Colocalizations of noncommutative spectra and bootstrap categories

Abstract

We construct a compactly generated and closed symmetric monoidal stable $\infty$-category $\mathtt{NSp'}$ and show that $\mathtt{hNSp'}^{op}$ contains the suspension stable homotopy category of separable $C^*$-algebras $\mathtt{\Sigma Ho^{C^*}}$ constructed by Cuntz-Meyer-Rosenberg as a fully faithful triangulated subcategory. Then we construct two colocalizations of $\mathtt{NSp'}$, namely, $\mathtt{NSp'}[\mathbb{K}^{-1}]$ and $\mathtt{NSp'}[\mathcal{Z}^{-1}]$, 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 $\mathtt{KK_\infty} := \mathtt{NSp'}[\mathbb{K}^{-1}]^{op}$ as a fully faithful triangulated subcategory. Hence $\mathtt{KK_\infty}$ should be viewed as the stable $\infty$-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 $\mathtt{hNSp'}[\mathbb{K}^{-1}]$ admits a completely algebraic description. We also construct a $K$-theoretic bootstrap category in $\mathtt{hKK_\infty}$ 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 $\mathtt{NSp'}$.