The noncommutative stable homotopy category $\mathtt{NSH}$ is a triangulated category that is the universal receptacle for triangulated homology theories on separable $C^*$-algebras. We show that the triangulated category $\mathtt{NSH}$ is topological as defined by Schwede using the formalism of (stable) infinity categories. More precisely, we construct a stable presentable infinity category of noncommutative spectra and show that $\mathtt{NSH}^{op}$ sits inside its homotopy category as a full triangulated subcategory, from which the above result can be deduced. We also introduce a presentable infinity category of noncommutative pointed spaces that subsumes $C^*$-algebras and define the noncommutative stable (co)homotopy groups of such noncommutative spaces generalizing earlier definitions for separable $C^*$-algebras. The triangulated homotopy category of noncommutative spectra admits (co)products and satisfies Brown representability. These properties enable us to analyse neatly the behaviour of the noncommutative stable (co)homotopy groups with respect to certain (co)limits. Along the way we obtain infinity categorical models for some well-known bivariant homology theories like $\mathrm{KK}$-theory, $\mathrm{E}$-theory, and connective $\mathrm{E}$-theory via suitable (co)localizations. The stable infinity category of noncommutative spectra can also be used to produce new examples of generalized (co)homology theories for noncommutative spaces.Comment: 26 pages; v2 major revision with some improved results, a mistake in matrix homotopy computation removed, title slightly changed (extended); v3 discussion of semigroup C*-algebras removed and made a part of arXiv:1403.4130, Section 3 rewritten accordingly and the title shortened to reflect the changes; v4 final revision incorporating the referee's corrections; v5 added/updated appendix/reference
Abstract. We construct a compactly generated and closed symmetric monoidal stable ∞-category NSp ′ and show that hNSp ′ op contains the suspension stable homotopy category of separable C * -algebras ΣHo C * constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of NSp ′ , namely, NSp, both of which are shown to be compactly generated and closed symmetric monoidal. We prove that Kasparov KK-category of separable C * -algebras sits inside the homotopy category ofop as a fully faithful triangulated subcategory. Hence KK ∞ should be viewed as the stable ∞-categorical incarnation of Kasparov KK-category for arbitrary pointed noncommutative spaces (including nonseparable C * -algebras). As an application we find that the bootstrap category in hNSp] admits a completely algebraic description. We also construct a K-theoretic bootstrap category in hKK ∞ that extends the construction of the UCT class by Rosenberg-Schochet. Motivated by the algebraization problem we finally analyse a couple of equivalence relations on separable C * -algebras that are introduced via the bootstrap categories in various colocalizations of NSp ′ .
Abstract. Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal ∞-categorical models for separable C * -algebras SC * ∞ and noncommutative spectra NSp using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of SC * ∞ and NSp with respect to strongly self-absorbing C * -algebras. We analyse the homotopy categories of the localizations of SC * ∞ and give universal characterizations thereof. We construct a stable ∞-categorical model for bivariant connective E-theory and compute the connective E-theory groups of O ∞ -stable C * -algebras. We also introduce and study the nonconnective version of Quillen's nonunital K ′ -theory in the framework of stable ∞-categories. This is done in order to promote our earlier result relating topological T-duality to noncommutative motives to the ∞-categorical setup. Finally, we carry out some computations in the case of stable and O ∞ -stable C * -algebras.
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