Derivators

Derivators have been introduced by Grothendieck, as one of his last mathematical contributions (and by Heller and others independently) to simplify, extend, and conceptually clarify the notions of derived and triangulated categories. 

Extended to the context of fibered (multi-)categories (e.g. any kind of sheaves of abelian groups on spaces, schemes, stacks etc.) the notion allows for a neat solution to problems of cohomological descent as well as homological descent. This gives an elegant way of extending the six-functor formalism of Grothendieck (which encodes, among other things, dualities like e.g. Serre duality, Poincar'e-Verdier duality) to stacks.