Presheaves over the sets which need not be well ordered, functional separation of their inductive limits and their representation by sections
Given a real analytic manifold Y, denote by the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor on the category of sheaves on and study its properties. Roughly speaking, is a sheaf on . As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.