comment!

Have you figured this out yet, or should I cheat by looking up some stuff?

I haven’t figured out the answer to my original question. Instead I have changed the question until it had an anwer! ;-)

More seriously, I came to think that my original idea to get hold of a 2-algebra here was not the most natural one. I proceeded by the following steps:

We are looking for a categorification of the algebra of functions over some space taking values in some field $k$. Because that’s hard in general, we might start with the toy example of a discrete space $S$, which is just a finite set of points. A function on $S$ is a morphism $S \overset{f}{\to} k$ in $\mathrm{Set}$. Hence its categorification should be a morphism in $\mathrm{Cat}$, i.e. a functor. Hence we must lift $S$ to some category. This is naturally done by stringifying it and throwing in some arrows between the points of $S$. This yields a quiver $Q$, which I’ll identitfy with its path category $Q$.

Next we must find a suitable target category. You have argued in HDA2 for $\mathrm{Hilb}$, which is $\mathrm{Vect}$ with extra structure. BD&R have shown that this goes in the right direction, mathematically, and several ideas from string theory, too, support this as a good idea.

So let me consider a functor $Q \to Vect$ as the first approximation to the categorification of the algebra of functions in $k^S$. These form the functor category $(\mathrm{Vect}_k)^Q$ which is a nice 2-algebra.

However, when we decategorify this 2-algebra by taking isomorphism classses of objects we see that we get $\mathbb{N}$-valued functions on $S$. This is not quite what we wanted. Maybe we don’t need to require to really get $k$-valued functions, but we would like to divide and subtract somehow.

As I have argued, taking the derived category $\mathbf{D}(\mathrm{Vect}^Q)$ might be a good idea. When done right it introduces a notion of subtraction, but still not one of division. Hence we should not be content with $\mathbf{D}(\mathrm{Vect}^Q)$ as our candidate categorification of $k^S$.

I will, however, have good use for $\mathbf{D}(\mathrm{Vect}^Q)$ further below, but not as a categorification of a function algebra but as a candidate category of 2-section of a vector 2-bundle over $Q$!

This comes about as follows:

The state of affairs becomes maybe less murky once we use an equivalent reformulation of $\mathrm{Vect}^Q$. Namely this category is ‘the same’ (insert the proper adjective here) as the category $kQ-\mathrm{Mod}$ of (left,say) modules of the path algebra $kQ$ of the quiver Q.

But once we are talking about modules, there is an obvious notion of potentially invertible product that suggests itself, namely the tensor product $\otimes_{kQ}$ over $kQ$. Of course for that to work we need to switch from left modules to bimodues.

For this reason I am proposing that we should replace $kQ-\mathrm{Mod}$ by $kQ-\mathrm{Bimod}$, the category of bimodules over the quiver path algebra $kQ$.

I believe using $\otimes_{kQ}$ as the monoidal product this becomes a weak monoidal category. Some of the bimodules in here are ‘tilting elements’ meaning that they are weakly invertible. I believe there should hence be a subcategory of $kQ-\mathrm{Bimod}$ with these tilting elements as objects, which is in fact a weak 2-group. But I haven’t really checked the technical fine print. The decategorification of this subcategory however is well known as the Picard group of $kQ$.

So concerning our categorification program the step from $\mathrm{Vect}^Q \simeq kQ-\mathrm{Mod}$ to $kQ-\mathrm{Bimod}$ is much like the step from natural numbers to positive rational numbers, it seems.

Assuming this to be a good idea, we still need to throw in additive inverses. So let’s pull the derived category trick at this point and go from $kQ-\mathrm{Bimod}$ to its derived category $\mathbf{D}(kQ-\mathrm{Bimod})$.

The details of how to really ‘subtract’ in a triangulated derived category are still being worked out by Bridgeland and others, apparently, but I shall be content with the rough notion of subtraction that we have here. (BTW, is $\mathbf{D}(kQ-\mathrm{Bimod})$ triangulated when $\mathbf{D}(kQ-\mathrm{Mod})$ is?)

The product $\otimes_{kQ}$ now becomes the derived product $\overset{L}{\otimes}$ and with respect to this $\mathbf{D}(kQ-\mathrm{Bimod})$ is still monoidal. The weakly invertible objects are now of course called titling complexes. The isomorphism classes of these form what is called the derived Picard group of $kQ$.

There is a series of very nice papers, talks and lecture notes on this by Amnon Yekutieli.

So that’s currently my proposal:

A ‘good’ candidate for the categorification of the algebra $k^S$ of functions on a discrete space $S$ is the derived category $\mathbf{D}(kQ-\mathrm{Bimod})$ of bimodules of the path algebra of a quiver whose vertex set is $S$.

I also have some ideas on how to check that this makes good sense:

If $\mathbf{D}(kQ-\mathrm{Bimod})$ is a good ‘2-algebra’, then, in the spirit of 2-NCG/2-SQM, we should find that states of a categorified quantum mechanics are 2-sections of ‘line 2-bundles’ with respect to this 2-algebra, where a ‘line 2-bundle’ here is a ‘1-dimensional’ module of $\mathbf{D}(kQ-\mathrm{Bimod})$.

There is an obvious candidate for what such a section should look like ‘locally’, it should just be an object in $\mathbf{D}(kQ-\mathrm{Mod})$, since that’s what $\mathbf{D}(kQ-\mathrm{Bimod})$ naturally acts on via $\overset{L}{\otimes}$.

In other words, a line 2-bundle over $Q$ would be something that locally looks like $\mathbf{D}(kQ-\mathrm{Mod})$.

I have described this idea of how to define vector 2-bundles in more detail in the section ‘Vector 2-bundles’ of these unfinished notes.

The fun thing is, that this turns out to be essentially right! Objects of $\mathbf{D}(kQ-\mathrm{Mod})$ are indeed known to describe states in 2-quantum mechanics, namely states of strings attached to D-branes.

And the ‘gauge group’ acting on these line 2-bundles, under which our 2-SQM should be physically invariant, is nothing but the derived Picard (2-)group which, again, is indeed known to describe certain duality operations on string configurations.

The string theory side suggests that we might want to go one more step further and replace $\mathbf{D}(kQ-\mathrm{Bimod})$ somehow with derived categories of left/right modules with different path algebras acting on both sides. This is what describes (generalized) ‘Seiberg dualities’ aka tilting equivalences aka mutations in D-brane physics.

And also it is essentially known how everything I wrote down here for representations of quivers translates into categories of coherent sheaves over these Calabi-Yau.

On the other hand, the category-theoretic side here might suggest something about string theory: If the line 2-bundle picture that I have drawn here is to be taken seriuously, then it would follow that a state of D-branes with strings between them is only ‘locally’ in some sense an element of $\mathbf{D}(kQ-\mathrm{Bimod})$. Globally several such elements might have to be glued together, since these would be local 2-section, not 2-functions.

I was very lucky to get lots of feedback on this idea from Aaron. He, too, does entertain the idea that there should be some sort of twisting going on for string fields (related to elliptic cohomology) but doesn’t see why the line 2-bundle that I have sketched here is what should capture this.

Possibly he is quite right, and I will have to further think about this stuff.

I am however intrigued by how closely a 2-connection on a line 2-bundle looks like the enriched elliptic objects by Stolz and Teichner.

Namely, if we restrict attention to complexes concentrated in degree 0, then a 2-conneciton would essentially assign elements of the weak Picard 2-group to bigons, i.e. bimodules to edges and bimodule morphisms to surfaces. This is quite like an enriched elliptic object.

One difference is that in the picture sketched above we would naïvely use the ordinary tensor product for horizontal composition, while Stolz and Teichner instead use Connes fusion. As far as I understand, however, Connes fusion is like the ordinary tensor product but ‘corrected by a twist’, somehow, such to ensure that all modules in the game are at the same time Hilbert spaces.

That, too, is very briefly discussed in the above notes.