Etale Homotopy Types and Bisimplicial Hypercovers

Abstract

An \'etale homotopy type $T(X, z)$ associated to any pointed locally fibrant connected simplicial sheaf $(X, z)$ on a pointed locally connected small Grothendieck site $(\mc{C}, x)$ is studied. It is shown that this type $T(X, z)$ specializes to the \'etale homotopy type of Artin-Mazur for pointed connected schemes $X$, that it is invariant up to pro-isomorphism under pointed local weak equivalences (but see \cite{Schmidt1} for an earlier proof), and that it recovers abelian and nonabelian sheaf cohomology of $X$ with constant coefficients. This type $T(X, z)$ is compared to the \'etale homotopy type $T_b(X, z)$ constructed by means of diagonals of pointed bisimplicial hypercovers of $x = (X, z)$ in terms of the associated categories of cocycles, and it is shown that there are bijections \pi_0 H_{\hyp}(x, y) \cong \pi_0 H_{\bihyp}(x, y) at the level of path components for any locally fibrant target object $y$. This quickly leads to natural pro-isomorphisms $T(X, z) \cong T_b(X, z)$ in $\Ho{\sSet_\ast}$. By consequence one immediately establishes the fact that $T_b(X, z)$ is invariant up to pro-isomorphism under pointed local weak equivalences. Analogous statements for the unpointed versions of these types also follow.