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 noncommutative CW-complexes in the sense of Eilers et al. (1998). We construct the stable 1-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 1-category of spectral presheaves on M. To prove this, we first prove a general result which states that any compactly generated stable 1-category is naturally equivalent to the 1-category of spectral presheaves on a full spectral subcategory spanned by a set of compact generators. This is an 1-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 M. That is, we define a category M_s strictly enriched in a certain monoidal model category of spectra Sp^M. We give a direct proof that the category of Sp^M-enriched presheaves M_s^op ! Sp^M with the projective model structure models NSp and conclude that M_s is a strict model for M.