The n-Category Café

Hi all.

I’m at the conference too, so I’ll put in a few words of what I think this might mean (and say more after the fact).

So John Roberts in the late 70’s realised that n-dimensional cohomology should have an n-category as the coefficient object. For the usual abelian group-valued cohomology, this is just the groupoid with only its n-morphisms non-trivial, consisting of the group $A$ (written as, say $\Sigma^nA$). Ross Street went further and wrote a paper saying cohomology should have as `base space’ an omega-category and should have as coefficients an $\omega$-category. We all know $\omega$-groupoids are secretly topological spaces, so we’ve been doing some of this part already.

Explicitly (this is talked about in John and Mike’s cohomology notes) this is all just a big, big generalisation of Schreier theory and Galois theory - realised by Grothendieck. So in dimension one we have that opfibrations over a category $X$ are equivalent to $[X,\mathbf{Cat}]$ (at least, that’s what I recall and I’m pressed for time - please someone tell me if I have egg on my face).

It’s just that we are used to dealing with spaces = homotopy types = groupoids. There is work on `directed homotopy theory’, in which paths are thought of (I gather) as paths in categories, not groupoids. There of course, things are not necessarily invertible.

It’s much too late to say that nonabelian cohomology is the raison d’etre for n-categories, since we’ve learned by now that n-categories are great for lots of things. I’d rather say nonabelian cohomology is a raison d’etre.

But one can argue that nonabelian cohomology was the raison d’etre, since we can see it looming over Grothendieck’s letter Pursuing Stacks, and also John Roberts’ paper that introduced n-categories in mathematical physics:

• John E. Roberts, Mathematical aspects of local cohomology, in Algèbres d’Opérateurs et Leurs Applications en Physique Mathématique, CNRS, Paris, 1979, pp. 321–332.

And, it’s still fun to see nonabelian cohomology as the raison d’etre for n-categories - it’s a useful viewpoint.

As David Roberts points out, the idea is to subsume many different kinds of cohomology into one big generalization: the cohomology

$$H(X,Y)$$

of a weak $\omega$-category $X$ with coefficients in a weak $\omega$-category $Y$. You can think of this as the set

$$[X,Y]$$

of weak natural isomorphism classes of weak $\omega$-functors $f : X \to Y$. Or, even better, you can think of it as the weak $\omega$-category

$$hom(X,Y)$$

The point is that the set $[X,Y]$ tends to classify interesting `things over X’ up to equivalence, while $hom(X,Y)$ describes the $\omega$-category of such things.

Topologists are most familiar with the example where $X$ is a space and $Y = B G$ is the classifying space for $G$-bundles, where $G$ is a topological group. We can think of $X$, $G$ and $B G$ as $\omega$-groupoids. Then $[X,Y]$ is the set of isomorphism classes of principal $G$-bundles over $X$, while $hom(X,Y)$ is the $\omega$-groupoid of such bundles.

This example shows that nonabelian cohomology with coefficients in an $\omega$-groupoid is nothing new if you’re a topologist. The real fun starts when we hit $\omega$-categories.

But, Grothendieck was really interested in a big further generalization. In algebraic geometry, cohomology with coefficients in a group is regarded as a pathetically dull special case of cohomology with coefficients in a sheaf of groups! So, when we replace groups by $\omega$-categories, we should also replace sheaves by their $\omega$-categorified versions, which Grothendieck called “stacks”.

All this seems quite intimidating the first time you meet it, and also the second time, and the third time. That’s why I explained it very gently in my paper with Mike Shulman. We don’t go much beyond the $\omega$-groupoid case, but I think we explain some aspects of this business more thoroughly than anyone ever had.