Etale Homotopy Types and Bisimplicial Hypercovers

TL;DR

This paper introduces a new étale homotopy type for pointed simplicial sheaves on Grothendieck sites, demonstrating its invariance, relation to classical types, and connection to sheaf cohomology, with comparisons to bisimplicial hypercover constructions.

Contribution

It defines a novel étale homotopy type for simplicial sheaves, proves its invariance under local weak equivalences, and relates it to existing types and sheaf cohomology.

Findings

The new étale homotopy type specializes to Artin-Mazur's type.

It is invariant under pointed local weak equivalences.

It is naturally isomorphic to the bisimplicial hypercover-based type.

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.