The n-Category Café

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 $\omega$-category $X$ (whose descent we are interested in) is, after being regarded as a cosimplicial simplicial set $N\circ X$ 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 $X$ that come from evaluating an $\omega$-category valued presheaf $A$ – modelling an $\infty$-(pre)stack – on a simplicial presheaf $Y$ – being some cover space.

So in practice we have

$$(N\circ X)_n = SPSh_C(Y_n, A) \in SSet \,,$$

where $C$ denotes the underlying site, $SPSh(C)$ denotes the $SSet$-enriched category of simplicial presheaves on $C$ and where we regard the simplicial presheaf components of $Y$

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

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 $A$ here, but will be more likely to have or are willing to have extra conditions on the chosen cover $Y_\bullet$.

So first notice the following simple observation

Lemma The cosimplicial simplicial set $X = N A(Y_\bullet)$ is Reedy fibrant if $Y_\bullet : \Delta^{op} \to SPSh(C)$ 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 $SPSh_C$ (injective or projective, local or global) the standard simplicial enrichment produces a simplicially enriched model category. For these, the $SSet$-valued hom sends cofibrations to fibrations in its first argument.

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

$$(colim_{[n] \stackrel{\gt}{\to} [k]} Y_k) \to Y_n$$

is a cofibration. Applying the $SSet$-valued hom into $A$ to this yields that

$$SPSh_C(Y_n,N\circ A) \to (lim_{[n] \stackrel{\gt}{\to} [k]} SPSh_C(Y_k,N\circ A))$$

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

endofproof

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

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

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

Lemma

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

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

Here by a good cover I mean a cover such that all $Y_n$ 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 $n$-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 $I \hookrightarrow J$ two index sets and $U_j$ a $j$-indexed family of representables, the canonical inclusion of cproducts

$$\coprod_i U_i \hookrightarrow \coprod_k U_k$$

is a cofibration.

But under the very assumption that $Y_\bullet$ is the Čech nerve of a good cover in the above sense, the inclusion of degenerate $n$-simplices into all $n$-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 $\omega$-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.

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

The $n$Lab 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 $n$Lab 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.