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 ∞-category of noncommutative CW-spectra, which we denote by NSp. Let M be the full spectral subcategory of NSp spanned by “noncommutative suspension spectra” of matrix algebras. Our main result is that NSp is equivalent to the ∞-category of spectral presheaves on M. To prove this we first prove a general result which states that any compactly generated stable ∞-category is naturally equivalent to the ∞category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an ∞-categorical version of a result by Schwede and Shipley [ScSh1]. In proving this we use the language of enriched ∞-categories as developed by Hinich [Hin2, Hin3]. We end by presenting a “strict” model for M. That is, we define a category Ms strictly enriched in a certain monoidal model category of spectra Sp. We give a direct proof that the category of Sp-enriched presheaves Mop s → Sp M with the projective model structure models NSp and conclude that Ms is a strict model for M.