A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology it suffices to verify it for Zariski squares and the empty cover of the empty scheme. Similarly, homotopy descent for the Nisnevich topology boils down to Nisnevich squares and the empty cover.

There is an analog of this result for smooth manifolds: the homotopy descent property for the open cover topology on smooth manifolds boils down to the descent for Mayer-Vietoris squares and descent for covers of disjoint unions by their components, i.e., {U_i → ∐_k U_k}. Similar statements can be proved for topological or PL-manifolds, CW-complexes, or even arbitrary topological spaces if we use the numerable site.

Of course, the descent property for Mayer-Vietoris squares is already implicitly present in those variants of the axioms of generalized cohomology theories that include the Mayer-Vietoris property as one of the axioms. Similarly, Milnor's axiom encodes the descent property for covers of disjoint unions. For the case of smooth manifolds, a recent paper by Kreck and Singhof “Homology and cohomology theories on manifolds” has an explicit formulation of the axioms of cohomology theories on smooth manifolds, which also contains the analogs of the above descent properties.

The Brown representability theorem (in any of its several formulations) can then be seen as the analog (on the level of homotopy groups) of the classical fact that homotopy-invariant sheaves on manifolds, CW-complexes, or topological spaces (with weak equivalences inverted and using numerable covers) are precisely the representable sheaves.

However, it doesn't seem to be possible (though I will be happy to be disproved on this point) to deduce the above result on the level of Grothendieck topologies from any of the above-cited classical results, and I also cannot find anything on this matter in the literature.

Is there a published reference that proves that the Grothendieck topology of open covers on smooth manifolds is generated by covers with two elements (i.e., Mayer-Vietoris squares) and covers of disjoint unions by their components?


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