### Bridge Building

#### Posted by David Corfield

If anyone wanted to bridge the gap between the two cultures, Terry Tao’s post – Cohomology for dynamical systems might provide a good place to start. Remember our last collective effort at bridge-building saw us rather unsuccessfully try to categorify the Cauchy-Schwarz inequality.

Regarding this current prospective crossing point, we hear that the first cohomology group of a certain dynamical system is useful for the ‘ergodic inverse Gowers conjecture’, and that there are hints that higher cohomology elements may be relevant. The post finishes with mention of non-abelian cohomology.

It wouldn’t be surprising if algebraic topology provided the common ground. A while ago we heard Urs describe Koslov’s work on *combinatorial algebraic topology*.

## Re: Bridge Building

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.