nLab
suspension spectrum

Definition

For $X$ a pointed topological space, its suspension spectrum is the spectrum ${\Sigma }^{\mathrm{\infty }}X$ whose degree-$n$ space is the $n$-fold reduced suspension of $X$:

$$({\Sigma }^{\mathrm{\infty }}X{)}_n={\Sigma }^{n}X$$(\Sigma^\infty X)_n = \Sigma^n X

Properties

As an infinity-functor ${\Sigma }^{\mathrm{\infty }}:{\mathrm{Top}}_{*}\to \mathrm{Spec}$ the suspension spectrum functor exhibits the stabilization of Top.

$$({\Sigma }^{\mathrm{\infty }}⊣{\Omega }^{\mathrm{\infty }}):{\mathrm{Top}}_{*}\stackrel{\stackrel{{\Sigma }^{\mathrm{\infty }}}{\leftarrow }}{\underset{{\Sigma }^{\mathrm{\infty }}}{\to }}\mathrm{Spec}$$(\Sigma^\infty \dashv \Omega^\infty)\colon Top_* \stackrel{\overset{\Sigma^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}} Spec