There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 stable $\infty$-categories are known to be equivalent:


1. The $\infty$-category $Fun(\mathbb N, \mathcal C)$ of filtered objects in $\mathcal C$ (where $\mathbb N$ is the poset of natural numbers).
2. The $\infty$-category $Fun(\Delta, \mathcal C)$ of cosimplicial objects in $\mathcal C$.
3. The $\infty$-category $Ch(\mathcal C)\_{\leq 0}$ of nonpositively-homologically-graded chain complexes in $\mathcal C$.


$(1) \Leftrightarrow (2)$ is due to Lurie (see HA 1.2.3) and $(1) \Leftrightarrow (3)$ is due to [Ariotta](https://arxiv.org/abs/2109.01017). $(2) \Leftrightarrow (3)$ is meant to be reminiscent of the classical [Dold-Kan theorem](https://ncatlab.org/nlab/show/Dold-Kan+correspondence).


There's another place where chain complex structures can come from though, namely from actions by the circle group $S^1$. For instance, in the [HKR theorem](https://ncatlab.org/nlab/show/Hochschild-Kostant-Rosenberg+theorem), the differential on the de Rham complex arises directly from the $S^1$-action on Hochschild cohomology. (I'm not familiar enough with the literature to have a reference for this, but I gather that the idea is to look at the map $X \oplus \Sigma X = \Sigma^\infty\_+ S^1 \wedge X \to X$ coming from a circle action; the second component is a map $\Sigma X \to X$ which is exactly the data needed for a differential; that it squares to zero comes from the associativity of the circle action, since the top cell of $S^1 \times S^1$ splits off.)


In other words, we are led to consider


4. The $\infty$-cateogry $Fun(\mathbb C\mathbb P^\infty, \mathcal C)$ of $\mathcal C$-objects with $S^1$-action.


Now, I think that (4) lives in the unbounded world -- the right things to compare to are


1. ' $Fun(\mathbb Z, \mathcal C)$ (filtrations extending in both directions)
2. ' (omitted -- but Kan's [combinatorial spectra](https://ncatlab.org/nlab/show/combinatorial+spectrum) might be relevant)
3. ' $Ch(\mathcal C)$ (unbounded chain complexes)


**Question:** Are the $\infty$-categories (1') and (3') equivalent to (4), for an arbitrary stable $\infty$-category $\mathcal C$?



Is there a Dold-Kan theorem for circle actions?
No, they are not equivalent, even for $C = Sp$.


Indeed, the category of spectra with $S^1$-action is also the category of $\mathbb S[S^1]$-modules, and is compactly generated by a single object.


On the other hand, compact objects of $Fun(\mathbb Z, Sp)$ are retracts of finite colimits of representables, and any finite set $S$ of representables cannot generate the whole thing - e.g. because if $n$ is below all the elements in $S$, then $F(n) = 0$ for any $F$ generated under colimits by $S$ . So it is not compactly generated by a single object (you have to change the proof a bit, but the same holds for $Fun(\mathbb N, Sp)$)


