nLab split hypercover



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A split hypercover is a cofibrant resolution of a representable in the projective local model structure on simplicial presheaves [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc} over a site CC.

It is a hypercover satisfying an extra condition that roughly says that it is degreewise freely given by representables.


Regard XCX \in C under the Yoneda embedding as an object X[C op,sSet] proj,locX \in [C^{op}, sSet]_{proj,loc}. Then a morphism (YX)[C op,sSet](Y \to X) \in [C^{op}, sSet] is a split hypercover of XX if

  1. YY is a hypercover in that

    1. YY is degreewise a coproduct of representables,

      Y= [n]ΔΔ[n] i nU i n,with{U i nC}Y = \int^{[n] \in \Delta} \Delta[n] \cdot \coprod_{i_n} U_{i_n} \;\,, \;\;\; with \{U_{i_n} \in C\} ;

    2. with YXY \to X regarded as a presheaf of augmented simplicial sets, for all nn \in \mathbb{N} the morphism Y n+1(cosk nY) n+1Y_{n+1} \to (\mathbf{cosk}_n Y)_{n+1} into the n+1n+1-cells of the nn-coskeleton is a local epimorphism with respect to the given Grothendieck topology on CC

  2. YY 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 YY becomes a cofibrant object in [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc}, 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.