$E$-theory is compactly assembled

We show that the equivariant $E$-theory category ${\mathrm{E}}_{\mathrm{s}\mathrm{e}\mathrm{p}}^G$ for separable $C^{\ast }$-algebras is a compactly assembled stable $\mathrm{\infty }$-category. We derive this result as a consequence of the shape theory for $C^{\ast }$-algebras developed by Blackadar and Dardarlat and a new construction of ${\mathrm{E}}_{\mathrm{s}\mathrm{e}\mathrm{p}}^G$. As an application we investigate a topological enrichment of the homotopy category of a compactly assembled $\mathrm{\infty }$-category in general and argue that the results of Carrión and Schafhauser on the enrichment of the classical $E$-theory category can be derived by specialization.