# nLab suspension of a chain complex

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### Theorems

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#### Homotopy theory

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Introductions

Definitions

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Paths and cylinders

Homotopy groups

Basic facts

Theorems

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# 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

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

Last revised on November 23, 2013 at 21:09:41. See the history of this page for a list of all contributions to it.