nLab
complex

Definition

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

$$d_X:X\to TX$$d_X : X \to T X

such that

$$X\stackrel{d_X}{\to }TX\stackrel{T{d}_X}{\to }TTX$$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 $\mathrm{Gr}(A)$ of graded objects in an additive category $C$ is called a chain complex.

• For $d_X:X\to TX$ a complex, the shifted differential object $d_{TX}:TX\stackrel{-T(d_X)}{\to }TTX$ is again a complex.