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Verity on descent for strict omega-groupoid valued presheaves

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 ω-groupoids.

Contents

Verity’s theorem

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 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.

Idea

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 (,1)(\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

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

StrωGrpdNGrpd Str \omega Grpd \stackrel{N}{\hookrightarrow} \infty Grpd

the strict ω\omega-groupoids form a subcategory of all ∞-groupoids. This is to be regarded as a refinement of the Dold-Kan correspondence which embeds strict ω\omega-groupoids with abelian group structure equivalently modeled as chain complexes into all \infty-groupoids

Ch +StrAbωGrpdStrωGrpdNGrpd. Ch_+ \stackrel{\simeq}{\to} StrAb \omega Grpd \hookrightarrow Str \omega Grpd \stackrel{N}{\hookrightarrow} \infty Grpd \,.

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 ω\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 StrωGrpdStr \omega Grpd-valued \infty-stacks (see the corresponding section at descent for the details).

The question remained open how that definition of descent on StrωGrpdStr \omega Grpd-valued presheaves relates to the general one of Grpd\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 StrωGrpdStr \omega Grpd-valued presheaf AA is the correct one if the hypercover π:YX\pi : Y \to X along which one checks descent is sufficiently well behaved – in that the cosimplicial \infty-groupoid A(Y)A(Y) is Reedy fibrant.

Revised on December 19, 2011 10:26:16 by Urs Schreiber (83.91.122.110)