Flexible sheaves

We define a notion of flexible functor from a category X to the category of topological spaces (or other similar categories). This is a generalization of the notion of functor, where the condition of compatibility with composition is replaced by the data of homotopies, homotopies between the homotopies, and so on. When X is a site, we define a sheaf condition and show that a flexible functor (truncated, of CW-type, over a quasi-compact site) admits an associated flexible sheaf. We develop many of the standard techniques of topology in this context. Then we consider the case where X is the category of schemes of finite type over a field of characteristic zero with the faithfully flat topology. We make several calculations in this case. We define a notion of representable flexible sheaf in this case. We how that if Y is a constant topological space, or a sheaf on X of a certain type related to the cristalline topology of a smooth scheme, then the sheaf of pointed morphisms into a representable flexible sheaf is again representable. This generalizes the construction of moduli spaces for vector bundles with integrable connection. (If (U, u) is a pointed representable flexible sheaf and (Y, y) is a pointed space, thenMap((Y, y), (U, u)) is a sort of “Betti” nonabelian cohomology of (Y, y) In the present version I have left these sections out.