The n-Category Café

That’s an excellent question but in a sense orthogonal to the point of my post: to get a geometric function theory (i.e., to allow pushforwards with good properties) you shouldn’t take perverse sheaves (or sheaves for that matter) but the full derived category of sheaves (which in nice situations is also the derived category of perverse sheaves). Perverse sheaves are the analogues of very nice functions, but don’t in themselves form a nice function space.

To me perverse sheaves are the right analogs of nice functions since they correspond (in characteristic zero) to holonomic D-modules - ie to nice (maximally overdetermined) systems of differential equations, which you can think of as categorified substitutes for individual solutions of these systems.
Another answer (in any characteristic) is that perverse sheaves are precisely the coefficient systems for cohomology which obey the Morse index theorem on stratified spaces -i.e., local Morse-theoretic measurements of perverse sheaves (aka nearby cycles) live in a single cohomological degree.

If you look at page 2 of Lurie’s paper on stable $(\infty,1)$-categories, you’ll see the prototypical example is the $(\infty,1)$-category of infinite loop spaces. These in turn are stable $(\infty,0)$-categories — where now ‘stable’ is used in the periodic table sense.

(Lurie calls $(\infty,1)$-categories ‘$\infty$-categories’, and $(\infty,0)$-categories are the same as $\infty$-groupoids, but I’m trying to clarify a certain pattern here.)

So, the question you’re asking amounts to this: is a thing of stable things a stable thing? More precisely: if you take a nice collection of $(\infty,n)$-categories, they naturally form an $(\infty,n+1)$ category. If you take a nice collection of stable $(\infty,n)$-categories — where I’m using ‘stable’ in the periodic table sense — do they naturally form a stable $(\infty,n+1)$ category?

Modulo the extreme vagueness of this question, I think the answer is ‘yes’. In fact even plain old categories, not ‘stable’, form a stable 2-category (i.e. a symmetric monoidal 2-category).

So in fact the real question is not whether every $(\infty,1)$-category that’s stable in Lurie’s sense is stable in the periodic table sense — I bet it is. The real question is the converse! But for that I’d need to understand his notion better.

in re: algebras of greatest interest can be realized as convolution algebras of the form Fun(X× YX)

It took me a while to realize that oo-algebras of conciderable interest are oo-convolution algebras, i.e. string field theory algebras regarded as spaces of functions, sections or forms on a space of strings with product or bracket given by integrating over all DEcompositions of a string into two

The Mueger paper we refer to is arXiv:math.CT/0111205, Ostrik’s is arXiv:math.QA/0202130.

I think the fact that Morita equivalence of convolution algebras holds at the categorified level is just a reflection of the fact that vector bundles are more sensitive than functions - the extra structures you get when you categorify are more sensitive to the geometry. Functions on pt/G are uninteresting, but vector bundles on pt/G are representations of G. This is the idea behind Tannakian theory - you can construct a group out of its category of representations by looking at the automorphisms of the forgetful functor, a fact which does not decategorify well.

As for the question on cohomology/Euler characteristics: pushing forward from X to a point we get a complex of sheaves on a point, which is just a complex of vector spaces, or by passing to cohomology (if we’re over C say) just a graded vector space. So as you say, it is exactly the de Rham complex, which (if we forget about things like ring structure) is equivalent to giving de Rham cohomology. The point I wanted to make there is that the “correct” generalization of graded vector space in families is a complex of sheaves..

Over C (or Q), stable $(\infty,1)$ categories are the same thing as what are known as “pre-triangulated dg categories”. These are categories, enriched over differential graded vector spaces (ie over complexes), whose homotopy category is triangulated. The main examples are any kind of derived category you may run into – the derived category is a triangulated category, but it comes by passing to $H^0$ from a dg category (where we have a COMPLEX of Homs, not just a graded vector space).

I have no idea what a perfect stack is, but if there’s any justice in the world, a ‘perfect module’ should be closely related to a perfect ring. There are a bunch of equivalent definitions of a perfect ring: for example, it’s one for which every module $M$ has a projective covering, i.e. a projective module $P$ with a morphism $P \to M$ that’s an essential epimorphism.

I’m too lazy to define ‘essential’ here — click the link — nor do I deeply understand the point of this concept. But, I think I’ve run into projective coverings while learning about quiver representations. I think that for $ADE$ quivers, at least, the irreducible representations have ‘projective covers’ that are indecomposable representations. If you’ve played with quiver representations, that’ll have some emotional resonance. If not, I guess not.

I don’t see how these ideas are related to ‘perfect stacks’, if at all.

The terminology “perfect stack” comes from the notion of a perfect complex: this is a complex of sheaves that’s locally equivalent to a finite complex of vector bundles. For example on a smooth variety these are all complexes of coherent sheaves. So things that are not perfect are either “big” (quasicoherent sheaves that are not coherent) or “cohomologically big” — the skyscraper at a singular point of a variety is not perfect (its self-Tor is nonzero in infinitely many degrees).

It is straightforward to see that perfect complexes are precisely the dualizable objects of the derived category of sheaves - these are also known as “rigid” in the symmetric monoidal literature. Thomason realized they were also the compact objects in the derived category (for varieties - more generally for quasicompact quasiseparated schemes..)

A perfect stack is one where there are enough perfect complexes — they generate the full quasicoherent derived category. They are the stacks where Thomason’s observation holds. This means that these are the spaces whose derived categories are algebraically manageable.

Our original terminology was “stack with air”, based on a comment by Neeman quoted as our epigraph —- the idea was to convey that this is something so common you might as well take it for granted, like air, but that you’re in bad shape without. The key point is that basically anything you care about is a perfect stack - eg any of the finite orbifolds, any variety, any moduli stack you might run into in characteristic zero, etc etc.

There is no problem formulating perfection on the model categorical level — it’s really a notion on the underlying homotopy category. But one of the main points is one CAN’T do a lot of algebra, in particular many operations we need, in the model setting - $(\infty,1)$-categories have precisely the right amount of structure. The results of Lurie’s DAG 2,3 enable one to really perform (non)commutative algebra with these objects in ways that just don’t work on the level of homotopy categories or of model categories.

Another difference between the two notions of groupoidification used here is the way the “integral kernel” is encoded. In Ben-Zvi-Francis-Nadler we do

pull-tensor-push

whereas in Baez-Dolan-Trimble we just do

pull-push.

In BZ-F-C a groupoidified linear map is not just a span, but a span with stuff (a coherent sheaf) on its correspondence object.

I am thinking of the way this “stuff” is encoded in the span itself as follows, described for the case of pull-push of ordinary functions.

So let $X$ be a space (regarded as a category) and let $g : X \to Vect$ be the fiber-assigning functor of the trivial vector bundle $E \to X$ with fiber the ground field $\mathbb{C}$.

There are two ways to talk about $\mathbb{C}$-valued functions on $X$ then: a) as sections of $E$ and b) as morphisms from $E$ to $E$. This may seem like a silly distinction on first sight, but both descriptions differ crucially in terms of the abstract arrow-theory which encodes them. In particular, it is the second picture where the fact that functions form a monoid arises from just abstract-nonsense.

The idea is then: when we have a span

$$X \stackrel{s}{\leftarrow} Q \stackrel{t}{\rightarrow} X$$

and want to pull functions on $X$ to $Q$, tensor them there with a function $K$ on $Q$ and then push the result down to functions on $X$, the function $K$ on $Q$ is secretly actually of nature b) while those on $X$ are of nature a).

More concretely: a morphism $E_1 \to E_2$ of vector bundles on $Q$ comes from a functor

$$K : Q \to [I,Vect] \,,$$

where $I = \{a \to b\}$ is the interval category. If we assume the two endpoints of the interval to always land on the ground field $\mathbb{C}$ then such a $K$ is precisely a $\mathbb{C}$-valued function on $Q$. It naturally fits into the picture

$$\array{ && X \times X \\ & \swarrow &\downarrow^K& \searrow \\ X && [I,Vect] && X \\ \downarrow^g &{}^{d_0}\swarrow& &\searrow^{d_1}& \downarrow^g \\ Vect &&&& Vect } \,.$$

Here $d_0, d_1 : [I,Vect] \to Vect$ are restriction to the two endpoints of the interval.

Now the claim is that the span of groupoids which encodes the integral kernel given by $K$ by just pull-push (not pull-tensor-push) is obtained by pull back along this diagram of the universal $Vect$-bundle $\mathbf{E} Vect \to Vect$ and the universal $[I,Vect]$-bundle $\mathbf{E}[I,Vect] \to [I,Vect]$, which themselves are given by the pullbacks

$$\array{ \mathbf{E} Vect &\to& pt \\ \downarrow && \downarrow^{pt \mapsto \mathbb{C}} \\ [I,Vect] &\stackrel{d_0}{\to}& Vect \\ \downarrow^{d_1} \\ Vect }$$

and

$$\array{ \mathbf{E} [I,Vect] &\to& pt \\ \downarrow && \downarrow^{pt \mapsto Id_\mathbb{C}} \\ [I,[I,Vect]] &\stackrel{d_0}{\to}& [I,Vect] \\ \downarrow^{d_1} \\ [I,Vect] } \,.$$

This yields a span of (discrete in the present example) groupoids

$$\array{ && K^* \mathbf{E}[I,Vect] \\ & \swarrow && \searrow \\ g^* \mathbf{E}Vect &&&& g^* \mathbf{E}Vect } \,.$$

Here $g^* \mathbf{E}Vect$ is just the fibration over $X$ which indeed is the vector bundle $E \to X$ represented by the fiber-assigning functor $g$.

Now, a section of the original vector bundle $E$ we can think of (by the same kind of reasoning, actually) as a span

$$\array{ && \Psi \\ & \swarrow && \searrow \\ \mathbb{C} &&&& g^* \mathbf{E}Vect } \,,$$

There is one point of $\Psi$ sitting over $1 \in \mathbb{C}$ and over each element in the total space of the vector bundle which is in the image of the corresponding section.

Claim: the action of the integral kernel $K$ over $Q$ on the section represented by $\Psi$ is the composition of these spans

$$\array{ &&&& K(\Psi) \\ &&& \swarrow && \searrow \\ && \Psi &&&& K^* \mathbf{E} [I,Vect] \\ & \swarrow && \searrow && \swarrow && \searrow \\ \mathbb{C} &&&& g^* \mathbf{E}Vect &&&& g^* \mathbf{E}Vect }$$

in that the pullback groupoid $K(\Psi)$ is over $1 \in \mathbb{C}$ a collection of points over the total space of the trivial bundle $E$ which, if you take their groupoid cardinality as fiberwise groupoids over $\mathbb{C}$, produce the section of $E$ which is the result of the desired pull-tensor-push operation. Only that the “tensor” part has become implicit in the pull-push.

Now there are also the details of the computation of

$$holim \left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}G } \right) \simeq H\backslash \backslash G // H$$ at $n$Lab: homotopy limit.

Thanks for writing me, Urs – it’s always nice to get letters!

I’m a little more distracted than usual because on top of the usual family commitments, I’m sick right now, and my mind feels a little sluggish. I’ll carve out some time soon though.

Wow.

If we were good and put this blog to some good use, we’d start some kind of journal club on this Ben-Zvi categorical TFT stuff, chapter by chapter.

If only the day had more hours. I didn’t even manage to get back to David with comments on his draft… :-/

But okay, since there is still the night, too, maybe I can set up something.

BTW, with a little bit of luck I’ll be able to make it to the Northwestern TFT meeting. Any other Café-dweller who’ll be there? It has the golden speaker list™ concerning $\infty$-QFT.