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.

Principle

Translated: the obstruction to exactness of a functor is encoded in higher homological invariants; compute these by resolving objects so that applying the functor becomes homologically tractable and then extract the obstruction degrees as homology/cohomology groups.

Demonstration

Tensor product − ⊗_R − is right-exact; its left-derived functors are Tor_i^R(−,−). Hom_R(−,−) is left-exact in its first argument; its right-derived functors are Ext^i_R(−,−). These are obtained by applying the functor to appropriate resolutions and reading off homology groups.

Misapplication

Attempting to read off derived functors from applying F to non-resolving complexes, ignoring boundedness or existence hypotheses, or equating the naive homology of an arbitrary complex with the derived functor without checking quasi-isomorphism invariance.

Consequence

Derived functors produce long exact sequences, spectral sequences, and other formal tools (e.g., Grothendieck spectral sequence) that control how functors interact with exact sequences; they generalize classical invariants like Tor and Ext and are fundamental to derived-category methods.

Reversal

The underived functor is what one gets by applying F directly to objects: it ignores higher obstruction data and therefore loses homological information captured by derived functors.

Boundary

Defined in abelian categories (or stable ∞-categories) where appropriate projective or injective resolutions exist or where derived categories exist; care is needed in non-abelian settings, for functors that are not additive, or when using unbounded complexes without suitable model or derived-category frameworks.

Semantic Tension

Tension between left and right derived constructions (choice of projective vs injective resolutions), between classical resolution-based derived functors and more modern derived-functor constructions in derived/∞-categories, and between computational convenience and conceptual functoriality.

Synthesis

A derived functor upgrades a classical functor to a homological invariant by resolving inputs so that the functor's failure of exactness is captured in higher-degree homology or cohomology, producing systematic invariants (R^iF, L_iF) that govern extensions, obstructions, and spectral-sequence phenomena.