nLab
suspension object

Definition

In an (∞,1)-category $C$, for any object $X$ its suspension object $\Sigma X$ is the homotopy pushout

$$\begin{array}{ccc}X& \to & *\\ \downarrow & & \downarrow \\ *& \to & \Sigma X\end{array},$$\array{ X &\to& {*} \\ \downarrow && \downarrow \\ {*} &\to& \Sigma X } \,,

where $*$ is the terminal object.

This is the mapping cone of the terminal map $X\to *$. See there for more details.

This concept is dual to that of loop space object.

Examples

• In Top, this is the reduced suspension of a space.

• In a category of chain complexes the suspension of a chain complex is given by shifting the degrees of the chain complex up by one.

Related concepts

• loop space object, free loop space object,

  • delooping

  • loop space, free loop space, derived loop space

• suspension object

  • suspension, reduced suspension