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July 23, 2009

Verity on Descent for Strict ω-Groupoid Valued Presheaves

Posted by Urs Schreiber

A while ago I had sent a question on a certain aspect of the notion of descent to Dominic Verity.

To my pleasant surprise, a few days later he sent me a detailed 13 page description and proof of the statement in question!

I was thinking of asking him to reproduce that document here, but then I hesitated for some reason. Now it so happens that Jim Stasheff emails me today, saying that Dominic Verity permits and that he himself suggests that I do a post.

Under these conditions of course I can’t resist and am very glad to do so. I made an nnLab entry of it

Verity on Descent for strict ω\omega–Groupoid valued Presheaves

Please see there for Dominic Verity’s document, some introductory comments and further links.

Posted at July 23, 2009 7:16 PM UTC

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Re: Verity on Descent for Strict ω-Groupoid Valued Presheaves

I’d like to mention some simple analysis of that condition that Dominic Verity finds needs to hold in order for Street’s definition of descent to agree with the “true” one.

Recall from the above theorem that the condition is that the cosimplicial ω\omega-category XX (whose descent we are interested in) is, after being regarded as a cosimplicial simplicial set NXN\circ X under the nerve, a fibrant object with respect to the canonical Reedy model structure on cosimplicial simplicial sets.

Now, in practice we are usually interested in XX that come from evaluating an ω\omega-category valued presheaf AA – modelling an \infty-(pre)stack – on a simplicial presheaf YY – being some cover space.

So in practice we have

(NX) n=SPSh C(Y n,A)SSet, (N\circ X)_n = SPSh_C(Y_n, A) \in SSet \,,

where CC denotes the underlying site, SPSh(C)SPSh(C) denotes the SSetSSet-enriched category of simplicial presheaves on CC and where we regard the simplicial presheaf components of YY

Y :Δ opPSh(C)SPSh(C) Y_\bullet : \Delta^{op} \to PSh(C) \hookrightarrow SPSh(C)

as a simplicial simplicial presheaf that is componentswise simplicially constant. (This is as usual in this business, compare the discussion at descent.)

Also in practice, we are not so likely to have or want to have extra conditions on AA here, but will be more likely to have or are willing to have extra conditions on the chosen cover Y Y_\bullet.

So first notice the following simple observation

Lemma The cosimplicial simplicial set X=NA(Y )X = N A(Y_\bullet) is Reedy fibrant if Y :Δ opSPSh(C)Y_\bullet : \Delta^{op} \to SPSh(C) is Reedy cofibrant with respect to any of the different standard model structures on simplicial presheaves.

Proof.

This follows from the fact that for all the standard model structures on SPSh CSPSh_C (injective or projective, local or global) the standard simplicial enrichment produces a simplicially enriched model category. For these, the SSetSSet-valued hom sends cofibrations to fibrations in its first argument.

Apply this to the Reedy (co)fibrancy conditions: Y Y_\bullet being Reedy cofibrant means that for all nn \in \mathbb{N} the canonical morphism

(colim [n]>[k]Y k)Y n (colim_{[n] \stackrel{\gt}{\to} [k]} Y_k) \to Y_n

is a cofibration. Applying the SSetSSet-valued hom into AA to this yields that

SPSh C(Y n,NA)(lim [n]>[k]SPSh C(Y k,NA)) SPSh_C(Y_n,N\circ A) \to (lim_{[n] \stackrel{\gt}{\to} [k]} SPSh_C(Y_k,N\circ A))

is a fibration, for all nn \in \mathbb{N}. But this is precisely the condition for the simplicial simplicial set X=NA(Y )X = N\circ A(Y_\bullet) to be Reedy fibrant.

endofproof

So it is sufficient for Dominic Verity’s theorem to apply that the cover Y Y_\bullet is Reedy cofibrant. In that case we may compute the descent for an \infty-prestack that is given in terms of an ω\omega-category valued presheaf by using Street’s formula (which in turn yields the expected descent condition).

So consider some conditions under which a cover Y Y_\bullet is Reedy cofibrant.

The cofibrancy condition mentioned in the above proof says in words that Y Y_\bullet needs to be such that the inclusion of the collection of all degenerate nn-cells into all nn-cells is a cofibration.

Lemma

In the (local or global) injective model structure on simplicial presheaves all Y Y_\bullet are Reedy cofibrant.

In the (local or global) projective model structure on simplicial presheaves at least those Y Y_\bullet that are Čech nerves of good covers U= iU iU = \sqcup_i U_i are Reedy cofibrant.

Here by a good cover I mean a cover such that all Y nY_n of its Cech nerve are coproducts of representables.

Proof.

In the injective model structure (global or local) the cofibrations are the objectwise cofibrations. The map in question, the inclusion of degenerate nn-simplices into all simplices, is objectwise a monomorphism of sets that are regarded as simplicial sets, hence objectwise a monomorphism of simplicial sets. All monomorphisms are cofibrations of simplicial sets in the relevant (standard) model structure on simplicial sets.

For the projective model structure on simplicial presheaves, one notices that

- all representable objects are cofibrant (straightforward from the definition).

- thereby for IJI \hookrightarrow J two index sets and U jU_j a jj-indexed family of representables, the canonical inclusion of cproducts

iU i kU k \coprod_i U_i \hookrightarrow \coprod_k U_k

is a cofibration.

But under the very assumption that Y Y_\bullet is the Čech nerve of a good cover in the above sense, the inclusion of degenerate nn-simplices into all nn-simplices is an inclusion of this kind.

endofproof

Possibly a stronger statement is true also for the projective model structure, but I am not sure yet.

In any case, this shows that for the practical application of determining the descent condition for ω\omega-category valued presheaves regarded as \infty-prestacks the condition that Dominic Verity finds is sufficient to make Street’s descent condition be the correct descent condition is a rather mild one. When using the injective model structure on simplicial presheaves it holds for all covers. For the projective model structure it holds at least for the good Čech covers.

Posted by: Urs Schreiber on August 10, 2009 12:30 PM | Permalink | Reply to this

Re: Verity on Descent for Strict ω-Groupoid Valued Presheaves

I have finally inserted the above statements into a discussion of descent for strict \infty-Lie groupoids in the nnLab entry on \infty-Lie groupoids.

Posted by: Urs Schreiber on July 26, 2010 10:25 AM | Permalink | Reply to this

Re: Verity on Descent for Strict ω-Groupoid Valued Presheaves

The nnLab entry on Dominic Verity’s theorem is one of the dozen or so (see the list given here) whose long entry titles unfortunately got truncated in the nnLab migration process.

To access it, you need to go to

Verity on descent for strict omega-groupoid valued prsheave

without the “s” at the end.

Posted by: Urs Schreiber on September 2, 2009 9:51 AM | Permalink | Reply to this

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