Verity on ∞Categories From Topology
Posted by John Baez
Anyone who wants to keep up with the latest research on $n$categories needs to know what the Australians are doing. It helps to be on the Australian Category Seminar mailing list… which is free if you visit. (Maybe it’s free otherwise, too — I don’t know.)
Dominic Verity has been doing wonderful things lately, and here’s what he’s talking about tomorrow…

Dominic Verity, Cobordisms and weak complicial sets, Australian Category Theory Seminar, February 13, 2008, Macquarie University, seminar room E7A 333.
Abstract: It is generally held that the totality of all $n$manifolds (with corners) should support a canonical weak $n$categorical structure, under which all composites of cells are describable as “gluings” of manifolds along common boundary components. At low dimensions, not only do we know that this is indeed the case, but we also know how to describe free weak 2 or 3categories in terms of such higher categories of “cobordisms embedded in cubes”.
In this talk we describe how to build a weak complicial set (qua simplicial weak $\omega$category) $Cob^k$, whose $n$simplices correspond to embeddings of manifolds of some fixed codimension $k$ into the geometric $n$simplex. Here we work in the PLcategory in order to avoid some of the technicalities involved when working with manifolds with corners in the smooth setting. However, it is likely that our main results bear direct translation to that latter context.
From the point of view of the theory of weak complicial sets, $Cob^k$ is an interesting structure for a number of reasons. Firstly, its underlying simplicial set is a Kan complex. Secondly, it possess a stratification which makes it into a weak complicial set but which is nontrivially distinct from the “all simplices are thin” structure that comes with Kanness. Finally, we have strong reasons to conjecture that this nontrivial complicial stratification can be extended, by making thin all simplices which are “morally” equivalences, to a complicial stratification whose thin simplices bear a characterisation in terms of simple homotopy equivalences (and which are thus $s$cobordismlike).
If anyone reading this attends that talk, any comments on it would be much appreciated here!
Re: Verity on ∞Categories From Topology
Sounds very much like this evolved from or was directly inspired by what Michael Hopkins was talking about here. (Or vice versa.)