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(2,1)-quasitopos?
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A split hypercover is a cofibrant resolution of a representable in the projective local model structure on simplicial presheaves over a site .
It is a hypercover satisfying an extra condition that roughly says that it is degreewise freely given by representables.
Regard under the Yoneda embedding as an object . Then a morphism is a split hypercover of if
is a hypercover in that
is degreewise a coproduct of representables,
;
with regarded as a presheaf of augmented simplicial sets, for all the morphism into the -cells of the -coskeleton is a local epimorphism with respect to the given Grothendieck topology on
is split in that the image of the degeneracy maps identifies with a direct summand in each degree.
The splitness condition on the hypercover is precisely such that becomes a cofibrant object in , according to the characterization of such cofibrant objects described here.
Over the site CartSp, the Cech nerve of an open cover becomes split as a height-0 hypercover precisely if the cover is a good open cover.
Last revised on May 1, 2021 at 05:00:14. See the history of this page for a list of all contributions to it.