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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!

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    $\begingroup$ There is some discussion of $S^1$-actions on ∞-categories in Lurie's paper "Rotation invariance in algebraic K-theory". $\endgroup$
    – Marc Hoyois
    Commented Jul 11, 2017 at 0:43
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    $\begingroup$ It may be useful not to view $S^1$ as a simplicial group, but rather as a group object in the ∞-category of spaces (also called an ∞-group). Of course, simplicial groups are one model for such things, but they have extra features that are not relevant to the situation at hand. An ∞-group $G$ can act on anything simply because the automorphisms of anything form an ∞-group, and taking (co)invariants is always a $BG$-indexed (co)limit in the ambient ∞-category. This doesn't tell you how to compute, but the definitions are very simple and independent of the kind of objects you're considering. $\endgroup$
    – Marc Hoyois
    Commented Jul 11, 2017 at 0:57
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    $\begingroup$ As Marc says, if you're just interested in setting up the theory, there's no difference between acting on infty-categories and acting on any other gadget: the ('Borel') homotopy theory of widgets with a G-action is the homotopy theory of local systems of widgets on the space BG, which is probably defined to be the homotopy theory of G-shaped diagrams (that settles your first bullet point.) Marc answered the second bullet. The third bullet also holds in general- you can calculate BG-shaped limits via that cosimplicial diagram in any infty-category. $\endgroup$
    – Dylan Wilson
    Commented Jul 11, 2017 at 14:00
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    $\begingroup$ (for G-actions on categories and spaces there's another answer to your first bullet point via straightening/unstraightening: the homotopy theory of G-shaped diagrams is then equivalent to the homotopy theory of (co)cartesian fibrations over the infty-group in question) $\endgroup$
    – Dylan Wilson
    Commented Jul 11, 2017 at 14:02

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