Noncommutative CW-spectra as enriched presheaves on matrix algebras

Abstract

Motivated by the philosophy that $C^{\ast }$-algebras reflect noncommutative topology, we investigate the stable homotopy theory of the (opposite) category of $C^{\ast }$-algebras. We focus on $C^{\ast }$-algebras which are noncommutative CW-complexes in the sense of Eilers et al. (1998). 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 (2003). In proving this, we use the language of enriched 1-categories as developed recently by Hinich.

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 ${\mathtt{Sp}}^{\mathtt{M}}$. We give a direct proof that the category of ${\mathtt{Sp}}^{\mathtt{M}}$-enriched presheaves ${\mathcal{M}}_s^{\mathtt{op}}\to {\mathtt{Sp}}^{\mathtt{M}}$ with the projective model structure models $\mathtt{NSp}$ and conclude that ${\mathcal{M}}_s$ is a strict model for $\mathcal{M}$.