# 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

${d}_{X}:X\to TX$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 $\mathrm{Gr}\left(A\right)$ 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 ${d}_{TX}:TX\stackrel{-T\left({d}_{X}\right)}{\to }TTX$ 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)