(E\)-theory is compactly assembled

We show that the equivariant \(E\)-theory category \(\mathrm{E}_{\mathrm{sep}}^{G}\) for separable \(C^{*}\)-algebras is a compactly assembled stable \(\infty\)-category. We derive this result as a consequence of the shape theory for \(C^{*}\)-algebras developed by Blackadar and Dardarlat and a new construction of \(\mathrm{E}_{\mathrm{sep}}^{G}\). As an application we investigate a topological enrichment of the homotopy category of a compactly assembled \(\infty\)-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.