Categorical consequences of nuclearity for morphisms in A and E

Characterize the categorical properties of the morphisms (f) in A and e(f) in E that follow from nuclearity of a *-homomorphism f: A → B in the separable G–C*-algebra category G*_{sep}; determine which compactness, exactness, or other structural features in A and E are implied by nuclearity of f.

Background

The paper constructs the ∞-categorical enhancements A (asymptotic morphisms) and E (equivariant E-theory) and analyzes compactness notions of morphisms in these categories. Nuclear maps are central in C*-algebra theory, often implying regularity and approximation properties.

Understanding how nuclearity of f lifts to structural properties of (f) and e(f) would connect classical analytic features with the categorical framework introduced here, potentially yielding new criteria for compactness or phantom behavior in A and E.