A Remark on Sheaf Theory for Non-hausdorr Manifolds

Much of sheaf theory can be developed for arbitrary topological spaces. This applies, for example , to the deenition of 'sheaf' itself, to the existence of injective resolutions, to the properties of the operations f and f associated to a continuous map f : Y ?! X, etc, etc. On the other hand, there is a very basic part of the theory which seems to depend crucially on the Hausdorr property (together with local compactness and paracompactness). Here one could think of the properties of soft and ne sheaves, of compact supports, of the operation f ! and its right adjoint f ! ('Verdier duality'), etc. It is for this reason that, for a large part of the theory, all the standard text make the overall assumption that the underlying spaces must be locally compact, Hausdorr, and of nite cohomological dimension (cf. 7, 10, 2]). There are geometric situations, however, such as foliation theory, where one naturally encounters sheaves on non-separated manifolds. For example, a central role is played by the holonomy groupoid of a foliation, and by Haeeiger's classifying groupoid ? q. These groupoids are smooth and very well behaved in many respects, but they are not all Hausdorr. This fact impedes not only the (transverse) sheaf theory on foliations, but also the study of the convolution algebra and the (reduced) C-algebra associated to the foliation. Motivated by foliations, we wish to indicate in this short note how sheaf theory can be extended, in an essentially unique way, to spaces which are locally suuciently nice, but are not necessarily separated. The crucial step is a suitable adaptation of 'compact supports' in such spaces. After this adaptation, all the usual constructions and arguments (of 2], say) up to Verdier duality (which includes Poincar e duality) and beyond, remain valid for this more general class of spaces. This note was originally written as an appendix to an earlier version of 6], where we apply the extended sheaf theory to the study of the cyclic type homologies of non-separated smooth groupoids, such as holonomy groupoids of a foliations. 1. Overall assumptions. For any space X in this paper we do assume that X has an open cover by subsets U X which are each paracompact, Hausdorr, locally compact, and of cohomological dimension bounded by a number d (depending on X but not on U). 2. c-soft sheaves. Let X be a space …