A Pair of Quasi-Inverse Functors for an Extension of Perverse Sheaves

In their article "Elementary construction of perverse sheaves", R.MacPherson and K. Vilonen show that on a Thom-Mather space X the category PervX of perverse sheaves is equivalent to the category C(F, G, T) whose objects are data of perverse sheaves on the complementary of the closed strata S, a local system on S and some gluing data. To show this equivalence of categories, they define a functor C going from the category PervX to the category C(F, G, T). This definition is based on the notion of perverse link. They do not define a quasi-inverse of this functor. moreover they have to consider first the case where S is contractible and then they extend the equivalence to the topological case using the stack theory. In this paper we propose to consider what we call a perverse closed set which is a bit different from a perverse link in order to define a quasi-inverse to the functor C. Moreover we treat directly the topological case without using stack theory.