The n-Category Café

Journal Club entry by Bruce Bartlett

Well it’s my turn to give a write-up in the Journal Club. The topic for this week is perfect stacks, and it concerns Section 3 of David Ben-Zvi, John Francis and David Nadler’s paper Integral transforms and Drinfeld centers in derived algebraic geometry.

As I understand it, we need the concept of perfect stacks because they represent the most general context in which the important theorems of geometric infinity-function theory hold — namely, that algebraic and geometric operations on their categories of sheaves are nicely compatible.

This is even more succinctly put in the heading area of the paper:

Compact objects are as necessary to this subject as air to breathe.

That, by the way, is apparantly a statement made by R.W. Thomson, to A. Neeman, to the authors via the book Triangulated categories, to me via the paper we are studying, and now to you, dear reader!

Now I’m pretty shoddy on some of the more homological/algebro-geometric aspects that I need to understand this stuff properly, being more accustomed to working with down-to-earth things such as smooth manifolds and differential forms, as opposed to spectra of rings and such like. Urs and Alex perhaps come from a similar viewpoint. Chris might know this stuff quite well. Anyhow, I’m going to try and keep Urs’s $\mathrm{\infty }$-stacky incarnation of smooth geometry via generalized smooth spaces and $\omega$-groupoids on the side, and ask questions as we go along about how “perfectness” translates into this context, if indeed it does.

As I understand it, a derived stack $X$ is roughly speaking something which assigns a $\mathrm{\infty }$-groupoid $X(U)$ to each ‘space’ $U$

in a manner which is compatible with gluing. In other words, a derived stack is an ‘$\mathrm{\infty }$-sheaf’. In the case of this paper, we fix a ‘commutative derived ring’ $k$, and then the category of ‘spaces’ in question is the opposite of the $\mathrm{\infty }$-category of derived $k$-algebras.

The concrete application closest to my sphere of understanding is the case where $k=ℂ$, so that the category of `spaces’ is the opposite of the category of connective commutative differential graded algebras over $ℂ$.

To go over into Urs’s framework, I think the category of ‘spaces’ is just the category $\mathrm{Man}$ of good old smooth manifolds, while the concept of $\mathrm{\infty }$-groupoid translates into ‘globular strict $\omega$-groupoid’.

So in Urs’s setup, a `derived stack’ is nothing but a smooth $\omega$-groupoid.

Question 1. Is this correct?

Okay, now suppose we’ve got one of these derived stacks $X$. The natural notion of a function on one of these guys is the $\mathrm{\infty }$-category of quasi-coherent sheaves over $X$, written $\mathrm{QC}(X)$. The way to define $\mathrm{QC}(X)$ is as follows. Any stack $X$ can be expressed as a gluing or colimit of affine derived schemes:

I’m not exactly sure what an affine derived scheme is, but in Urs’s setup I think this means that any generalized smooth space can be expressed as a colimit of (the stacks represented by) smooth manifolds:

Come to think of it, I’m not quite sure what that means either, but I know it has something to do with the ‘coYoneda lemma’ which I stumbled onto on the nLab.

Anyhow, an affine thingy is something of the form $U=\mathrm{Spec}A$ for some algebra thingy $A$, so we can define $\mathrm{QC}(U)$ for affine thingys to be to be the $\mathrm{\infty }$-category of $A$-modules. And then we can define $\mathrm{QC}(X)$ for a general stack $X$ as the appropriate gluing together of these module categories:

Question 2. I’m actually a bit uncomfortable with this, since it’s not a concrete ‘definition’ in the classical sense of the word, since the original expression of $X$ as a colimit wasn’t canonical, and neither is the limit above. I know these sorts of definitions are rife in homotopy theory though, so I guess I must just accept it? I would be happier if there was a nice canonical concrete expression.

Question 3. In Urs’s picture, does the concept of $\mathrm{QC}(X)$ make sense? Is it the $\mathrm{\infty }$-category of vector-bundles on $X$ (possibly equipped with connection) Or is it more of a ‘local-system’ thing?

Ok, so here’s what it means for a derived stack to be perfect. We want $\mathrm{QC}(X)$ to be ‘generated’ by some smaller subcategory consisting of ‘finite objects’. There are two candidates for what ‘generated’ might mean (we’ll get to this) and there are three candidates for what ‘finite objects’ might mean.

The good news is that we don’t have to know a lot about quasicoherent sheaves to discuss this matter.

The three candidates for what ‘finite object’ might mean are as follows.

• An object $M\in \mathrm{QC}(X)$ is perfect if its restriction $f^{*}M$ to any affine chart $f:U\to X$ is perfect, in the sense that it lies in the smallest subcategory of $\mathrm{QC}(U)$ containing $A=\mathrm{Spec}(U)$ and which is closed under finite colimits and retracts.
• An object $M\in \mathrm{QC}(X)$ is compact if $\mathrm{Hom}(M,\cdot )$ commutes with all colimits (equivalently, with all coproducts).
• An object $M\in \mathrm{QC}(X)$ is strongly dualizable (or dualizable for short) if it has a dual $M^{\vee }$ in the categorical sense — that is, if there exists an object $M^{\vee }$ together with unit and trace maps $u:1\to M^{\vee }\otimes M$ and $\tau :M\otimes M^{\vee }\to 1$ satisfying the snake diagrams.

The cool thing is that these three definitions are clearly quite natural, and they ask questions of the geometry of $X$, the categorical structure of $\mathrm{QC}(X)$, and the monoidal structure of $\mathrm{QC}(X)$ respectively.

I wish I could get some more familiarity with how these three concepts work out in specific examples. From the nLab, I see that the concept of a ‘compact’ object can be applied to any category, except in general one should ask that $\mathrm{Hom}(M,\cdot )$ only commutes with all filtered colimits. There’s a nice example: a topological space $X\in \mathrm{Top}$ is compact if and only if the hom-functor

commutes with all filtered colimits. I’d like to understand this example!

I wish I could understand this concept of filtered colimit better. I need to go back to point-set topology and understand what a ‘filter’ means. I wish John would write an expository article explaining filters (in topological spaces), and how they generalize to filtered colimits.

Anyhow, the interesting fact is that these three notions (perfecness, compactness, dualizable) all coincide in the affine situation. Moreover, ‘strongly dualizable’ and ‘perfect’ are apparantly local properties, so therefore they always coincide. The only problem-child is ‘compact object’, which is not a local property and hence might not glue nicely. More on that later.

Having discussed the various candidates for the notion of ‘finite object’, we must still discuss what we are going to mean when we say that they ‘generate’ our category $\mathrm{QC}(X)$. The two candidates are:

• $\mathrm{QC}(X)$ is the inductive limit of the category of finite objects (basically, this says that $\mathrm{QC}(X)$ is the formal completion of the category of finite objects).
• $\mathrm{QC}(X)$ is compactly generated by the finite objects — that is, there is a set of finite objects $C_i\in C$ such that if $\mathrm{Hom}(C_i,M)\cong 0$ for all $i$, then $M\cong 0$.

The choice made in this paper is as follows.

Definition. A derived stack $X$ is said to be perfect if it has affine diagonal and if the $\mathrm{\infty }$-category $\mathrm{QC}(X)$ is the inductive limit

of the full $\mathrm{\infty }$-subcategory $\mathrm{Perf}(X)$ of perfect complexes. A morphism $X\to Y$ is said to be perfect if its fibers $X{\times }_{Y}U$ over affines $U\to Y$ are perfect.

The main point is that we could have used any of the other notions, because of the following nice fact.

Proposition. For a derived stack $X$ with affine diagonal, the following are equivalent:

• $X$ is perfect.
• $\mathrm{QC}(X)$ is compactly generated, and its compact and dualizable objects coincide.

The proof of this apparantly comes down to checking the following lemma:

Lemma. If $N$ is a retract of a dualizable object $M$, then $N$ is also dualizable.

I tried to prove this using string diagrams, but I got stuck. I think it’s because I’m not incorporating the closed structure (ie. the internal hom).

Question 4. Give a nice string diagram proof of this statement.

So let’s summarize where we are (we’re only at the end of Section 3.1!):

• We’ve defined the derived category $\mathrm{QC}(X)$ of quasicoherent sheaves on a derived stack $X$.
• We’ve said what it means for $\mathrm{QC}(X)$ to be perfect. This boiled down to making one of a number of different natural choices. Happily, they turn out all to be equivalent, precisely when $\mathrm{QC}(X)$ is perfect!
• The good news for n-category cafe denizens is that all of these choices appear to be reasonably ‘elementary’ and don’t require sky-high formalism. This appears to go for the proofs too. Thus the concept of ‘perfectness’ has something fundamental and important to say about monoidal-ish categories (see Question 4 above, as well as the fact that being ‘closed under retracts’ ties in very nicely with the 2-Hilbert space formalism). Let’s try to get to grips with it.

The remaining two sections deal with base change and the projection formula for perfect morphisms (section 3.2), and that many common classes of stacks are perfect (section 3.3). I think one can only appreciate these sections properly once one has been working with schemes and algebro-geometric stacks for a while… then when one reads the results, one will say “oh, that’s nice, because that particular property was always a problem in the so-and-so approach”. Sadly I am not in that boat.