On the space of morphisms between \'etale groupoids

TL;DR

This paper studies the space of pointed morphisms between étale groupoids, introducing a Banach manifold structure under certain conditions, thus providing a geometric framework for these morphism spaces.

Contribution

It constructs a Banach manifold structure on the set of morphisms between étale groupoids, linking it to an étale groupoid of morphisms with a well-defined orbit space.

Findings

The set of pointed morphisms can be given a Banach manifold structure.

This structure makes the morphism space the objects of an étale groupoid.

The orbit space of this groupoid corresponds to the space of morphisms between the original groupoids.

Abstract

Given two \'etale groupoids $\Cal G$ and $\Cal G'$, we consider the set of pointed morphisms from $\Cal G$ to $\Cal G'$. Under suitable hypothesis we introduce on this set a structure of Banach manifold which can be considered as the space of objects of an \'etale groupoid whose space of orbits is the space of morphisms from $\Cal G$ to $\Cal G'$.