nLab complex

Definition

In a additive category with translation $T : C \to C$ a complex is a differential object

$$d_X : X \to T X$$

such that

$$X \stackrel{d_X}{\to} T X \stackrel{T d_X}{\to} T T X$$

is the zero morphism.

Examples

• A complex in the category $Gr(C)$ 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.

Related concepts

• chain complex

References

For instance section 11 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves,