Hyperhomology functor

A series of functors $\mathbf{L}_n F$ on the category of complexes connected with some functor $F$. In fact, let $F\colon A \to B$ be a covariant additive functor from an Abelian category $A$ with a sufficient number of projective objects into an Abelian category $B$. Further, let $K_{\bullet}$ be a chain complex with values in $A$ and let $L_{\bullet \bullet}$ be a Cartan–Eilenberg resolution of $K_{\bullet}$, consisting of projective objects. Then the bicomplex $F(L_{\bullet\bullet})$ determines the homology groups $H_n(F(L_{\bullet\bullet})) = \mathbf{L}_n F(K_{\bullet})$ and two spectral sequences (cf. Spectral sequence) converging to them with initial terms

$$ {}'E^2_{p,q} = H_p(L_q F(K_{\bullet})) \quad \text{and} \quad {}''E^2_{p,q} = L_p F(H_q(K_{\bullet})) . $$

These homology groups and spectral sequences depend functorially on $K_{\bullet}$ and are known, respectively, as the hyperhomology functors for $F$ and the spectral hyperhomology functors for $F$. The hyperhomology functor $\mathbf{L}_{\bullet} F$ is a homology functor on the category of complexes in the following important cases: when $F$ commutes with inductive limits; when the objects in the category $A$ have projective resolutions of length $\le n$; or when it is considered on the category of complexes with positive degrees.

Hypercohomology functors are defined dually.

References

[1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221