Noncommutative CW-spectra as enriched presheaves on matrix algebras

Motivated by the philosophy that $C^*$-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of $C^*$-algebras. We focus on $C^*$-algebras which are non-commutative CW-complexes in the sense of [ELP]. We construct the stable $\infty$-category of noncommutative CW-spectra, which we denote by $\mathtt{NSp}$. Let $\mathcal{M}$ be the full spectral subcategory of $\mathtt{NSp}$ spanned by "noncommutative suspension spectra" of matrix algebras. Our main result is that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. To prove this we first prove a general result which states that any compactly generated stable $\infty$-category is naturally equivalent to the $\infty$-category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an $\infty$-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched $\infty$-categories as developed by Hinich [Hin2,Hin3]. We end by presenting a "strict" model for $\mathcal{M}$. That is, we define a category $\mathcal{M}_s$ strictly enriched in a certain monoidal model category of spectra $\mathtt{Sp^M}$. We give a direct proof that the category of $\mathtt{Sp^M}$-enriched presheaves $\mathcal{M}_s^{op}\to\mathtt{Sp^M}$ with the projective model structure models $\mathtt{NSp}$ and conclude that $\mathcal{M}_s$ is a strict model for $\mathcal{M}$.