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.

Principle

Derived functors convert failure of exactness into computable homological invariants: by resolving objects and applying the original functor, cohomology groups record obstructions and extension data that the underived functor misses.

Demonstration

Example: Ext^i(-,-) are the right derived functors R^i Hom(−,−), and Tor_i(-,-) are the left derived functors L_i(− ⊗ −); these arise by applying Hom or tensor to injective or projective resolutions and taking cohomology.

Misapplication

Attempting to compute derived functors without appropriate resolutions, using non-resolving classes when the category lacks enough projectives/injectives, or treating derived functors as objectwise equalities rather than cohomological constructions.

Consequence

Derived functors produce long exact sequences, spectral sequences, and obstruction classes that link short exact sequences in the source category to homological data in the target; they systematize cohomological computations and invariants.

Reversal

The reversal is the underived functor acting on projective or injective objects where higher derived functors vanish: in that restricted setting the derived construction collapses to the original functor and no new cohomology appears.

Boundary

Derived functors require abelian or derived settings and typically enough projectives or injectives (or model/derived enhancements); in non-abelian contexts one must use homotopical replacements (derived functor in homotopy category) rather than naive resolutions.

Semantic Tension

Derived functor notion competes with homotopy-invariant or total derived constructions: the classical R^i/L_i picture suffices in many abelian cases, but in homotopical or ∞-categorical contexts one prefers total derived functors or derived functors of the homotopy category for better invariance.

Synthesis

A derived functor extends an exactness-limited functor to a graded family measuring its failure to be exact: through resolutions and cohomology it encodes extension, obstruction and higher-order information that is crucial to homological algebra.