Derived Functor

A construction that assigns to a (typically non-exact) additive functor F between abelian categories a sequence of functors R^iF or L_iF (right or left derived functors) measuring the failure of exactness, obtained by applying F to an injective or projective resolution and taking cohomology or homology, or equivalently by passing to the total derived functor in the derived category.

A construction that associates to a (left or right) exact functor between abelian (or suitably derived) categories a sequence of higher functors (R^iF or L_iF) obtained by applying F to resolutions (injective or projective) and taking cohomology; it measures how far F fails to be exact.