Functors between Kasparov categories from etale groupoid correspondences

For an etale correspondence Omega:G -> H of etale groupoids, we construct an induction functor Ind(Omega) :KKH -> KKG between equivariant Kasparov categories. We introduce the crossed product of an H-equivariant correspondence by Omega, and use this to build a natural transformation alpha Omega:K-*(G(sic)Ind(Omega)-)double right arrow K-*(H(sic)-). When Omega is proper these constructions naturally sit above an induced map in K-theory K-*(C-*(G))-> K-*(C-*(H)). (c) 2024 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).