Flexible sheaves

1 This is an unfinished explanation of the notion of " flexible sheaf " , that is a homotopical notion of sheaf of topological spaces (homotopy types) over a site. See " Homotopy over the complex numbers and generalized de Rham cohomology " (submitted to proceedings of the Taniguchi conference on vector bundles, preprint of Toulouse 3, and also available at my homepage 2) for a more detailed introduction, and also for a further development of the ideas presented here. The present paper was finished in December 1993 while I was visiting MIT. Since writing this, I have realized that the theory sketched here is essentially equivalent to Jardine-Illusie's theory of presheaves of topological spaces (although they talk about presheaves of simplicial sets which is the same thing). This is the point of view adopted in " Homotopy over the complex numbers and generalized de Rham cohomology ". 1 Added in August 1996: I am finally sending this to " Duke eprints " because it now seems that there may be several useful features of the treatment given here. First of all, the direct construction of the homotopy-sheafification (by doing a certain operation n + 2 times) seems to be useful, for example I need it in the preprint " Topological realization of a simplicial presheaf ". Second, the construction of canonical inverses for homotopy equivalences seems to be interesting even in the case of topological spaces. Third, the strictification of a flexible presheaf into a presheaf (the operation denoted K below) is probably a paradigm for what could eventually be done in the realm of n-stacks; and in fact the whole treatment here should serve as a paradigm for the treatment of n-stacks. Finally, V. Navarro-Aznar recently pointed out that there are cases where the flexible point of view might be essential, particularly in his theory of presheaves of complexes (the flexible version of which was worked out to levels 2 and 3 in [22], and this was in fact one of the origins for our notion of flexible sheaf). For example one might want to look at complexes with values in an additive category which doesn't admit very big limits, such as the category of motives [9]. In this case the strictification procedure no longer works and there may well be flexible presheaves which are not equivalent to strict presheaves. A couple of references have been added in …