The n-Category Café

Tim gave the Friday morning talk at Barcelona. In the first hour, he began by saying that much of the material he had been wanting to cover had been touched on at various points in the conference - but that he wanted to highlight and point out various important constructions which are not as well-known as perhaps they should be. It was a very interesting talk, but I am not confident to write about it, because my simplicial skills are lagging behind somewhat (I seriously need a go at that menagerie). I’ll just mention the following topics which Tim felt important to stress: Kan complexes, the Moore complex of a simplical group, decollages, crossed complexes (and T-complexes), the “W bar” construction, and the Puppe sequence.

In his second, “presentation” talk, he presented the slides on Formal homotopy quantum field theories and 2-groups. Formal homotopy quantum field theories (HQFT’s) were invented by Turaev, and they are essentially TQFT’s in a background space $B$, up to homotopy. Rodrigues showed that an $n$-dimensional HQFT can be regarded as a monoidal functor

where $nCob(B)$ is the category whose objects are $(n-1)$ closed manifolds equipped with a map into $B$, and whose morphisms are cobordisms equipped with a map into $B$, considered up to homotopy (in $B$) fixing the boundary.

So… a HQFT is midway between an “abstract” TQFT (with no background space) and a Stolz-Teichner style “smooth” TQFT embedded in $B$. That makes them an important thing to understand.

Tim reviewed classification results for these gismos. For instance if $B$ is a $K(\pi, 1)$ for some group $\pi$, then Turaev showed that a 2d HQFT in $B$ is the same thing as a crossed $\pi$-algebra. My understanding is that a crossed $\pi$-algebra can be thought of as a Frobenius algebra object in $Rep \Lambda \pi$ - the category of representations of the loop groupoid of $\pi$. Is this correct?

Similarly Brightwell and Turner showed that if $B$ is a $K(A, 2)$, then a 2d HQFT over $B$ is the same thing as a Frobenius algebra equipped with an action of $A$. Again, my understanding is that this is the same thing as a Frobenius algebra object in $Rep A$. Is this correct?

Then Tim spoke about his work with Turaev, on extending these results to all 2-types. We know that a 2-type corresponds algebraically to a crossed module, so one begins by fixing a crossed module $\mathcal{C} = (C, P, \partial)$ and working from there. The main theorem is that formal 2d HQFT’s over a 2-type represented by $\mathcal{C}$ correspond to Frobenius algebras equipped with an action of the 2-group $\mathcal{C}$ by automorphisms.

The disclaimer “formal” is there to indicate that since they work directly with the 2-group $\mathcal{C}$ and a triangulation, they are not sure if they get out all HQFT’s in this way. Though I don’t understand the details, based on my experience with TQFT’s my rough intuition is that you should indeed generically get out all HQFT’s in this way - except possibly for the non-semisimple ones, which are non-generic, and can’t be achieved from a triangulation construction… is that right?

Anyhow something which was very interesting for me here is how an algebra can be acted on by a 2-group. I hadn’t thought of that before.