# Noncommutative CW-spectra as enriched presheaves on matrix algebras

### Gregory Arone

Stockholm University, Sweden### Ilan Barnea

Haifa University, Israel### Tomer M. Schlank

Hebrew University of Jerusalem, Israel

## 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 $∞$-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 (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 $M_{s}→Sp_{M}$ with the projective model structure models $NSp$ and conclude that $M_{s}$ is a strict model for $M$.

## Cite this article

Gregory Arone, Ilan Barnea, Tomer M. Schlank, Noncommutative CW-spectra as enriched presheaves on matrix algebras. J. Noncommut. Geom. 16 (2022), no. 4, pp. 1411–1443

DOI 10.4171/JNCG/481