The n-Category Café

I don’t think I shall even try to reproduce everything Michael Hopkins said about simplicial sets and complicial sets. I think all technical details are better looked up in the literature (see the paper by Street mentioned above).

I am out of time for today anyway.

The main point is that by looking at $(d+m)$-manifolds over $m$-simplices (i.e. equipped with a surjective map onto a standard $m$-simplex) one constructs a stratified simplicial set which corresponds to the $\omega$-category $\mathcal{M}^d(n)$.

“Stratified simplicial sets” (described at least by Verity and Street) are simplicial sets together with a collection of $m$-simplices which are marked as thin.

A thin $m$-simplex is to be thought of as representing an identity $m$-morphism. Stratified simplicial sets are hence a way to talk about the nerve $NC$ of an $n$-category $C$. $m$-simplices in $NC$ are $m$-morphisms of $C$, and the identity $m$-morphisms are labeled as thin.

The very last few minutes of the talk were about a way to pass from $\mathcal{M}^d(n)$ to an ordinary topological space. The conclusion was a big theorem saying that this topological space is equivalent to one studied by Galatius, Madsen, Tillmann and Weiss.

This part of the lecture was explicitly announced to be somewhat speculative and vague. The main message for me (answering a question I had) was that Hopkins et al. do have an idea of a systematic way to say what an $n$-tiered TFT is (whereas in the existing literature the constructions always seem a little ad hoc).