Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
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.
Revised on August 31, 2010 03:56:17
by Urs Schreiber