The n-Category Café

Tao’s “dynamical system” $(X,(G,\cdot))$ is the action groupoid of $G$ acting on $X$ and I think the cohomology groups he describes are the corresponding groupoid cohomology groups.

Yes, this can be interpreted in general nonabelian cohomology.

Nonabelian cohomology, quite generally, is cohomology on $\infty$-groupoids with coefficients in $\infty$-groupoids, using homotopical cohomology theory.

One way to model this is using $\omega$-groupoids (sufficient for Tao’s application) and the folk model structure on them.

then an $n$-cocycle on a groupoid $C$ with coefficients in the abelian group $A$ is an $\omega$-anafunctor from $C$ to $\mathbf{B}^n A$, i.e. a span

$$\array{ \hat C &\to & \mathbf{B}^n U(1) \\ \downarrow \\ C }$$ where the left leg is an acylic fibration, i.e. an $\omega$-functor which is $k$-surjective for all $k$.

That ordinary group cohomolgy is reproduced this way is nicely described in the work by Ronnie Brown, Phillip Higgins and Rafael Sivera, in the context of Nonabelian algebraic topology. I know they have a comprehensive monograph in preparation which explains this in detail, one can ask them for a pdf copy. But maybe this is also described in one of their published articles.

Groupoid cocycles such as Tao considers appear in particular in the study of Dijkgraaf-Witten theory in the context of the twisted Drinfeld double. An interpretation of this entirely in the above context of cohomology of $\infty$-groupoids with coefficients in $\infty$-groupoids is here.

I’m afraid Urs’ comment may scare away people who aren’t experts on $\omega$-groupoids. People usually need to climb the dimensional ladder slowly at first.

So, to build a bridge between cultures, it would be nice to give Terry Tao some good concrete examples where the sort of cohomology class he was asking about actually comes up: an element of $H^2(G,X,U)$.

I wanted to provide an example, but I didn’t get around to it because I was stunned by his notation $H^n(G,X,U)$. I’d never seen before, so I decided I should get back to my actual work.

Now Urs seems to be bridging the notational gap. So, let me guess: Tao has got a group $G$ acting on a set $X$, and an abelian group $U$. So, is $H^n(G,X,U)$ the cohomology of the action groupoid $X//G$ with coefficients in $U$?

I guess in his expository post Tao is talking about the case where $X$ and $G$ are topologically discrete (indeed finite). So he’s not worrying about the topology of $X//G$ yet. But in many interesting examples of dynamical systems, $X$ and often $G$ will have some topology. So, I imagine we’ll shortly need to consider the case of a topological group $G$ acting on a topological space $X$, making $X//G$ into a topological groupoid.

Or, maybe Tao is heading for the world of measure theory rather than topology.

The paper – Nonconventional ergodic averages and nilmanifolds – by Kra and Host is mentioned as a source by Tao. Appendix C contains information about the cohomology of dynamical systems.

Perhaps more accessible is his earlier post – What is a gauge?. Example 2 concerns dynamical systems. Surely this material is amenable to Café treatment.

Hi, thanks for discussing my post!

While I am ultimately interested in whether higher cohomology has applications to dynamical systems in the measure-theoretic or topological categories, for simplicity I wanted to first start with discrete dynamical systems, i.e. a discrete group G acting on a discrete set X.

My main question is whether higher cohomology has any relationship with the extension problem - i.e. extending a group action of G on X to some principal U-bundle over X, where U is an abelian group. (Nonabelian groups are also very interesting, but for simplicity let’s start with the abelian case. Again, take U to be discrete to avoid complications.)

The first cohomology group $H^1(G,X,U)$ classifies all U-extensions of the G-system X up to conjugation (so I guess it should be equivalent to $Ext(U,G,X)$ or something like that, though one should caution that U is not itself a G-system so one does not precisely have a short exact sequence here). My question was whether the higher cohomology groups, say $H^2(G,X,U)$, are also related to some sort of extension problem. Presumably things like the universal coefficient theorem for cohomology would be relevant here, but other people should be more expert than I in answering this.

Perhaps the situation for group cohomology is better understood - what do higher cohomology groups in that situation have to do with the extension problem for groups?