Suspension spectrum (${\Sigma }^{\infty }X$)

For a based space $X$, its suspension spectrum has level $j$ equal to $S^j\wedge X$. Its derived adjunction with the infinite loop space is $[{\Sigma }^{\infty }X,E{]}_{\mathrm{st}}=[X,{\Omega }^{\infty }E{]}_{\ast }$.

The sphere spectrum is $\mathbb{S}={\Sigma }^{\infty }{S}^0$. It is the unit of the smash product of spectra. Its integer suspensions of spectra represent stable homotopy groups.