This entry is about a theorem by Dominic Verity that characterizes the descent condition for (∞,1)-sheaves/∞-stacks that take values not in arbitrary ∞-groupoids, but in strict ∞-groupoids.
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.
Abstract
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 ∞-categories (or more precisely strict ∞-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.
As for all combinatorial simplicial model categories the $(\infty,1)$-topos presented by this model structure is the full SSet-enriched subcategory on fibrant and cofibrant objects.
By a theorem by Dugger-Isaksen-Hollander on the projective local model structure on simplicial presheaves the fibrant simplicial presheaves are those that
take values in ∞-groupoids (i.e. the simplicial sets assigned by them are Kan complexes)
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 ∞-groupoids. Under the oriental-nerve
the strict $\omega$-groupoids form a subcategory of all ∞-groupoids. This is to be regarded as a refinement of the Dold-Kan correspondence which embeds strict $\omega$-groupoids with abelian group structure equivalently modeled as chain complexes into all $\infty$-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 $\omega$-groupoid valued $\infty$-stacks inside all $\infty$-stacks. This may be thought of as nonabelian homological algebra that uses not chain complexes of sheaves but crossed complexes.
In his work
Ross Street had proposed a formulation of the descent condition for such $Str \omega Grpd$-valued $\infty$-stacks (see the corresponding section at descent for the details).
The question remained open how that definition of descent on $Str \omega Grpd$-valued presheaves relates to the general one of $\infty Grpd$-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 $Str \omega Grpd$-valued presheaf $A$ is the correct one if the hypercover $\pi : Y \to X$ along which one checks descent is sufficiently well behaved – in that the cosimplicial $\infty$-groupoid $A(Y)$ is Reedy fibrant.