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 Lueck.

Cited by in corpus (10)

• Homological algebra in bivariant K-theory and other triangulated categories
• Dualities in equivariant Kasparov theory
• Semantics for a Quantum Programming Language by Operator Algebras
• Mackey 2-functors and Mackey 2-motives
• Gorenstein homological algebra and universal coefficient theorems
• Higher nonunital Quillen K'-theory, KK-dualities and applications to topological $\mathbb{T}$-dualities
• Noncommutative stable homotopy and stable infinity categories
• Algebraic K-theory, K-regularity, and T-duality of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras
• A stable $\infty$-category for equivariant $KK$-theory
• Algebraic quantum kk-theory