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Let $G$ be a Lie group and $\mathcal{D}$ be a differentiable stack (I am also ok to start with a topological group and topological stack).

I have seen someone mentioning somewhere that the notion of group action on stacks appeared first in Group Actions on Stacks and Applications by M. Romagny. The below definition of group action on a differentiable stack is from Group actions on stacks and applications to equivariant string topology for stacks by Gregory Ginot and Behrang Noohi.

A Lie group action on a differentiable stack is given by a morphism stacks $\alpha: G\times \mathcal{D}\rightarrow \mathcal{D}$ satisfying some conditions. Though they did not specify, I am believe that by $G$ they mean the stack $[*/G]$, so an action of a Lie group $G$ on a differentiable stack $\mathcal{D}$ is a morphism of stacks $\alpha: [*/G]\times \mathcal{D}\rightarrow \mathcal{D}$ satisfying some conditions (correct me if I am wrong).

Questions :

  1. Is there any notion of Lie group $G$ action on a Lie groupoid $[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$? Would a pair of maps $(G\times \mathcal{G}_1\rightarrow \mathcal{G}_1, G\times \mathcal{G}_0\rightarrow \mathcal{G}_0)$ giving an action of Lie group on the manifolds $\mathcal{G}_1,\mathcal{G}_0$ compatible with source, target etc maps of Lie groupoid, a good notion of Lie group action on a manifold?
  2. Is the notion of Lie group action on a differentiable stack mentioned above deduced/inspired from some notion of Lie group action on a Lie groupoid, in the sense that this notion of Lie group action on Lie groupoid is Moria invariant giving an action of Lie group on a differentiable stack?
  3. Is this definition of Lie group action on a differentiable stack directly/indirectly related to the notion of action of a group object on an object of a category as mentioned in Definition $2.15$ of Notes on Grothendieck topologies, fibered categories and descent theory?
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  • $\begingroup$ Thanks @YCor for the edit :) $\endgroup$
    – Praphulla Koushik
    Commented Feb 26, 2020 at 0:26
  • $\begingroup$ I am not sure that $[*/G]$ is the correct object to consider. If you want to recover the action of a group on a manifold in the case that $\mathcal D$ is just a manifold, you really need $G$ to be a Lie group, that is a manifold with extra structure maps (like $G\times G\to G$). I don't think $[*/G]$ would do the job. $\endgroup$
    – Sebastian Goette
    Commented Feb 26, 2020 at 19:45
  • $\begingroup$ @SebastianGoette That seem to be correct.. :) :) Thank you.. How should I interpret the map $G\times \mathcal{D}\rightarrow \mathcal{D}$ as? $\endgroup$
    – Praphulla Koushik
    Commented Feb 27, 2020 at 3:13
  • $\begingroup$ I was hoping to see a good answer to your question by somebody else ... $\endgroup$
    – Sebastian Goette
    Commented Feb 28, 2020 at 9:54
  • $\begingroup$ @SebastianGoette Oh. Please let me know if you have any favorite reference for this set up? $\endgroup$
    – Praphulla Koushik
    Commented Mar 1, 2020 at 16:04

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