and
nonabelian homological algebra
In a additive category with translation $T : C \to C$ a complex is a differential object
such that
is the zero morphism.
A complex in the category $Gr(A)$ of graded objects in an additive category $C$ is called a chain complex.
For $d_X : X \to T X$ a complex, the shifted differential object $d_{T X} : T X \stackrel{-T(d_X)}{\to} T T X$ is again a complex.
For instance section 11 of