Functors between Kasparov categories from étale groupoid correspondences

Abstract

For an étale correspondence $Ω\colon G \to H$ of étale groupoids, we construct an induction functor $\mathrm{Ind}_Ω\colon \mathrm{KK}^H \to \mathrm{KK}^G$ between equivariant Kasparov categories. We introduce the crossed product of an $H$-equivariant correspondence by $Ω$, and use this to build a natural transformation $α_Ω\colon K_*( G \ltimes \mathrm{Ind}_Ω-) \Rightarrow K_*(H \ltimes -)$. When $Ω$ is proper these constructions naturally sit above an induced map in K-theory $K_*(C^*(G)) \to K_*(C^*(H))$.