$E$-theory is compactly assembled

TL;DR

This paper proves that the equivariant E-theory category for separable C*-algebras is a compactly assembled stable ∞-category, connecting shape theory and topological enrichment to advance understanding in operator algebra theory.

Contribution

It establishes the compactly assembled nature of the equivariant E-theory category and introduces a new construction linking shape theory with E-theory.

Findings

E-theory category is a compactly assembled stable ∞-category

A new construction of the E-theory category is developed

Connections to topological enrichment and classical E-theory results are made

Abstract

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 Carri\'on and Schafhauser on the enrichment of the classical $E$-theory category can be derived by specialization.