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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 (correct me if I am wrong). 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 Feb 26 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 Feb 26 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 Feb 27 at 3:13
  • $\begingroup$ I was hoping to see a good answer to your question by somebody else ... $\endgroup$ – Sebastian Goette Feb 28 at 9:54
  • $\begingroup$ @SebastianGoette Oh. Please let me know if you have any favorite reference for this set up? $\endgroup$ – Praphulla Koushik Mar 1 at 16:04

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