Colocalizations of noncommutative spectra and bootstrap categories

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 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 analyze a couple of equivalence relations on separable C⁎-algebras that are introduced via the bootstrap categories in various colocalizations of NSp′.