equivalences in/of $(\infty,1)$-categories
The collection of spectra form an (∞,1)-category $Sp(\infty Grpd)$ which is in fact a stable (∞,1)-category. Indeed, it is the universal property stabilization of the $(\infty,1)$-category ∞Grpd, equivalently of the simplicial localization of the category Top at the weak homotopy equivalences.
$Sp(\infty Grpd)$ plays a role in stable homotopy theory analogous to the role played by the 1-category Ab of abelian groups in homological algebra, or rather of the category of chain complexes $Ch_\bullet(Ab)$ of abelian groups.
In the context of (∞,1)-categories a spectrum is a spectrum object in the (∞,1)-category $L_{whe} Top_*$ of pointed topological spaces.
Recall that spectrum objects in the (infinity,1)-category $C$ form a stable (∞,1)-category $Sp(C)$.
The stable (∞,1)-category of spectrum objects in $L_{whe} Top_*$ is the stable $(\infty,1)$-category of spectra
With the smash product of spectra $Sp(L_{whe}Top_*)$ becomes a symmetric monoidal (infinity,1)-category.
an algebra object in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is an associative ring spectrum;
a commutative algebra object in $Sp(L_{whe}Top_*)$ with respect to this monoidal structure is a commutative ring spectrum;
The prime spectrum of a monoidal stable (∞,1)-category for p-local and finite spectra is labeled by the Morava K-theories. This is the content of the thick subcategory theorem.
the stable $(\infty,1)$-category of spectra is described in section 9 of
Its monoidal structure is described in section 4.2
That this is a symmetric monoidal structure is described in section 6 of