  We discuss two known sheaf-cosheaf duality theorems: Curry's for the face
posets of finite regular CW complexes and Lurie's for compact Hausdorff spaces,
i.e., covariant Verdier duality. We provide a uniform formulation for them and
prove their generalizations. Our version of the former works over the sphere
spectrum and for more general finite posets, which we characterize in terms of
the Gorenstein* condition. Our version of the latter says that the
stabilization of a proper separated $\infty$-topos is rigid in the sense of
Gaitsgory. As an application, for stratified topological spaces, we clarify the
relation between these two duality equivalences.
