Functoriality for groupoid and Fell bundle $C^*$-algebras

TL;DR

This paper introduces a functorial framework linking morphisms of étale groupoids and Fell bundles to $C^*$-algebras, establishing categorical equivalences and colimit preservation for inductive systems.

Contribution

It defines new morphisms for étale groupoids and Fell bundles, creating a functorial connection to $C^*$-algebras and proving categorical equivalences and colimit preservation.

Findings

Categorical equivalence between Cartan pairs and twists

Functorial induction of *-homomorphisms from Fell bundle morphisms

Preservation of colimits in inductive systems of groupoids and Fell bundles

Abstract

We define a class of morphisms between \'etale groupoids and show that there is a functor from the category with these morphisms to the category of $C^*$-algebras. We show that all homomorphisms between Cartan pairs of $C^*$-algebras that preserve the Cartan structure arise from such morphisms between the underlying Weyl groupoids and twists, and attain an equivalence of categories between Cartan pairs with structure preserving homomorphisms and a their associated twists. We define analogous morphisms for Fell bundles over $C^*$-algebras and show these functorially induce ${}^*$-homomorphisms between the Fell bundle $C^*$-algebras. We also construct colimit groupoids and Fell bundles for inductive systems of such morphisms, and show that the functor to $C^*$-algebras preserves these colimits.