If we want to understand the K-theory of the C*-algebras of étale groupoids, we can learn a lot from understanding functoriality of the construction. Continuous groupoid homomorphisms do not work, as morphisms of groups and of spaces give rise to $*$-homomorphisms in opposite directions.

Étale correspondences are a class of morphisms of étale groupoids fit for this purpose, and cover the widely used case of Morita equivalences. I will exhibit many more classes of morphisms encompassed by correspondences, such as (backwards) continuous maps of spaces and homomorphisms that are local homeomorphisms.

In addition to inducing a map in K-theory, I show that a (proper) étale correspondence $\Omega \colon G \to H$ induces a map $H_*(\Omega) \colon H_*(G) \to H_*(H)$ of ample groupoid homology. This homology is strongly linked to the K-theory and I show that under mild conditions $H_*(\Omega)$ being an isomorphism implies that $K_*(C^*_r(G)) \cong K_*(C^*_r(H))$. This has applications to the computation of K-theory groups for left regular C*-algebras associated to inverse semigroups and left cancellative small categories.

Bio: Alistair Miller is a postdoc at the University of Southern Denmark interested in groupoids, C*-algebras and their K-theory. He obtained his PhD from Queen Mary University of London in 2022 under the supervision of Xin Li.