2021/02/26 by Bunke, Ulrich, Engel, Alexander, Land, Markus
#Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2102.13372
For a countable group G we construct a small, idempotent complete, symmetric monoidal, stable ∞-category KKGsep whose homotopy category recovers the triangulated equivariant Kasparov category of separable G-C^*-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable ∞-category KKG which receives a symmetric monoidal functor kkG from possibly non-separable G-C^*-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying G. We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite K-homology theory on proper and locally compact G-topological spaces, allowing for coefficients in arbitrary G-C^*-algebras. Finally, we extend the functor kkG from G-C^*-algebras to G-C^*-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.