The statement of Verdier duality was fairly fancy–it used the jazzed up language of derived categories, and was phrased in terms of the existence of an adjoint to a suitable derived functor. Though by the end of it hopefully it’ll become clear that, at least for manifolds, the mysterious adjoint functor has a fairly concrete interpretation that will turn out to be plain old Poincare duality.
But before we get there, we need to discuss some sheaf theory. I have usually thought of sheaves in the context of algebraic geometry, and I wanted to review (at least for my own benefit) some of the classical theory of sheaves on ordinary non-pathological topological spaces, like locally compact Hausdorff ones. In addition, it will at any rate be necessary to introduce the compactly supported cohomology functors to get Verdier duality. So I’ll do a couple of preparatory posts.
1. Soft sheaves
Let be a locally compact space. If and is closed, we define for the inclusion. This allows us to make sense of a section “over a closed subset.” The interpretation via the espace étale is helpful here: if is the espace étale of , then a section over is the same thing as a section of the projection over . It follows from the latter interpretation that if we have sections on closed subsets that agree on , one obtains uniquely a closed subset on . Now we want to show that we can recover this notion from the familiar idea of sections over an open set.
where ranges over open sets containing . (more…)