8 - Derived Functors

Summary

Here we talk about derived functors. To make the definitions precise, we introduce 2-categorical notation. Suppose K and E are abstract categories, F : K → E is a functor and S ⊆ K is a multiplicatively closed set of morphisms. In this context we define the left and right derived functors of F w.r.t. S.These derived functors RF, LF : KS → E have universal properties, making each unique up to a unique isomorphism. Then we provide a general existence theorem for right and left abstract derived functors, in terms of the existence of suitable resolving subcategories J, P ⊆ K, respectively.In Section 8.4 we specialize to triangulated derived functors. Here K and E are triangulated categories, F : K → E is a triangulated functor and S ⊆ K is a multiplicatively closed set of cohomological origin. The right and left derived functors RF, LF : KS → E are defined like in the abstract setting, and their uniqueness is also proved the same way. Existence requires resolving subcategories P and J that are full triangulated subcategories of K. The chapter is concluded with a discussion of contravariant triangulated derived functors.