By a theorem of Cisinksi, every combinatorial model category defines a strong derivator. For a Grothendieck category A, there are several combinatorial model structures defined on A, thus its derived category is the base of a strong derivator. In this talk, we present an alternative path to this result assuming further that A has enough projective objects. This approach has the benefit of simplicity (and less prerequisites) and gives a very explicit description of homotopy Kan extensions, in particular homotopy limits and colimits. We will present these results. Further, as an application, we will show how to extend the description of local cohomology via Koszul complexes from closed subsets to arbitrary systems of supports, i.e. stable for specialization subsets. Time permitting, we will discuss how this point of view applies to the co/homology of groups.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
