nLab
split hypercover

Idea

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

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

Definition

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

1. $Y$ is a hypercover in that

   1. $Y$ is degreewise a coproduct of representables,

      $Y={\int }^{[n]\in \Delta }\Delta [n]\cdot {\coprod }_{i_n}{U}_{i_n},\mathrm{with}\{U_{i_n}\in C\}$ ;

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

2. $Y$ 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 $Y$ becomes a cofibrant object in $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$, 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

• Dan Dugger, Sharon Hollander, Dan Isaksen, Hypercovers and simplicial presheaves (arXiv:math.AT/0205027)