$E$-theory is compactly assembled

TL;DR

This work establishes that the equivariant $E$-theory categories for separable $C^{*}$-algebras arise from compactly assembled stable $\infty$-categories, grounded in Blackadar–Dadarlat shape theory and a novel construction of $\mathrm{E}^{G}_{\mathrm{sep}}$. It develops a robust $\infty$-categorical framework—including compact maps, shapes, exhaustions, and phantom maps—and builds three equivalent models of equivariant $E$-theory via universal properties and localization techniques. The authors prove that the central categories $\mathrm{A}^{G}_{\mathrm{sep}}$ and $\mathrm{E}^{G}_{\mathrm{sep}}$ are compactly assembled and illustrate consequences for topological enrichment and morphism-topologies, connecting to Carrión–Schafhauser-type enrichments. The results yield a refined understanding of when morphisms are $E$-theory equivalences and provide tools for analyzing strong phantom maps, with implications for rigidity and bootstrap phenomena in equivariant noncommutative homotopy theory.

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

Theorems & Definitions (308)

• Theorem 1.1
• Definition 1.2
• Remark 1.3
• Example 1.4
• Theorem 1.5
• Definition 1.6
• Definition 1.7
• Proposition 1.8
• Definition 1.9
• Theorem 1.10