Etale Homotopy Types and Bisimplicial Hypercovers

摘要

An \'etale homotopy type $T(X, z)$ associated to any pointed locally fibrantconnected simplicial sheaf $(X, z)$ on a pointed locally connected smallGrothendieck 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 pointedconnected schemes $X$, that it is invariant up to pro-isomorphism under pointedlocal weak equivalences (but see \cite{Schmidt1} for an earlier proof), andthat it recovers abelian and nonabelian sheaf cohomology of $X$ with constantcoefficients. This type $T(X, z)$ is compared to the \'etale homotopy type$T_b(X, z)$ constructed by means of diagonals of pointed bisimplicialhypercovers 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_0H_{\bihyp}(x, y) at the level of path components for any locally fibrant targetobject $y$. This quickly leads to natural pro-isomorphisms $T(X, z) \congT_b(X, z)$ in $\Ho{\sSet_\ast}$. By consequence one immediately establishes thefact that $T_b(X, z)$ is invariant up to pro-isomorphism under pointed localweak equivalences. Analogous statements for the unpointed versions of thesetypes also follow.