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.