nLab
suspension of a chain complex

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

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

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

Related concepts

• mapping cone

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

• suspension object

  • suspension, reduced suspension