## October 25, 2006

### 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

(1)$\mathrm{TFT} : d\mathrm{Cob} \to \mathrm{Vect}$

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 theory is 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).

Posted at October 25, 2006 9:01 PM UTC

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### 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:

We discuss an approach to constructing a weak $n$-category of cobordisms. First we present a generalisation of Trimble’s definition of $n$-category which seems most appropriate for this construction; in this definition composition is parametrised by a contractible operad. Then we show how to use this definition to define $n$-category $\mathbf{nCob}$, whose $k$-cells are $k$-cobordisms, possibly with corners. We follow Baez and Langford in using “manifolds embedded in cubes” rather than general manifolds. We make the construction for 1-manifolds embedded in 2- and 3-cubes. For general dimensions $k$ and $n$ we indicate what the construction should be.

Posted by: urs on December 15, 2006 10:26 PM | Permalink | Reply to this

### Morton on extended Cobordisms

Since it seems I have begun listing literature on extended cobordism categories here, and since John just mentioned brand new work in this direction, I’ll include this one, too:

Jeffrey Morton, A Double Bicategory of Cobordisms With Corners

Abstract:

Interest in cobordism categories arises in areas from topology to theoretical physics, and in particular in Topological Quantum Field Theories (TQFT’s). These categories have manifolds as objects, and cobordisms between them as morphisms, have - that is, manifolds of one dimension higher whose boundary decomposes into the source and target. Since the boundary of a boundary is empty, this formulation cannot account for cobordisms between manifolds with boundary. This is needed to describe open-closed TQFT’s, and more generally, “extended TQFT’s”. We describe a framework for describing these, in the form of what we call a “Verity Double Bicategory”, after Dominic Verity, who introduced them. This is similar to a double category, but with properties holding only up to certain 2-morphisms. We show how a broad class of examples is given by a construction involving spans in suitable settings, and how this gives cobordisms between cobordisms when we start with the category of manifolds.

Posted by: urs on December 18, 2006 12:31 PM | Permalink | Reply to this
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