The n-Category Café

Hi Bruce,

we talked about this by email: I am not sure where the $\bar X$ comes from. But let me make the following general comment, which should be obvious, but just so we are sure that we are all on the same page:

The statement that is being categorified here is the following very simple statement:

for $X$ and $Y$ finite sets and for $C(X)$ and $C(Y)$ the $k$-vector spaces of $k$-valued functions on $X$ and $Y$, the basic fact of matrix calculus says tjat there are natural isomorphisms of vector spaces

$$C(X) \otimes C(Y) \simeq C(X \times Y) \simeq LinMaps(C(X),C(Y)) \simeq C(X)^* \otimes C(Y) \,.$$

It’s really that simple. The whole concept of “perfect derived stack” just says: consider a derived stack which happens to be well behaved enough such that the proof of this simple fact goes through literally for quasi-coherent sheaves!

When I read Integral transforms I do it in reverse order: the key theorem is prop. 4.6 on p. 28, which establish one piece of the above statement. Notice that the computation that proves this proposition looks like a trivial computation in linear algebra. It’s very simple. Then from there you can go back and say: okay, we want that this simple proof works, so now we declare $X$ and $Y$ to be objects nice enough so that it does work. So $X$ and $Y$ are to be perfect stacks, which pretty much says nothing but that: on them this simple linear algebra style reasoning makes sense literally.

What’s a geometric stack?

That’s a stack $X$ which has a cover $Y \to X$ where $Y$ is an “ordinary space”, or rather, where $Y$ is a representable stack.

So if the underlying site is that of topological spaces, $Y$ is required to be a topological space and, geometric stacks are then called [[topological stack]].

If the underlying site is one of $algebras^{op}$, then $Y$ is required to be a dualized algebra (an affine algebraic space), and a geometric stack is called an [[algebraic stack]].

These $n$Lab entries with lots of details have kindly been created by Zoran Škoda yesterday.

(But yesterday some software bug prevented me from posting this here, for some reason.)