Categorical aspects of bivariant K-theory

Abstract

This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them with extra structure such as a tensor product and a triangulated category structure. We discuss the construction of the Baum–Connes assembly map via localisation of categories and explain how this is related to the purely topological construction by Davis and Lück.