The problem of regularity for sheaves on the subanalytic site

Let X be a real analytic manifold. Sheaf theory is only suited to study objects which can be defined by local properties. For example, the property of being temperate is not local, and there is no sheaf of temperate functions. We may overcome this difficulty by introducing a Grothendieck topology on X. In this talk, we study the subanalytic site, that is the category of open subanalytic subsets of X endowed with a Grothendieck topology, and the problem of regularity, which is motivated by the notion of microsupport. More precisely, we prove a Kashiwara-Schapira Conjecture that relates the notion of regularity for the sheaves on the subanalytic site and that for D-modules.