Post date: 16-jul-2015 15:52:42
In the context of relative (or homotopical) categories, i.e. a category equipped with a class of weak equivalences (satisfying further properties), we define derived functors as certain lift of a Kan extension along the homotopy category. Intuitively, a derived functor of a given functor F between homotopical categories is the best approximation of F by a homotopical functor, i.e. one that preserves weak equivalences. In most of the cases, the derived functor is constructed by some sort of "deformation", we will show that this construction indeed produces a derived functor, and moreover, this satisfies the amazing property of being an absolute Kan extension. From this we conclude that adjunctions of "deformable functors" descent to adjunctions between the homotopy categories. We discuss how this applies in the context of model categories and in homological algebra as our main examples, though this degree of generality will be necessary in the next talks.