Deligne Localized Functors

In this paper we present the notion of ``Deligne localized functors'', an avatar of the derived functors, whose definition is inspired by Deligne in [SGA 4,XVII]. Their definition involves the notions of Ind and Pro categories, they always exist and are characterized in terms of universal properties. The classical localized functor, in the sense of Grothendieck and Verdier, exists if suitable conditions are verified for the Deligne localized functors. We apply these notions to triangulated and derived categories.