Is there essentially unique notion of module over monoidal stable $\infty$-categories?

Question

There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/R$.

I would like to replace here $\operatorname{Commutative rings}$ with some ($\infty$-?)category $\bf X$ of symmetric monoidal (should I say E$_\infty$?) stable $\infty$-categories. Then presumably "internal abelian groups" is not relevant, as stable $\infty$-categories have their own intrinsic sort of "abelian group" structure. On the other hand, also presumably, just taking slice over $R$ would not give any notion of module readily: for whatever kind of morphism $f:R'\to R$ of monoidal $\infty$-categories, I don't see any sensible way to define some action of $R$ on the fibre of $f$ (or is there any?).

So my question is what kind of structures in ${\bf X}/R$ would provide good notion of $R$-module in the stable $\infty$-setting? Or maybe there are different notions of module and some of them must be captured in entirely different ways?

Answer 1

The ∞-categorical analog of the fact you mention can be found in Higher Algebra, corollary 7.3.4.14:

Let $\operatorname{CAlg}$ be the category of $E_\infty$-rings and $A\in \operatorname{CAlg}$. Then there is a canonical equivalence

$$\operatorname{Sp}(\operatorname{CAlg}_{/A})\simeq \operatorname{Mod}_A$$

between the stabilization of the ∞- category of $E_∞$-rings with a map to $A$ and the ∞-category of $A$-modules.

Note that when $A$ is a discrete commutative ring, $\operatorname{Mod}_A$ is the usual derived category of $A$ (so not the 1-category of $A$-modules).

In fact a similar result holds for algebras over any coherent operad (ibid. Theorem 7.3.4.13). So one would be tempted to take $\operatorname{Sp}(\mathcal{C}_{/A})$ to be the definition of "$A$-modules" for $A\in\mathcal{C}$. It is not clear at all that this is a well behaved notion though: in the case where $\mathcal{C}$ is the ∞-category of animated rings (i.e. the ∞-category obtained by inverting the weak equivalences in the category of simplicial rings), this does not recover the natural notion of module: see for example the discussion in Section 25.3.3 of Lurie's Spectral Algebraic Geometry.