On the relationship between certain descent notions II

Dominic Verity – 22 July 2009

In the literature one can find a number of different limit notions which one might refer to as a "descent construction". Genrally speaking, these may all be regarded as a kind of lax, pseudo or homotopy limit of a cosimplicial diagram of objects in some theory of "spatially enriched" categories. While each of these notions certainly deserves to bear the descent name, it is not necessarily immediately clear how they may be related in any more specific mathematical sense. Recently I was asked by Urs Schreiber if I knew how a couple of these descent notions might be related formally, and so spent a little time contemplating this problem. My hope is that this talk might achieve two things, firstly I hope to provide a little of the intuition which leads us to define and study such descent constructions. Then I would like to discuss a specific answer to Urs' question, which gives a precise relationship between Ross Street's descent construction for strict omega-categories (or more precisely strict omega-groupoids in this case) and the simplicial descent construction used to characterise the fibrant objects in model categories of simplicial sheaves.