(also nonabelian homological algebra)
additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
Let $\mathcal{A}$, $\mathcal{B}$ be two abelian categories.
A homological $\delta$-functor from $\mathcal{A}$ to $\mathcal{B}$ is for each $n \in \mathbb{N}$ a functor
equipped for each short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ with a natural transformation
such that for each such short exact sequence there is, naturally a long exact sequence
The archetypical example is the chain homology functor
from the category of chain complexes of some abelian category (for $\mathbb{N}$-graded complexes).
The universal example are (non-total) right derived functors.
The notion is due to
A textbook account is for instance section 2.1 of