Sheaves on subanalytic sites, some examples and constructions

We will start by recalling the definitions of sheaves on topological spaces and (topological) sites. We will show (with some examples) that there are functional spaces which are not sheaves in the classical sense but become sheaves when we consider less open subsets and less coverings. We will focus our attention on a particular topological site, the subanalytic site. In this framework we will define the six Grothendieck operations and thanks to them the specialization. Then we will see as we can construct functorially sheaves of function asymptotically developable.