The n-Category Café

Journal club post by Alex Hoffnung

Hi-

I have not been asked to write a “book report” since 3rd or 4th grade. Somehow this was more difficult than I remember. Anyway, I hope the following serves as a useful jump off for our journal club.

I am going to attempt to report on the introductory sections of the Ben-Zvi, Francis, Nadler (BZFN) paper. So how does one describe an introduction which already gives a detailed section by section description of the whole of the paper? Let’s find out. (Here is a hint: restate or copy shamelessly much of what the authors have already kindly told us.)

Let me start by saying what BZFN are attempting to do. The main applications stated are in the simpler form the calculation of Drinfeld centers of monoidal categories of sheaves and construction of topological field theories.

The reason to first state the applications which seem to be way off in our future is just to get down the jargon and then put it to the side for a while, hopefully lessening our burden while we trudge through the preliminaries.

So what is a Drinfeld center (of a monoidal category (of sheaves)) and roughly how does one calculate it? I guess (from reading the abstract) that the Drinfeld center should be an $\infty$-subcategory of the category of quasicoherent sheaves and is the same as the Hochschild cohomology category. Of course, this still leaves a lot of explaining to do. Hopefully, someone who wants to report on Section 5 can jump in here and explain this. If not, I will come back later and try to understand/explain these things.

Second, what are topological field theories (TFT’s) and roughly how does one construct such a thing? A TFT is a symmetric monoidal functor from a certain cobordism $\infty$-category to a symmetric monoidal $\infty$-category. In Section 6, we will see an explicit construction of 2d TFT where the circle is sent to the $\infty$-category of quasicoherent sheaves on the loop space of a perfect stack. Similarly, I hope this can be a jumping off point for someone to talk about Section 6.

Having vaguely introduced the applications, what are the main results that this paper will present as our tools? We want to identify the category of sheaves on a fiber product with two algebraic constructions:

i) the tensor product of the categories of sheaves on the factors,

and

ii) the category of linear functors between the category of sheaves on the factors.

Here the authors tell us that (ii) allows us to realize functors as integral transforms. So far I have been laying out an outline for this paper where we consider these factors mentioned above to be schemes or stacks (except maybe briefly in the applications). Lets quickly finish the story for this case, make note of the fact that we want to take everything above and make it a bit fancier by considering derived stacks, then use the comment above on (ii) to consider an even simpler example which DBZ was kind enough to explain in a previous post. Hopefully, this will help us find some intuition for why and how we want to understand (i) and (ii) before getting scared off by things like derived stacks.

If we let X be a scheme or a stack, then we can obtain a stable $\infty$-category QC(X), whose homotopy category is the unbounded quasicoherent derived category $D_{qc}$(X). QC(X) becomes a symmetric monoidal stable $\infty$-category. So we want to calculate the $infty$-categories built out of QC(X) from tensor products, linear functors and other algebraic constructions in terms of geometry on X.

So that last paragraph (taken almost word for word from the paper) is about where my boundaries begin to be pushed. Now I think the best thing to do is take a step back and remember that David was kind enough to write us a short note on geometric function theory a while back. In the interest of keeping this self-contained and accessible to as many people as possible (including me) we should recap the lessons of this note.

The toy model: function theory on finite sets. This is a great introduction to some of the main characters in our story, in particular, spans or correspondences, the push-pull, integral transforms, functions on (fiber) products and maps between different structures of functions on algebraic objects (in this case sets). Eventually functions will mean quasicoherent sheaves and sets should mean something like scheme, stack, groupoid, or derived stack.

We start by considering a span of finite sets

and, for instance, the vector spaces of complex functions on each of these sets. It is very natural to pull back a function on X to a function on Z by the function from Z to X. Fortunately, we can also push-forward a function on Z to a function on Y by summing over fibers, allowing us to obtain linear maps from $\mathbb{C}[X]$ to $\mathbb{C}[Y]$. This turns out to be just a special case of a richer phenomenon. What I have just described doesn’t fully exploit the span. We can choose a function on Z and when summing over fibers of elements of Y convolve by this function, thus obtaining a new linear map from $\mathbb{C}[X]$ to $\mathbb{C}[Y]$. My first description implicitly involved convolution with the constant function $1$ on Z. So, in general, functions on Z are integral transforms taking functions on X to functions on Y for any span of the sort given above.

Now, Ben-Zvi goes further to explain the how products play into this game. (I should mention that this story of the toy model was first told to me by Tony Licata) This will give us a nice base to think about the more difficult theorems we are attempting to understand in this journal club. But first, I would like to at least make a small excursion to mention the relation to groupoidification (since I have a vested interest in the subject and it has been discussed here and here extensively). This toy model which we intend to extend to geometric $\infty$-function theory is also a toy example for groupoidification. Spans of sets are just really boring examples of spans of groupoids and using the concept of groupoid cardinality described at the links above and zeroth (co)homology of groupoids, we can employ the same pull-push operation and convolution to obtain linear maps. Also, one could continue reading Ben-Zvi’s note to consider functions on finite orbifolds and leading up to his own synopsis of the paper we are currently reading.

Lets understand Ben-Zvi’s comments on products and then get back to the paper at hand. Given finite sets X and Y, there is of course a nice span

and using the construction above we get

between integral kernels and linear transformations.

When we have a cospan

we get a relative version of this construction

.

The theorems we intend to try to understand use an exact analogue of this correpondence.

So now we should take the plunge into derived stacks and try to repeat the story just told.

1) stacks and higher stacks arise from quotients (or more general colimits) on schemes

2) derived stacks arise from fiber products (or more general limits) on schemes and stacks

I would like to eventually include some links to definitions and examples here. Any help with that would be great.

Now we begin repeating what we have said above in this derived setting. To a derived stack X, we assign a stable $\infty$-category QC(X) in a manner which extends the definition for an ordinary stack or scheme.

In particular, let X = Spec R, an affine derived scheme, then QC(X) is the $\infty$-category of R-modules $Mod_R$ whose homotopy category is the usual unbounded derived category of R-modules. As before, the tensor product provides QC(X) with the structure of a symmetric monoidal stable $\infty$-category.

Now we should state the generalized versions of the previous stated applications and main results. The main applications being the calculation of the Drinfeld centers (and higher $\E_n$-centers) of $\infty$-categories of sheaves and functors. BZFN give as an example, the identification of the Drinfeld center of the quasicoherent affine Hecke category with sheaves on the moduli of local systems on a torus. And as a second application, explaining how the results obtained fit into the framework of 3-dimensional topological field theory (of Rozansky-Witten type). In particular, BZFN verify categorified analogues of the Deligne and Kontsevich conjectures on the $\E_n$-structure of Hochschild cohomology.

We are clearly back to the land of scary. So I think the best thing to do is to follow BZFN and give a brief overview of each section, which can serve as an outline to be filled in over the course of this journal club.

Why perfect stacks? Urs has begun to tackle this question here . We will be mainly interested in studying the $\infty$-category $QC(X)$, for X a derived stack, but this is in general too unwieldy algebraically. The problem being that it may contain objects that cannot be constructed in terms of concrete, locally finite objects. The “perfect” solution is to consider a smaller $\infty$-subcategory $QC(X)^\cdot$ of “generators” which are “finite” in some sense. This leads us to the idea of an ind-category. This is a nice idea, where one considers a small category of “managable” objects which has diagrams whose inductive limits are morally the larger objects which may not be quite so managable. Of course, these objects cannot be limits of any sort since they do not live in this smaller “managable” category, so one considers the diagrams themselves as placeholders for these objects. There is another apporach which involves the vanishing of right orthogonals, and I do not really understand this. Maybe someone can step in here and give me a hand.

In Section 3.1, BZFN discuss appropriate notions of “finiteness”

-perfect objects (geometry)

-dualizable objects (monoidal structure)

-compact objects (categorical structure)

I would like to discuss how these relate to the corresponding notions in parenthesis, but I am not quite sure how this goes yet.

A quasicoherent sheaf $M\in QC(X)$ is a PERFECT COMPLEX if it locally restricts to a perfect module (a finite complex of free modules). Equivalently, M is DUALIZABLE with respect to the monoidal structure on QC(X). We (or maybe I) should probably create nlab pages for the capitalized words.

We are now in a position to define a PERFECT derived stack X. The two conditions are that it has an affine diagonal and that QC(X) is given as the inductive limit of the full $\infty$-subcategory of perfect complexes. Then we can, of course define PERFECT morphisms.

On a perfect stack X, compact objects of QC(X) are the same as perfect compexes (which are the same as dualizable objects).

I think there should be some discussion on the importance of compact objects. Hopefully, I can come back and include that here soon.

So we can reformulate the definition of a perfect derived stack as having an affine diagonal, QC(X) being compactly generated (i.e. no right orthogonal to the compact objects) and that compact and dualizable objects coincide.

BZFN remark that compactly generated categories can be expressed as categories of modules.

Moving on to tensors and functors (Section 4) we can see analogues of the constructions in the toy example. For X, X’ schemes over a ring k, the dg category of k-linear continuous (colimit preserving) functors is equivalent to the dg category of integral kernels:

(a theorem of Toen). A similar result holds for dg categories of perfect (equivalently, bounded coherent) complexes on smooth projective varieties.

The main technical results are as follows:

- the tensors and functors of $\infty$-categories of quasicoherent sheaves are calculated in the symmetic monoidal $\infty$-category $Pr^L$ of presentable $\infty$-categories with morphisms left adjoints (Section 2)

- the tensors and functors of $\infty$-categories of erfect complexes are calculated in the symmetric monoidal $\infty$-category Idem of k-linear idempotent complete stalk small $\infty$-categories (Section 4.1)

Finally we want to state the theorems:

For $X \rightarrow Y \leftarrow X’\] maps of perfect stacks, there is a canonical equivalence

between the categories of sheaves on the derived fiber product and the tensor product of the $\infty$-categories of sheaves on the factors.

There is also a canonical equivalence

for $\inft$-categories of perfect complexes.

For $X\rightarrow Y$ a perfect morphism to a derived stack with affine diagonal and $X' \rightarrow Y$ arbitrary, there is a canonical equivalence

between the $\infty$-category of sheaves on the derived fiber product and the $\infty$-category of colimit-preserving QC(Y)-linear functors.

When X is a smooth and proper perfect stack, there is also a canonical equivalenc

for $\infty$-categories of perfect complexes.

The rest of this section gives an overview of the applications of these results. We will get to this soon.

I have attempted to recap what I read in the introduction here. My main goal was to provide a template for our journal club. I hope that we can transport parts of this over to the nLab and fill in the blanks will more detailed exposition as we go. I had some technical trouble such as using subscripts. Maybe someone can tell me what I am doing wrong.