# Groupoids vs Pseudogroups

(Warning: I'm not an expert in the topic) Let's work in a "geometric" category, for example the category $\mathfrak{Diff}$ of "manifolds" (without the requirements of connectedness and second countability and even of finite dimensionality), and consider also the subcategory of "usual" finite dimensional (Edit: possibly disconnected, Hausdorff, second countable) manifolds $\mathrm{Diff}$. We can than consider two kinds of objects:

• Lie groupoids $\mathcal{G}=(\mathcal{G}_1 \rightrightarrows \mathcal{G}_0)$ where $\mathcal{G}_0$ and $\mathcal{G}_1$ are smooth manifolds (in $\mathfrak{Diff}$) and the source $\mathrm{src}$ and target $\mathrm{trg}$ maps are submersions.
• Pseudogroups $\Gamma$ of diffeomorphisms of some manifold $M$ in $\mathrm{Diff}$.

Remark 1. From a pseudogroup $\Gamma$ on $M$ (for example the local symplectomorphisms if $M$ is equipped with a symplectic form, or the local diffeomorphisms preserving a foliation...) an (abstract) groupoid $\mathcal{G}=\mathrm{Germ(\Gamma)}$ can be obtained via the construction of germs, and I believe it's also a (usually infinite dimensional) Lie groupoid. In this case $\mathcal{G}_0$ is some space of germs of manifolds, and $\mathcal{G}_1$ consists of germs of local diffeomorphisms belonging to $\Gamma$. (Edit: for the benefit of the readers let me say D.Carchedi in his answer below has pointed out that this $\mathcal{G}$ is actually finite dimensional).

Remark 2. Given a Lie group action $\varphi\colon K\times M \to M$, the "action groupoid" $G_\varphi=(K\times M \rightrightarrows M)$ in $\mathrm{Diff}$ can be constructed. If I'm not mistaken, also an "action pseudogroup" $\Gamma_\varphi$ can be constructed, which consists of all the restrictions of the diffeomorphisms of the form $\varphi_g$, $g\in K$, to open subsets of $M$.

Remark 3. We can apply the germ construction to $\Gamma_{\varphi}$, obtaining a new groupoid $\mathcal{G}_{\varphi} = \mathrm{Germ} (\Gamma_{\varphi})$. Given $\gamma\in \mathcal{G}$ and $x,y \in M$ such that $\mathrm{src} (\gamma)=\mathrm{germ}_M (x)$ and $\mathrm{trg} (\gamma)=\mathrm{germ}_M (y)$, we can think of the element $\gamma$ both as a $\mapsto$ arrow between points (with no internal structure) $x,y$ (since $\varphi_g \colon x\mapsto y$) and as a $\to$ arrow between objects $x$ and $y$ (once we identify $x$ and $y$ with their germs of neighbourhoods in $M$).

Remark 4. Groupoids can be viewed as a generalization of group actions (and, more generally, of equivalence relations). In the theory of orbifolds (either in the differentiable category or in others) the morphisms $G_1$ of a groupoid $G_1\rightrightarrows G_0$ in $\mathrm{Diff}$ representing the orbifold $X=[G_1\rightrightarrows G_0]$ are viewed as abstract "glueing data" to obtain a "space" $X$ by patching the pieces of the (possibly very disconnected) space $G_0$ (please correct me if this view is mistaken). So, ideally, a morphism $(g\colon x\to y) \in G_1$ corresponds to the assertion that $\varphi_g\colon x \mapsto y$ for some map $\varphi_g\colon U \to V$ between local patches around $x$ and $y$ (producing identifications in an "étale" way).

Remark 5. On the contrary to remark 4, in the theory of moduli (we leave for a moment the differentiable setting) when a groupoid $G_1\rightrightarrows G_0$ represents a moduli stack $\mathcal{M}$, elements (closed points) of $G_1$ are seen as maps (isomorphisms) between the "structured things" that $\mathcal{M}$ parametrizes.

Q1. Concerning Remark 3, which is the relation btween $G_{\varphi}$ and $\mathcal{G}_{\varphi}$? Are they equivalent?

Q2. Given a Lie groupoid $G=(G_1\rightrightarrows G_0)$, say with finite dimentional object space $M:=G_0$, which conditions (if any) on $G$ ensure that there is a pseudogroup $\Gamma$ on $M$ so that $G$ is equivalent to $\mathrm{Germ}(\Gamma)$ ? So, for example, can Remark 4 be made rigorous? In other words, can the $\mapsto$ viewpoint (examplified by Remark 4) be interchanged with the $\to$ viewpoint (examplified by Remark 5) ? Is étaleness of $G$ sufficient or necessary?

Q3. More generally, given a groupoid $\mathcal{G}=(\mathcal{G}_1 \rightrightarrows \mathcal{G}_0)$, is there a manifold $\mathcal{S}$ in $\mathfrak{Diff}$ such that $\mathcal{G}$ is equivalent to the groupoid $\mathrm{Germ} (\Gamma)$ of germs of some pseudogroup $\Gamma$ of diffeomerphisms of $\mathcal{S}$ ?

Q4. What about other "geometries" (i.e. other sites in which to consider internal groupoids and -assuming the notion can be generalized- pseudogroups)?

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(can't understand why some LaTeX doesn't work) – Qfwfq Jun 16 '11 at 13:13
(Some LaTeX doesn't work because it interferes with an unrelated preprocessor. The trick is not to try to understand it, and just put backticks around it when it breaks.) – Emil Jeřábek Jun 16 '11 at 14:00
That doesn't sound like a mathematical approach. You just need to understand how Markdown and the TeX processor interact... I agree that the solution is to put backticks when necessary. – Yuji Tachikawa Jun 16 '11 at 15:49
(edited the title) – Qfwfq Jun 16 '11 at 15:57
Why on earth would you consider restricting to a category consisting of only connected manifolds? It prohibits natural constructions such as $\coprod U_i \to M$ for $U_i$ an open cover of $M$. – David Roberts Jun 17 '11 at 1:51

You've asked a mouthful. Let me do my best:

First, in your first remark, the Lie groupoid is NOT infinite dimensional. It is very finite dimensional. The arrows of $Germ(G)$ can be viewed as encoding a sheaf over $G_0,$ and hence can be given the unique topology making the source map into a local homeomorphism. This endows the arrow space with the structure of a (possibly non-Hausdorff) manifold of the same dimension as $M$. If you do not like non-Hausdorff manifolds, you can instead construct a Lie $2$-groupoid out of purely Hausdorff manifolds which is equivalent to it.

Q1. Offhand, I am not familiar with the construction of a pseudo-group out of a group action, unless the group acting is discrete (but if you do, please give me a reference). Lets restrict to discrete $K$. In general, the action groupoid $G_\varphi$ and the germ groupoid $\mathcal{G}_\varphi$ are not equivalent, but there is a strong relation; the latter is the "effective-part" of the former. If $K$ is finite, they are the same if and only if the action if faithful. In general, it turns out that the former may be viewed as a gerbe over the latter (when considering their associated differentiable stacks)

Q2 and Q3: The condition you are looking for is that $G$ is an effective etale Lie groupoid. Etale means that the source and target maps are local diffeomorphisms. Effective is more subtle:

Given any arrow $g$ in an etale Lie groupoid $\mathcal{G}$, we can find a neighborhood $U$ of $g$ in the arrow space $\mathcal{G}_1$ such that both the source and target maps restrict to an embedding over $U$. Hence $g$ induces a diffeomorphism from $s(U)$ to $t(U)$. This encodes a psuedo-group, and there is an obvious functor from $\mathcal{G}$ to the Lie groupoid associated to this pseudo-group. More or less by definition, an etale Lie groupoid is effective, if and only if this homomorphism is an isomorphism. Alternatively, effectivity means that for each point $x$ of $\mathcal{G}_0$, the action of the stabilizer group of $x$ on the germ of $\mathcal{G}_0$ around $x$ (given by the above construction) is faithful.

For your question on orbifolds, the answer is yes. See "Orbifolds, Sheaves, and Groupoids" by Moerdijk and Pronk.

Q4: If you make the question more specific, maybe I can attempt to answer it.

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You gave a very exaustive answer! - Concerning Q4, I was wondering if, for example, in algebraic geometry the answers to the previous questions would be essentially different. Is an effective étale groupoid still "equivalent" to a pseudogroup in this context? Or we must use étale topology instead of Zarisky topology to get analogous results? I wouldn't be surprised if the Zarisky topology were too coarse to obtain an isomorphism $g:s(U)\to t(U)$. – Qfwfq Jun 21 '11 at 14:59

You could look at

arXiv:1107.5511 Pseudogroups and their etale groupoids Mark V. Lawson, Daniel H. Lenz

and othjer works by Lawson in this area, for example the relation between inverse semigroups and ordered groupoids, originally studied by C. Ehresmann.

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You might want to look at the article: On the Structure of Lie Pseudo-Groups and other related articles, available here:

http://www.math.umn.edu/~olver/paper.html

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