In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. (Def: http://en.m.wikipedia.org/wiki/Derived_functor)

The technique of taking a left/right exact functor $\mathcal{F}$ between abelian categories and deriving a collection of functors $\{R^\bullet\mathcal{F}\}$ such that $R^0\mathcal{F} = \mathcal{F}$ with certain desirable compatible properties (e.g., long exact sequence). Reference: Wikipedia.

This operation, while fairly abstract, unifies a number of constructions throughout mathematics (e.g., derived functors of the $\hom$ and $\otimes$-functor between $R-\mathsf{Mod}$ are $\text{Ext}^n$ and $\text{Tor}^n$-functors of homological algebra, respectively).

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