Derived Functor - Naturality

Naturality

Derived functors and the long exact sequences are "natural" in several technical senses.

First, given a commutative diagram of the form

(where the rows are exact), the two resulting long exact sequences are related by commuting squares:

Second, suppose η : F → G is a natural transformation from the left exact functor F to the left exact functor G. Then natural transformations Riη : RiF → RiG are induced, and indeed Ri becomes a functor from the functor category of all left exact functors from A to B to the full functor category of all functors from A to B. Furthermore, this functor is compatible with the long exact sequences in the following sense: if

is a short exact sequence, then a commutative diagram

is induced.

Both of these naturalities follow from the naturality of the sequence provided by the snake lemma.

Conversely, the following characterization of derived functors holds: given a family of functors Ri: A → B, satisfying the above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object I of A, Ri(I)=0 for every positive i, then these functors are the right derived functors of R0.