nLab Brown-Gersten property

Contents

Contents

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 2U_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

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

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

(1)(X)holim(iI(U i)i 1,i 2I(U i 1U i 2)) \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 inPSh(SmthMfd,sSet)\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

    XU 1U 2, X \,\simeq\, U_1 \cup U_2 \,,

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

    (X) (U 1) (U 2) (U 1U 2) \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 XiIU iX \,\simeq\, \underset{i \in I}{\coprod} \, U_i, the homotopy descent-condition (1) reduces to saying that

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

    is a homotopy product.

    In particular, if X=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:

The version for the Nisnevich topology:

Detailed review of these two cases:

Generalization of these two cases to completely decomposable topologies:

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:

Last revised on August 7, 2022 at 17:55:53. See the history of this page for a list of all contributions to it.