In homological algebra, the hyperhomology or hypercohomology (${\displaystyle \mathbb {H} _{*}(-),\mathbb {H} ^{*}(-)}$) is a generalization of (co)homology functors which takes as input not objects in an abelian category ${\displaystyle {\mathcal {A}}}$ but instead chain complexes of objects, so objects in ${\displaystyle {\text{Ch}}({\mathcal {A}})}$. It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor ${\displaystyle \mathbf {R} ^{*}\Gamma (-)}$.

Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories.

One of the motivations for hypercohomology comes from the fact that there is no obvious generalization of cohomological long exact sequences associated to short exact sequences

${\displaystyle 0\to M'\to M\to M''\to 0}$

i.e. there is an associated long exact sequence

${\displaystyle 0\to H^{0}(M')\to H^{0}(M)\to H^{0}(M'')\to H^{1}(M')\to \cdots }$

It turns out that hypercohomology gives techniques for constructing a similar cohomological associated long exact sequence from an arbitrary long exact sequence since its inputs are given by chain complexes instead of just objects from an abelian category.

${\displaystyle 0\to M_{1}\to M_{2}\to \cdots \to M_{k}\to 0}$