Cobordisms and weak complicial sets

Dominic Verity – 20 February 2008

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 3-categories 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 co-dimension k into the geometric n-simplex. Here we work in the PL-category 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 non-trivially distinct from the "all simplices are thin" structure that comes with Kan-ness. Finally, we have strong reasons to conjecture that this non-trivial 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-cobordism-like).