Recent papers in derived algebraic geometry use a notion of $S^1$-actions on infinity categories. I think I understand what this "should" be and how to calculate with it; however, I can't find a much development of the foundational theory of actions of simplicial groups on $\infty$-categories in the literature. So, I feel my footing is a bit shaky; is there existing work, or work in progress, on this front?

Let me give a list of vague ideas for which I'm seeking formal statements, for example:

• Two (equivalent) Koszul dual definitions of a category $C$ (presentable stable -- how much flexibility can we have with the context?) with a action of a (reasonable) simplicial group $G$. In spaces, a space $X$ with a $G$-action can either be defined by a map $G \times X \rightarrow X$ (this would be a strict $G$-action) or a map $Y \rightarrow BG$ (I think this is non-strict?).

• A definition of invariant category $C^G$. The notion of invariants should depend on the ambient category of $\infty$-categories (e.g. differs in the setting of presentable categories with continuous functors vs. presentable categories with compact-object preserving continuous functors). One possible definition is that if $F$ is the functor associating to a category $C$ without a given group action the same category $C$ with the trivial group action, then the invariants functor should be the right adjoint to this. Of course, one has to make all appropriate definitions here...

• Equivalence with the usual presentation of the invariant category as a (small) limit in "reasonable" settings as (I've omitted the double/triple arrows) $$\cdots \text{Map}(G \times G, X) \rightarrow \text{Map}(G, X) \rightarrow X$$

Any comments would likely be helpful!