A few steps with Grothendieck derivators

The theory of derivators (going back to Grothendieck, Heller, and others) provides an axiomatic approach to homotopy theory. It adresses the problem that the rather crude passage from model categories to homotopy categories results in a serious loss of information. In the stable context, the typical defects of triangulated categories (non-functoriality of cone construction, lack of homotopy colimits) can be seen as a reminiscent of this fact. The simple but surprisingly powerful idea behind a derivator is that instead one should form homotopy categories of various diagram categories and also keep track of the calculus of homotopy Kan extensions.

In this talk we cover some basics of derivators culminating in a sketch proof that stable derivators provide an enhancement of triangulated categories. Possibly more important than this result itself are the techniques developed along the way as they lay the foundations for further research directions. The aim of this talk is to (hopefully) advertise derivators as a convenient, 'weakly terminal' approach to axiomatic homotopy theory.