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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.

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I like the fact you have groupoids in your office. :) I wish I had a couch for my groupoids to sit on! –  David Roberts Jan 13 '11 at 6:21
You may have a locally Lie groupoid, and such will usually have an associated holonomy Lie groupoid. See the papers I refer to below. –  Ronnie Brown Jan 26 '12 at 14:08
I don't know whether this helps: you can find more information on this groupoid in the book by Moore and Schochet "Global Analysis of Foliated Spaces". The "equivalence relation groupoid" is an example of a Borel groupoid. –  Indrava Roy Mar 30 '12 at 13:10
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2 Answers

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 ideosyncratic (there is a bit of a dictionary provided in appendix A or the arXiv version). Perhaps though for your purposes only section 1 will be necessary.

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Well, yes, I can topologize it that way. Thanks for the reference! –  Theo Johnson-Freyd Jan 14 '11 at 4:43
Ah, so one of the reasons I don't like that topologization is that when a leaf wraps densely around something, I'd rather the corresponding subset Hom(x,-) in the space of morphisms to nevertheless be a manifold. The example is a foliation on a torus by irrational lines; then the "leaf space" groupoid is the Lie groupoid for the R-action, so I'd like the source fibers to be lines, whereas topologizing as a subset of the pair space makes the open sets on each fiber complicated. –  Theo Johnson-Freyd Jan 16 '11 at 23:35
Hmm, yes. Pradines makes some point about focusing on foliation groupoids that satisfy some conditions (regular or simple) but I don't know what these entail or imply. –  David Roberts Jan 17 '11 at 6:40
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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.

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