### Hopkins Lecture on TFT: Infinity-Category Definition

#### Posted by Urs Schreiber

In the second part of his lecture on topological field theory (notes on the first part were reproduced here) Michael Hopkins sketched what he currently sees as the emerging picture for the $n$-tiered (aka “extended”) formulation of the definition of topological (quantum) field theory.

As I mentioned last time, in this picture one wants to refine the standard formulation in terms of 1-functors

by passing to $n$-functors into something like $n$-vector spaces.

In order to do so, M. Hopkins reviewed rudiments of the definition of weak $\omega$-categories in terms of **complicial sets**, due to Street and Verity.

Ross Street
*Weak omega-categories*

(pdf).

He then sketched how he imagines forming an **$\omega$-category of $d$-manifolds**, $\mathcal{M}_\bullet^d(n)$, such that together with a suitable $\omega$-category $R$ with an $E_\infty$ ring structure, one would say that

An

$n$-tiered $d$-dimensional topological field theoryis a morphism of $\omega$-categories(2)$\text{TFT} : \mathcal{M}^d(n) \to R \,.$

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).

## Re: Hopkins Lecture on TFT: Infinity-Category Definition

Just for the record: the work

Eugenia Cheng and Nick Gurski

Towards an n-category of cobordisms

addresses the same general question as above.

Abstract: