Spaces with local equivalence relations, and their monodromy

We elaborate a suggestion of Grothendieck, and study the invariant sheaves for a local equivalence relation on a space (e.g., a foliation). One of our purposes is to compare this to the standard model for the leaf-(quotient-)space of a foliation, given by the holonomy groupoid. To this end, we prove that, under suitable connectedness assumptions, Grothendieckʹs invariant sheaves can be described in terms of a closely related, but different, “monodromy” groupoid. cond purpose is to prove that every étale groupoid arises this way.