Colocalizations of noncommutative spectra and bootstrap categories

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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 ΣHoᶜ* constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of NSp', namely, NSp'[K⁻¹] and NSp'[Z⁻¹], 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 of KK∞:= NSp'[K⁻¹]op 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'[K⁻¹] 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'.