Suspension functor

Definition

The suspension functor $S$ is a functor from the category of topological spaces with continuous maps to itself, defined as follows:

• It sends each topological space $X$ to its suspension $SX$
• Given a continuous map $f:X\to Y$, the induced map $Sf:SX\to SY$ is the map naturally induced by quotienting out from the map $X\times I\to Y\times I$ given by $f\times id$.

Iteration

The iteration of the suspension functor $n$ times is equivalent to taking the join with a $(n-1)$-sphere.