Stable Homotopy Theory of Simplicial Presheaves

Abstract

Let C be an arbitrary Grothendieck site. The purpose of this note is to show that, with the closed model structure on the category S Pre( C ) of simplicial presheaves in hand, it is a relatively simple matter to show that the category S Pre( C ) stab of presheaves of spectra (of simplicial sets) satisfies the axioms for a closed model category, giving rise to a stable homotopy theory for simplicial presheaves. The proof is modelled on the corresponding result for simplicial sets which is given in [ 1 ], and makes direct use of their Theorem A.7. This result gives a precise description of the associated stable homotopy category Ho ( S Pre(C) ) stab , according to well known results of Quillen [ 6 ]. One will recall, however, that it is preferable to have several different descriptions of the stable homotopy category, for the construction of smash products and the like.