The Verdier Hypercovering Theorem

This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms [X, Y] in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where Y is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object X to be locally fibrant.

EXCISION FOR SIMPLICIAL SHEAVES ON THE STEIN SITE AND GROMOV'S OKA PRINCIPLE

A complex manifold X is said to satisfy the Oka–Grauert property if the inclusion [Formula: see text] is a weak equivalence for every Stein manifold S, where the spaces of holomorphic and continuous maps from S to X are given the compact-open topology. Gromov's Oka principle states that if X has a spray, then it has the Oka–Grauert property. The purpose of this paper is to investigate the Oka–Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka–Grauert property is equivalent to X representing a finite homotopy sheaf on the Stein site. This expresses the Oka–Grauert property in purely holomorphic terms, without reference to continuous maps.

On the homotopy theory of sheaves of simplicial groupoids

The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as extraordinary, sheaf cohomology.In [11] we began constructing a theory of classifying spaces for sheaves of simplicial groupoids, and that study is continued here. Such a theory is essential for the development of basic tools such as Postnikov systems, Atiyah-Hirzebruch spectral sequences, characteristic classes, and cohomology operations in extraordinary cohomology of sheaves. Thus, in some sense, we are continuing the program initiated by Illusie[7], Brown[2], and Brown and Gersten[3], though our basic homotopy theory of simplicial sheaves is different from theirs. In fact, the homotopy theory we use is the global one of [10]. As a result, there is some similarity between our theory and the theory of Jardine[8], which is also partially based on [10]