nLab
suspension of a chain complex

Idea

In a category of chain complexes ${\mathrm{Ch}}_{•}(𝒜)$, the suspension object of a chain complex $C_{•}$ is the complex

$$\Sigma C_{•}=C[1{]}_{•}$$\Sigma C_\bullet = C[1]_\bullet

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

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

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

$$d_n^{X[1]}≔-d_{n-1}^X.$$d^{X[1]}_n \coloneqq - d^X_{n-1} \,.

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

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

$$d_n^{X[p]}≔(-1{)}^p{d}_{n-p}^X.$$d^{X[p]}_n \coloneqq (-1)^p d^X_{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