Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions.
Here, let me describe this groupoid. Take a foliated manifold, which for my question might as well have constant rank. It defines various different groupoids with object-set the manifold. An important one, but not the one who's hanging out in my office, is the one whose morphisms are homotopy classes of paths that are tangent to the foliation. I believe a result of Crainic and Fernandes holds that this groupoid can be equipped with a structure of a Lie groupoid.
My groupoid is conceptually similar, but a result like Crainic and Fernandes' fails. Namely, in my groupoid is nothing more than an equivalence relation: the objects are again the points of the manifold, and two objects are isomorphic iff they are in the same leaf of the foliation, and then isomorphic in a unique way.
I believe that for general foliations, my groupoid cannot be equipped with a Lie structure. Indeed, I'm having trouble even topologizing it, although I guess I can topologize it as a quotient of the Crainic-Fernandes groupoid (which is a "source- simply connected cover" of my groupoid). But I want even more: I'd like a good language that describes things like its smooth structure, and I'd much rather refer to existing (presumably more general) literature than derive just what I need ad hoc.
I've seen things like "Lie groupoids where the morphisms space may not be Hausdorff" (sometimes called "differentiable groupoids"), and also "Lie groupoids where the morphism space may be a stack rather than a manifold (sometimes called "Weinstein groupoids"). Since I don't myself have much understanding of my groupoid except in terms of its underlying sets, I don't know whether either of these notions is useful. Any help would be appreciated.