Étale homotopy types and bisimplicial hypercovers

Abstract

Suppose $(C,x)$ is a pointed locally connected small Grothendieck site, and let $(X,z)$ denote any connected locally fibrant simplicial sheaf $X$ equipped with a "geometric" point $z$. Following Artin-Mazur, an étale homotopy type of $X$ may then be defined via the geometrically pointed hypercovers of $X$ to yield a pro-object of the homotopy category, but this is not the only possible definition. In Étale homotopy of simplicial schemes, Friedlander defined another étale homotopy type of a simplicial scheme $X$ by taking diagonals of geometrically pointed bisimplicial hypercovers. In this paper, these two types are shown to be pro-isomorphic by means of a direct comparison of the associated cocycle categories. Friedlander's construction of étale homotopy types as actual pro-simplicial sets relies on a rigidity property of the étale topology that may not always be available for arbitrary sites; the cocycle methods employed here do not have this limitation. By consequence, the associated homotopy types constructed from hypercovers and bisimplicial hypercovers are shown to be pro-isomorphic on any locally connected small Grothendieck site, and the comparison at the level of cocycles shows, in particular, that both abelian and non-abelian sheaf cohomology may be computed via bisimplicial hypercovers on arbitrary small Grothendieck sites.