The details are here:
Here is an abstract that served as an abstract for a talk on this at the Australian Category Seminar at Macquarie University on Wednesday 27th of May 2009.
In the literature one can find a number of different limit notions which one might refer to as a “descent construction”. Generally speaking, these may all be regarded as a kind of lax, pseudo or homotopy limit of a co-simplicial 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 ω-categories (or more precisely strict ω-groupoids in this case) and the simplicial descent construction used to characterise the fibrant objects in model categories of simplicial sheaves.
Generally, models for ∞-stack (∞,1)-toposes are provided by a model structure on presheaves with values in simplicial sets.
By a theorem by Dugger-Isaksen-Hollander on the projective local model structure on simplicial presheaves the fibrant simplicial presheaves are those that
and satisfy descent.
While general ∞-groupoids are useful due to their generality and conceptual simplicity, for many concrete computations it is useful to get a more concrete algebraic model and consider just strict ω-groupoids. Under the oriental-nerve
the strict -groupoids form a subcategory of all ∞-groupoids. This is to be regarded as a refinement of the Dold-Kan correspondence which embeds strict -groupoids with abelian group structure equivalently modeled as chain complexes into all -groupoids
It is a familiar process to restrict general ∞-stacks to those that factor through the entire inclusion: this is the topic of homological algebra and restricts the general notion of cohomology to that of abelian sheaf cohomology.
What we are interested in here is a notion in between the fully strictly abelian context and the fully general context: that of strict -groupoid valued -stacks inside all -stacks. This may be thought of as nonabelian homological algebra that uses not chain complexes of sheaves but crossed complexes.
In his work
The question remained open how that definition of descent on -valued presheaves relates to the general one of -valued presheaves under the above inclusion.
It is this question that Dominic Verity’s theorem answers.
In words, Verity’s theorem says that Ross Street’s descent conditon on a -valued presheaf is the correct one if the hypercover along which one checks descent is sufficiently well behaved – in that the cosimplicial -groupoid is Reedy fibrant.