Holmstrom Suspension

A general setting in which a suspension functor exists is that of a pointed model category

http://ncatlab.org/nlab/show/reduced+suspension

http://nlab.mathforge.org/nlab/show/suspension+object

Note that there are two different suspension functors for a pointed sset: the smash product with $S^1$and the Kan suspension $\Sigma X$. See Goerss-Jardine section III.5 and the summary paragraph at the start of chapter III.

nLab page on Suspension