Equivariant KK-theory and model categories

2025/06/19 by Datta, Anupam, Joachim, Michael
Mathematics · #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2506.16238

openalex publication_date 2025/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We cast Kasparov's equivariant KK-theory in the framework of model categories. We obtain a stable model structure on a certain category of locally multiplicative convex G-C^*-algebras, which naturally contains the stable ∞-category KK^G_sep as described by Bunke, Engel, Land (\citeBEL). Non-equivariantly, KK-theory was studied using model categories by Joachim-Johnson (\citeMJ). We generalize their ideas in the equivariant case, and also fix some critical errors that their work had.

Citations

• A symmetric monoidal fracture square
• The spectral model of (Real) K-theory
• Generalized homomorphisms and KK with extra structure
• Bivariant KK-Theory and the Baum-Connes conjecure
• Dwyer-Kan localization revisited
• C^*-Algebra relations
• Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category
• Pullback and Pushout Constructions in C*-Algebra Theory
• Shape theory for C^*-algebras.

Related

• A stable ∞-category for equivariant KK-theory
• Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category
• Equivariant Algebraic K-Theories
• The universal property of inverse semigroup equivariant KK-theory
• Aspects of equivariant KK-theory in its generators and relations picture