Holmstrom Derived functor

See Murfet

According to Weibel, (p. 29) a notion of left derived functor can be defined for any functor from a category to an abelian category, as long as the domain is closed under finite limits and has enough projective objects.

Derived functors are examples of Kan extensions, see MacLane.

Eilenberg: Bourbaki exp 46

Derived functors can be defined in the more general setting of homotopical algebra nLab

One can talk about versions of derived functors in settings without enough projectives or injectives. See eg Buchsbaum in LNM0061, and maybe ask Julia.

Tierney and Vogel in LNM0086 talks about deriving any functor from a cat with finite limits and a projective class, to an abelian cat. This encompasses several other notions.

http://www.ncatlab.org/nlab/show/satellite

http://ncatlab.org/nlab/show/derived+functor+on+a+derived+category

Def: A left Ken Brown functor is a functor which carries trivial cofibs between cofibrant objects to weak equivalences. I think it is true that a left KB functor admits an absolute total left derived functor (Maltsiniotis lectures in Seville). Also, consider a composition of a functor $F$ followed by another functor $F'$. Assume both are KB, and that $F$ takes cofibrant objects to cofibrant objects. Then $L(F' \circ F) \to L(F') \circ L(F)$ is an iso.

http://mathoverflow.net/questions/93716/cov-right-exact-additive-functors-that-dont-commute-with-direct-sums

nLab page on Derived functor