A remark on sheaf theory for non-Hausdorff manifolds

Abstract

Much of sheaf theory can be developed for arbitrary topological spaces. This applies, for example, to the denition 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 Hausdor 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, Hausdor, and of nite cohomological dimension (cf. [7, 10, 2]).