# nLab complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## 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 $Gr(A)$ 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.

## References

For instance section 11 of

Revised on September 1, 2012 18:21:09 by Urs Schreiber (82.113.121.88)