Is there a "geometric" language that describes the equivalence groupoid of a foliated manifold? 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.
 A: I'm being thick, but surely you can topologize the space of arrows as a subspace of $M\times M$ where $M$ is the space of objects? It may be a horrible space, but it exists. As far as naming the thing goes, I would call it a presentation of the stack-replacement of the space of leaves or similar. If $L = M/{\sim}$ is the space of leaves (which admittedly is 'bad', so let us assume it is 'good' for now), then your groupoid is the Cech groupoid of the canonical map $M \to L$. It is morally equivalent to the space of leaves if they are considered as stacks (leaving aside the question of whether your groupoid actually presents a stack for now). As far as a reference goes, how about the 1989 paper of Jean Pradines 'Morphisms between spaces of leaves viewed as fractions' Numdam/arXiv (the latter is an update of the original published version). The only drawback to that paper is that it uses language inherited from the Ehresmann school of category theory, which is quite idiosyncratic (there is a bit of a dictionary provided in appendix A of the arXiv version). Perhaps though for your purposes only section 1 will be necessary.
A: I do not exactly understand your problem but it is possible that some developments and expositions of other ideas of Pradines might help, namely 
R. Brown and  M. E.-S. A.-F. Aof,  ``The holonomy groupoid of a
locally  topological groupoid'', Top. and its  Appl., 47 (1992)
97-113.
R. Brown and O.  Mucuk, ``The monodromy groupoid of a Lie
groupoid'',  Cah. Top. G\'eom. Diff. Cat 36 (1995) 345-369.
R. Brown and O.  Mucuk, ``Foliations, locally Lie groupoids, and
holonomy'',  Cah. Top. G\'eom. Diff. Cat. , 37 (1996) 61-71.
These papers are related to 
Pradines, Jean,  "Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales." C. R. Acad. Sci. Paris Sér. A-B 263 1966 A907–A910.
where the first paper explains Théoreme 1, and the second paper explains Théoreme 2, of that Note, following explanations given to me by Jean in 1981 in Toulouse.  We have other papers on these ideas, e.g. on local subgroupoids. 
I thought these ideas of Jean were great and needed a full exposition. The first paper shows how a holonomy groupoid arises from an "iteration of local procedures", the local procedures being defined by a locally topological, or smooth, groupoid. Also this holonomy groupoid satisfies a universal property. 
A: If your foliation is "regular", which roughly means the leaves have constant dimension, then every chart of the manifold has the form $L \times T$, where $L$ is the longitudinal (leafwise) direction and $T$ the transeversal direction. Moreover, the change of coordinates is of the form $(x,y) \mapsto (f(x,y),h(y))$. (This is thanks to the Frobenius theorem!). I mean the second coordinate depends only on the transversal direction. This particular map $h : T_1 \to T_2$ is called a holonomy. (It's a "small" holonomy, meaning it works for points close enough to each other.)
So now take two points $p=(x_1,y_1)$ and $q=(x_2,y_2)$ in the same leaf $L$, so that the pair $(p,q)$ is an arrow of your groupoid. For simplicity, say $p$ and $q$ are close to each other. Take $T_1$ and $T_2$ transversals to $L$, at the points $p$ and $q$ respectively. So the pair $(p,q)$ has coordinates $L \times T_1 \times T_2 \times L$.
But actually $T_2$ in these coordinates is redundant, because $y_1$ and $y_2$ are identified via the "small" holonomy diffeomorphism $h$ (that's why we assumed $p$ and $q$ are close to each other, so that there is a change of coordinates map).
Now, if $p$ and $q$ are far from each other, then you just do the classical trick: First you connect them with a smooth path $\gamma$ which stays on the leaf $L$. Then, $\gamma$ being compact, you cover it with a finite number of foliation charts. So you get a "big" holonomy $h_1 \circ \ldots h_k$.
So a charts of your groupoid at $(p,q)$ is of the form $L \times T \times L$. There you have it, your groupoid is a Lie groupoid, and its dimension is twice the dimension of the leaf plus the dimension of the transversal.
Your groupoid is really the "graph" of the foliation (the equivalence relation as you say).
If your foliation is singular, which means that the dimension of the leaves drops (in a semi-continuous way), then the graph is just a topological groupoid. It is a quotient of the holonomy groupoid of the foliation, which is also a topological groupoid in general. In fact, both of these groupoids carry a diffeological structure, so you can still do differential geometry with them. This structure is defined by the notion of bisubmersion.
