# Lie group (topological group) action on differentiable stack (topological stack)

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?
• Thanks @YCor for the edit :) – Praphulla Koushik Feb 26 '20 at 0:26
• 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. – Sebastian Goette Feb 26 '20 at 19:45
• @SebastianGoette That seem to be correct.. :) :) Thank you.. How should I interpret the map $G\times \mathcal{D}\rightarrow \mathcal{D}$ as? – Praphulla Koushik Feb 27 '20 at 3:13
• I was hoping to see a good answer to your question by somebody else ... – Sebastian Goette Feb 28 '20 at 9:54
• @SebastianGoette Oh. Please let me know if you have any favorite reference for this set up? – Praphulla Koushik Mar 1 '20 at 16:04