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

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.