2010/02/18 by Michael D. Misamore, Misamore, Michael D.
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1002.3532
An étale homotopy type T(X, z) associated to any pointed locally fibrant connected simplicial sheaf (X, z) on a pointed locally connected small Grothendieck site (\mcC, x) is studied. It is shown that this type T(X, z) specializes to the étale 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 \citeSchmidt1 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 étale homotopy type Tb(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 π0 H\hyp(x, y) ≅ π0 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) ≅ Tb(X, z) in \Ho\sSet_∗. By consequence one immediately establishes the fact that Tb(X, z) is invariant up to pro-isomorphism under pointed local weak equivalences. Analogous statements for the unpointed versions of these types also follow.