# Suspension functor

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