## Abstract

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.

Original language | American English |
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Pages (from-to) | 1411-1443 |

Number of pages | 33 |

Journal | Journal of Noncommutative Geometry |

Volume | 16 |

Issue number | 4 |

DOIs | |

State | Published - 2022 |

### Bibliographical note

Publisher Copyright:© 2022 European Mathematical Society.

## Keywords

- Noncommutative CW-complexes
- enriched infinity categories
- enriched model categories
- noncommutative spectra
- stable infinity categories