  The noncommutative stable homotopy category $\mathtt{NSH}$ is a triangulated
category that is the universal receptacle for triangulated homology theories on
separable $C^*$-algebras. We show that the triangulated category $\mathtt{NSH}$
is topological as defined by Schwede using the formalism of (stable) infinity
categories. More precisely, we construct a stable presentable infinity category
of noncommutative spectra and show that $\mathtt{NSH}^{op}$ sits inside its
homotopy category as a full triangulated subcategory, from which the above
result can be deduced. We also introduce a presentable infinity category of
noncommutative pointed spaces that subsumes $C^*$-algebras and define the
noncommutative stable (co)homotopy groups of such noncommutative spaces
generalizing earlier definitions for separable $C^*$-algebras. The triangulated
homotopy category of noncommutative spectra admits (co)products and satisfies
Brown representability. These properties enable us to analyse neatly the
behaviour of the noncommutative stable (co)homotopy groups with respect to
certain (co)limits. Along the way we obtain infinity categorical models for
some well-known bivariant homology theories like $\mathrm{KK}$-theory,
$\mathrm{E}$-theory, and connective $\mathrm{E}$-theory via suitable
(co)localizations. The stable infinity category of noncommutative spectra can
also be used to produce new examples of generalized (co)homology theories for
noncommutative spaces.
