The String Coffee Table

First some words on stacks, by myself. See this for more.

I think of it this way.

Let $V$ be a vector space. Every such vector space can be thought of as sitting inside its dual space $$V^* = \mathrm{Hom}(V,K) \,,$$ where $K$ is the simple vector space obtained by regarding the ground field $K$ as a 1-dimensional vector space over itself.

Now categorifiy this setup. Let $C$ be some category. What is analog of the ground field $K$ then. It’s the category of (small, I guess) sets. So let’s say that the “dual” $C^*$ of $C$ is the (contravariant-)functor category $$C^* := \mathrm{Hom}_\mathrm{contr}(C,\mathrm{Set}) \,.$$ The Yoneda Lemma ($\to$) says that $C$ sits inside its dual $C^*$.

Ordinary topological or smooth spaces live in the category $C = \mathrm{Top}$ or $C = C^\infty$. Therefore, quite similar to how we obtain distributions as “generalized functions”, we can regard objects in $\mathrm{Top}^*$ or $(C^\infty)^*$ as “generalized spaces”.

And it’s easy to make things even more general, simply by replacing our archetypical category $\mathrm{Set}$ (our “ground field”) with something more fancy. The next best guess is to use instead $\mathrm{Cat}$, the 2-category of categories.

Usually we want a couple of things to behave nicely. So mostly we do not consider $\mathrm{Hom}_\mathrm{cont}(\mathrm{Top},\mathrm{Cat})$, but use instead the 2-category of groupoids, leading to $\mathrm{Hom}_\mathrm{cont}(\mathrm{Top},\mathrm{Grpd})$.

Moreover, we want the category that we take the dual of to behave nicely. It should be such that it has objects (spaces) which can sensibly be “covered” by other objects. If that works as expected we say that our category is a site ($\to$).

For $C$ a site, every object in $\mathrm{Hom}_\mathrm{cont}(C,\mathrm{Set})$ is a presheaf (in sets). If it satisfies some gluing condition the presheaf is actually a sheaf.

Similarly, for $C$ a site, every object in $\mathrm{Hom}_\mathrm{cont}(C,\mathrm{Grpd})$ is called a pre-stack. If it satisfies some gluing condition then the pre-stack is actually a stack.

All right. If you are reading this at all, you probably knew all this already.

2) Part II: sheaves on topological stacks and cohomology on topological stacks

The standard example for a (topological) stack is the global quotient stack of a point. Let $G$ be any (topological) group. The stack

is defined such that it maps any space $T$ to the groupoid of $G$-principal bundles over $T$

(This stack describes a global orbifold, which, like any orbifold, can alternatively be thought of as a groupoid ($\to$). But this groupoid is nothing but $G$ itself, regarded as a category with a single object. Thinking of it this way makes the following statements very obvious.)

In the example mentioned in part 1), we encountered a lifting gerbe in the guise of a $B U(1)$-bundle. Using the language of stack we can now do the same example with the gerbe made explicit.

From the morphism

we get a morphism of global quotient stacks

whose fiber is a $\mathbf{B}U(1)$ stack.

Let $X$ be an ordinary space, as before, but now considered as a stack (i.e. as the assignment $X(T) = \mathrm{Hom}(T,X)$).

Before we had that maps $X \to B SO(n)$ classified $SO(n)$-bundles on $X$. In terms of morphisms of stacks we find that

precisely specifies a $SO(n)$-bundle on $X$.

Hence we can consider our pullback diagram from the first part of the talk, now entirely interpreted in terms of stacks

What is $\tilde X$? It’s the $U(1)$-gerbe which obstructs the lift of our $SO(n)$-bundle to a $\mathrm{Spin}^\mathbb{C}(n)$-bundle.

(Xou can equivalently think of $X$ as given by the stack which comes from the groupoid obtained from a fibration $Y^{[2]} \to Y \to X$. In terms of this $\tilde X$ is precisely the groupoid of transition $U(1)$-bundles as in Hitchin’s or Murray’s description of bundle gerbes.)

All right, enough examples. Now some words on the construction that we are actually after.

We want

$\bullet$ a construction involving topological/smooth stacks $X$

$\bullet$ such that there is a notion of sheaf over $X$ (like we had sheaves over ordinary spaces before),

$\bullet$ hence a notion of a site $\mathbf{X}$ associated to such a stack $X$ (to be thought of as the sight of open subset of $X$),

$\bullet$ giving rise to categories $\mathrm{Pr}\mathbf{X}$, $\mathrm{Sh}\mathbf{X}$, $D^+(\mathrm{Sh}_\mathrm{Ab}\mathbf{X})$ of presheaves and sheaves over $\mathbf{X}$, as well as a derived category of abelian sheaves (sheaves with values in abelian groups) over $\mathbf{X}$;

$\bullet$ such that for any morphism

$\bullet$ we obtain adjoint functors

$\bullet$ as well as derived adjoint functors

(It’s not a typo that there is no “$R$” (for right derived functor) on the left. For some reason, which I cannot reproduce, we do not need an $R$ there.).

$\bullet$ We want a cohomology theory on the stack $X$ such that ordinary cohomology is reproduced in that

$\bullet$ and such we have a notion of cohomology twisted by a stack $\tilde X$

where $\mathbb{Z}_X$ is the constant $\mathbb{Z}$-valued sheaf on $\mathbf{X}$.

It is not difficult to come up with the rather obvious general idea how to define a site structure and a sheaf on a stack. The subtle part is to check that, in the topological/smooth context, all conditions are suitably satisfied and everything works as expected.

The main problem to be dealt with, however, is that generic morphism of stacks

won’t be representable. (See p. 6 of U. Bunke’s notes for a reminder of what it means for a morphism of stacks to be representable.) This implies that one has to do real work in order to define the induced morphisms of categories of presheaves

(Compare section 1.2.7 on p. 3 in U. Bunke’s notes.)

So what is the obvious idea for how to define the site $\mathbf{X}$? Let its objects be open sets in the sense

where $U \text{and} \phi \text{nice}$ means that both are representable and that $\phi$ has local sections if we are considering topological stacks, or that it is a submersion in the case that we are considering smooth stacks.

Morphisms in $\mathbf{X}$ are the obvious over-morphisms

As I mentioned, one important point is to find the right notion for the induced morphism

for any $F : X \to Y$. The right definition uses some limit construction which I won’t try to typeset here. See definition 2.3 in U. Bunkes notes.

Having defined morphisms on presheaves this way, we can sheafify them to morphisms $f_*$ on sheaves by setting

where $i : \mathrm{Sh}\mathbf{X} \to \mathrm{Pr}\mathbf{X}$ is the functor that forgets the sheaf condition, and $i^\sharp : \mathrm{Pr}\mathbf{X} \to \mathrm{Sh}\mathbf{X}$ is its adjoint, the sheafification functor.

There is work required in making all this precise and checking that everything behaves as it should. Here we haste to state the main result that follows once all this is done.

First of all, notice that one essentially equivalent way to define twisted deRham cohomology of an open subspace $U \overset{\phi}{\to}X$ is as the cohomology of

where the unknown $z$ is taken to have degree 2. We think of this as living in the derived category of abelian sheaves

One gets the more standard flavor of twisted deRham cohomology by somehow inverting $d/dz$ in this construction.

Anyway, the final statement is now this:

Let $G\to X$ (a stack) be a gerbe with band $U(1)$, classified by $H^3(X,\mathbb{Z})$. Then the $G$-twisted cohomology

is (non-canonically) isomorphic to the version of twisted deRham cohomology described above.

The non-canonical choice which is involved is, interestingly, the choice of connection (= “connective structure and curving”) on $G$ which realizes the curvature given by the class in $H(X,\mathbb{Z})$.

It’s the non-canonicalness of this choice which makes the twisted deRham cohomology unnatural (under morphisms of $G$), while $\mathrm{Tw}_G(\mathbb{R})$ is perfectly natural.

There are other advantages of this formulation. For instance it allows immediately to generalize the notion of twisted cohomology to other abelian groups. Replacing $\mathbb{R}_X$ with $\mathbb{Z}_X$ we obtain, in particular, a gerbe-twisted version of integral cohomology, something completely out of reach by means of the more naive (“ad hoc”) methods defining twisted cohomology.

That’s it for today. This talk more or less ended where the Vienna talk started, so most of the meat has been omitted. Have a look at the notes for all details.