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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 nn-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)TFT:dCobVect \mathrm{TFT} : d\mathrm{Cob} \to \mathrm{Vect}

by passing to nn-functors into something like nn-vector spaces.

In order to do so, M. Hopkins reviewed rudiments of the definition of weak ω\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 ω\omega-category of dd-manifolds, d(n)\mathcal{M}_\bullet^d(n), such that together with a suitable ω\omega-category RR with an E E_\infty ring structure, one would say that

An nn-tiered dd-dimensional topological field theory is a morphism of ω\omega-categories

(2)TFT: d(n)R. \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)(d+m)-manifolds over mm-simplices (i.e. equipped with a surjective map onto a standard mm-simplex) one constructs a stratified simplicial set which corresponds to the ω\omega-category d(n)\mathcal{M}^d(n).

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

A thin mm-simplex is to be thought of as representing an identity mm-morphism. Stratified simplicial sets are hence a way to talk about the nerve NCNC of an nn-category CC. mm-simplices in NCNC are mm-morphisms of CC, and the identity mm-morphisms are labeled as thin.

The very last few minutes of the talk were about a way to pass from d(n)\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 nn-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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Read the post Hopkins Lecture on TFT: Chern-Simons
Weblog: The n-Category Café
Excerpt: Basic and advanced concepts in Chern-Simons theory.
Tracked: October 27, 2006 4:46 PM
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Weblog: The n-Category Café
Excerpt: A general concept of extended QFT and its relation to the Kapranov-Ganter 2-character.
Tracked: November 17, 2006 5:10 PM

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 nn-category of cobordisms. First we present a generalisation of Trimble’s definition of nn-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 nn-category nCob\mathbf{nCob}, whose kk-cells are kk-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 kk and nn 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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Read the post Hopkins-Lurie on Baez-Dolan
Weblog: The n-Category Café
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Read the post Schommer-Pries on Classification of 2-Dimensional Extended TFT
Weblog: The n-Category Café
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