nLab
split hypercover

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

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.

Definition

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.

Properties

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.

Examples

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.

References

Revised on August 31, 2010 03:56:17 by Urs Schreiber (87.212.203.135)