Noncommutative CW-spectra as enriched presheaves on matrix algebras

Motivated by the philosophy that $C^$-algebras reflect noncommutative topology, we investigate stable homotopy theory of (opposite) category $C^$-algebras. We focus on $C^\*$-algebras which are CW-complexes in sense Eilers et al. (1998). construct $\infty$-category CW-spectra, denote $\mathtt{NSp}$. Let $\mathcal{M}$ be full spectral subcategory $\mathtt{NSp}$ spanned “noncommutative suspension spectra” matrix algebras. Our main result is equivalent to presheaves $\mathcal{M}$. To prove this, first a general states any compactly generated naturally set compact generators. This an $\infty$-categorical version Schwede and Shipley (2003). In proving use language enriched 1-categories as developed recently Hinich. end presenting “strict” model for That is, define $\mathcal{M}\_s$ strictly certain monoidal spectra $\mathtt{Sp}^\mathtt{M}$. give direct proof $\mathtt{Sp}^\mathtt{M}$-enriched $\mathcal{M}\_s^{\mathtt{op}}\to\mathtt{Sp}^{\mathtt{M}}$ with projective structure models conclude strict