#
nLab

suspension of a chain complex

### Context

#### Homological algebra

#### Homotopy theory

## Background

## Variations

## Definitions

## Paths and cylinders

## Homotopy groups

## Theorems

# Contents

## Idea

In a category of chain complexes $Ch_\bullet(\mathcal{A})$, the suspension object of a chain complex $C_\bullet$ is the complex

$\Sigma C_\bullet = C[1]_\bullet$

(or sometimes denoted $C[-1]_\bullet$, depending on an unessential choice of sign convention) obtained by shifting the degrees up by one:

$C[1]_n \coloneqq C_{n-1}$

with the differential the original one but equipped with a sign:

$d^{C[1]}_n \coloneqq - d^C_{n-1}
\,.$

Generally, for $p \in \mathbb{Z}$, $C[p]$ is the chain complex with

$C[p]_n \coloneqq C_{n-p}$

$d^{C[p]}_n \coloneqq (-1)^p d^C_{n-p}
\,.$

Revised on November 23, 2013 21:09:41
by

Tim Porter
(2.26.12.120)