Functoriality of groupoid quantales. I

We provide three functorial extensions of the equivalence between localic étale groupoids and their quantales. The main result is a biequivalence between the bicategory of localic étale groupoids, with bi-actions as 1-cells, and a bicategory of inverse quantal frames whose 1-cells are bimodules. As a consequence, the category InvQuF of inverse quantale frames, whose morphisms are the (necessarily involutive) homomorphisms of unital quantales, is equivalent to a category of localic étale groupoids whose arrows are the algebraic morphisms in the sense of Buneci and Stachura. We also show that the subcategory of InvQuF with the same objects and whose morphisms preserve finite meets is dually equivalent to a subcategory of the category of localic étale groupoids and continuous functors whose morphisms, in the context of topological groupoids, have been studied by Lawson and Lenz.