For $X$ a pointed topological space, its suspension spectrum $\Sigma^\infty X$ is the spectrum whose degree-$n$ space is the $n$-fold reduced suspension of $X$:
As an infinity-functor $\Sigma^\infty\colon Top_* \to Spec$ the suspension spectrum functor exhibits the stabilization of Top.
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Nicholas J. Kuhn, Suspension spectra and homology equivalences, Trans. Amer. Math. Soc. 283, 303–313 (1984) (JSTOR)
John Klein, Moduli of suspension spectra (arXiv:math/0210258, MO)
Suspension spectra of infinite loop spaces are discussed (in a context of Goodwillie calculus and chromatic homotopy theory) in