Easily visualizable examples of Lie groupoids?

Question

The goal is to understand the Poisson cohomology of Poisson manifolds, which according to the nCatLab, is "just" the Lie algebroid cohomology of the corresponding Poisson Lie algebroid. Chasing down the rabbit-hole-esque chain of definitions lands me at the definition of a Lie groupoid, which merges two things that I am familiar with (Lie groups and groupoids) into a single chimeric creature that I am having difficulty trying to comprehend. I can easily imagine a smooth structure on a group, or alternately a group structure on a manifold (tori come to mind), but the vague mental images I conjure up when I look at groupoids are not easily amenable to adorning a smooth structure with. Alternately, it's hard to think of what a manifold with a groupoid structure would "look like". Hence I would like a few low-dimensional examples of Lie groupoids which make clear how a manifold can carry a groupoid structure. Hopefully there could be some examples where the manifold of objects is finite, or countable (i.e. discrete). Any help would be appreciated.

EDIT: To focus the question a little more, I'd like low-dimensional manifolds satisfying the following criteria:

(i) manifold of objects discrete, manifold of maps continuous

(ii) manifold of objects continuous, manifold of maps discrete

(iii) manifold of objects and manifold of maps both continuous

where by "continuous" I mean non-discrete.

Answer 1

Not sure if this is an answer, but probably more than a comment. First off, there's a trivial example: for any manifold $M$ you can make a Lie groupoid $M \Rightarrow M$ by making both the source and target maps the identity. This already hints at the fact that Lie groupoid structures are much less "rigid" and more plentyful than Lie group structures. The way I would think about Lie groupoids is not by trying to 'visualize' one (I'm not sure this is useful for groups anyhow), but thinking about some important examples. To quote a few:

• Given a group $G$ acting on a manifold $M$, you can form a Lie groupoid (called the action groupoid) $G \times M \Rightarrow M$, where $(g, x) \in G \times M$ has source $x$ and target $gx$ (figure out the multiplication). This groupoid encodes the action of $G$ on $M$.
• Given a submersion $p : M \to N$ you can form a Lie groupoid $M \times_p M \Rightarrow M$ with source and target maps the respective projections on first and second factors. Essentially we are putting arrows being elements on the same $p$-fiber. In the case $M = N$ and $p$ is the identity this is the called the pair groupoid.
• A fun example is the fundamental groupoid $\Pi(M) \Rightarrow M$. Here the arrows from $x$ to $y$ are homotopy classes (w/ fixed endpoints) of paths from $x$ to $y$ in $M$. It's not obvious that this has a Lie groupoid structure, but it does!
• Given a vector bundle $E \to M$, it has a general linear groupoid $\text{GL}(E) \Rightarrow M$ where arrows from $x$ to $y$ are linear isomorphisms $E_x \to E_y$.

Naturally by making suitable choices in the above constructions you can come up with examples where all relevant manifolds can be visualized -- but I can't imagine a case where that's actually useful or interesting.