Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions.

- When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some Lie group $G$?
- When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(M\rightrightarrows M)$ for some manifold $M$?
- When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $G\ltimes M$ for some Lie group $G$ acting on a manifold $M$?

It is only reasonable to see what properties the Lie groupoids $(G\rightrightarrows *), (M\rightrightarrows ), G\ltimes M$ have so that we can say $\mathcal{G}$ should have atleast these properties.

My observations:

- Given a Lie group $G$, the Lie groupoid $(G\rightrightarrows *)$ is transitive Lie groupoid. Thus, it should be the case that $\mathcal{G}$ is necessarily transitive to be represented by Lie group. Surprisingly, any transitive Lie groupoid is representable by a Lie group, more precisely $\mathcal{G}\cong \mathcal{G}_x$ (isotropy group) for any $x\in \mathcal{G}_0$ . I do not know if we can say anything more in this. Is there any other characterization? Can we say anything more?
- For a manifold $M$, the Lie groupoid $(M\rightrightarrows M)$ is a proper Lie groupoid simply because, $(s,t):M\rightarrow M\times M$ is the diagonal map which is a proper map. So, $(M\rightrightarrows M)$ is a proper Lie groupoid. If we want $\mathcal{G}$ to be represented by manifold, we should have atleast this condition that $\mathcal{G}=(P\rightrightarrows X)$ is a proper Lie groupoid i.e., $(s,t):P\rightarrow X\times X$ is a proper map. For manifold $M$, $(s,t):M\rightarrow M\times M $ is injective. If we want $\mathcal{G}$ to be represented by manifold, we should have atleast this condition that $\mathcal{G}=(P\rightrightarrows X)$ is such that $(s,t):P\rightarrow X\times X$ is injective. David Roberts says here that, assuming that $(s,t):P\rightarrow X\times X$ is proper and injective sufficient to confirm $\mathcal{G}\cong X/P$ (Any quick proof for this is also welcome). Are there any other characterization? How do we know where to stop looking at properties for characterization? I mean how do we guess (before proving) that $(s,t)$ is proper and injective then, $\mathcal{G}$ is represented by manifold. Why
**proper, injective**determining "manifoldness" of Lie groupoid?

Are there similar characterizations for a Lie groupoid to be a translation groupoid i.e., of the form $G\ltimes M$ for some Lie group $G$ acting on manifold $M$? It is observed here that, if a Lie groupoid is proper and etale, then $\mathcal{G}$ is *locally* a translation groupoid. Are there any characterization for whole Lie groupoid to be translation groupoid? What should we look for in a Lie groupoid to expect it to be a translation groupoid?