nLab Brown-Gersten property

Idea

The Brown-Gersten property is the statement that for certain Grothendieck topologies the homotopy descent-property for simplicial presheaves (i.e. their $\infty$-stack-property) follows essentially (up to a simple condition on connected components) already as soon as the given simplicial presheaf $\mathcal{F}$ satisfies a Mayer-Vietoris-type property in that for covers by a pair of patches $U_1, U_2$ their fiber product-square is taken by $\mathcal{F}$ to a homotopy pullback-square.

This has traditionally been discussed over schemes (see below) but an analogous statement holds also over topological spaces and smooth manifolds (see further below).

Details

Over schemes

Detailed review in in Jardine 2015, §5.4.

Over smooth manifolds

Write

• SmthMfd for the category of smooth manifolds, regarded as a site via its Grothendieck topology of open covers;

• $PSh\big(SmthMfd, sSet)\big)$ for the category of simplicial presheaves over smooth manifolds.

Recall (e.g. from model structure on simplicial presheaves) that a simplicial presheaf $\mathcal{F} \,\in\, Psh(SmthMfd, sSet)$ is said to be a local object or satisfy homotopy descent if for

• all smooth manifolds $X \in SmthMfd$;

• all its open covers $\big\{ U_i \hookrightarrow X \big\}_{i \in I}$

the canonical morphism of simplicial sets from the value of $\mathcal{F}$ on $X$ into the homotopy limit of the simplicial diagram over values of $\mathcal{F}$ on the Čech nerve of the cover

(1)$$\mathcal{F}(X) \xrightarrow{\phantom{----}} \underset{}{holim} \bigg( \underset{i \in I}{\prod} \mathcal{F}(U_i) \rightrightarrows \underset{i_1, i_2 \in I}{\prod} \mathcal{F}(U_{i_1} \cap U_{i_2}) \cdots \bigg)$$

is a simplicial weak equivalence.

Theorem

(Brown-Gersten property over smooth manifolds, Pavlov 2022, Thm. 1.1)
A simplicial presheaf $\mathcal{F} \,in\, PSh(SmthMfd, sSet)$ satisfies homotopy descent (1) for all open covers, as soon as it does so

1. for open covers consisting of a pair of open subsets;

2. for open covers by pairwise disjoint patches (hence by connected components).

Notice here that

1. in the case of an open cover by pair of patches

   $$X \,\simeq\, U_1 \cup U_2 \,,$$

   the homotopy descent-condition (1) reduces to saying that the commuting square

   $$\array{ \mathcal{F}(X) &\longrightarrow& \mathcal{F}(U_1) \\ \big\downarrow && \big\downarrow \\ \mathcal{F}(U_2) &\longrightarrow& \mathcal{F}(U_1 \cap U_2) }$$

   is a homotopy cartesian square (a homotopy pullback).

2. in the case of a cover by disjoint patches, hence $X \,\simeq\, \underset{i \in I}{\coprod} \, U_i$, the homotopy descent-condition (1) reduces to saying that

   $$\mathcal{F}(X) \;\;\simeq\;\; \underset{i \in I}{\prod} \mathcal{F}(U_i)$$

   is a homotopy product.

   In particular, if $X = \varnothing$ is the empty space covered by itself, this means that

   $$\mathcal{F}(\varnothing) \;\;\simeq\;\; \ast$$

   is the terminal object (i.e. the simplicial presheaf which is constant on the singleton set).

In condensed mathematics

A Brown–Gersten-type property holds for condensed sets; as explained in the linked article, it suffices to verify the descent property for disjoint covers with zero or two elements, as well as singleton families given by surjections of profinite sets.

This property no longer holds for condensed ∞-groupoids, and hypercovers are now necessary. However, one can pass to the equivalent site of compact extremally disconnected Hausdorff spaces, where finite disjoint covers suffice.

References

The original discussion for the Zariski topology:

• Kenneth S. Brown, Stephen Gersten, Algebraic K-theory as generalized sheaf cohomology, in: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 266-292. Lecture Notes in Math. 341 Springer (1973) [doi:10.1007/BFb0067062]

Lecture notes on this case:

• J. F. Jardine, Brown-Gersten and Nisnevich descent, motivic descent, Lecture 13 of: Lectures on Local Homotopy Theory [pdf]

The version for the Nisnevich topology:

• Fabien Morel, Vladimir Voevodsky, $\mathbb{A}^1$-homotopy theory of schemes, Publications Mathématiques de l’Institut des Hautes Études Scientifiques 90 (1999) 45–143 [doi:10.1007/BF02698831]

Detailed review of these two cases:

• John F. Jardine, Section 5.4 of: Local homotopy theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]

Generalization of these two cases to completely decomposable topologies:

• Vladimir Voevodsky, Homotopy theory of simplicial presheaves in completely decomposable topologies, Journal of Pure and Applied Algebra 214 8 (2010) 1384-1398 [arXiv:0805.4578, doi:10.1016/j.jpaa.2009.11.004]

Discussion for numerable open covers of topological spaces and of smooth manifolds with application to smooth $\infty$-groupoids and their shape via cohesive path ∞-groupoids:

• Dmitri Pavlov, Numerable open covers and representability of topological stacks, Topology and its Applications 318 (2022) 108203 [arXiv:2203.03120, doi:10.1016/j.topol.2022.108203]