Dualisable categories in noncommutative geometry

Abstract

In this note, we highlight modern perspectives on noncommutative geometry using dualisable categories. To this end, we show that several categories naturally arising in functional analysis (derived categories of bornological and condensed modules), operator algebras (E-theory) and stable categories (noncommutative motives) are dualisable. This in particular leads to a definition of algebraic K-theory for analytic spaces. Furthermore, the dualisability of E-theory and motives leads us to pursue an analogy between topological and algebraic K-theory, inspired by assembly conjectures (Baum–Connes and Farrell–Jones).