TITLE: Groupoids vs Pseudogroups 
QUESTION [9 upvotes]: (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)?

REPLY [3 votes]: 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.