Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On the homotopy theory of sheaves of simplicial groupoids

André Joyala1 and Myles Tierneya2

a1 Département de mathématiques, Université du Québec à Montréal, Montréal, H3C 3P8, Canada

a2 Department of Mathematics, Rutgers University, New Brunswick, New Jersey, 08903, U.S.A.

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]

(Received February 08 1994)

(Revised May 19 1994)