  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 $\mathtt{NSp'}$.
