Base change for module categories? ($E_\infty$-modules in $\mathrm{Cat}$)

Question

I'm working on a project where I would like to consider the category of symmetric monoidal categories. Though I suspect it will be easier easier to consider the $\infty$-category of symmetric monoidal $\infty$-categories (I believe I want strong monoidal functors, but I'm open to the lax case too).

Given a symmetric monoidal category $(\mathcal{C},\otimes)$ we can define a "module category" (also called a "2-module" or an "actegory" in the literature) which is a category $\mathcal{M}$ equipped with an action $\mathcal{C} \times \mathcal{M} \to \mathcal{M}$ and various natural isomorphisms witnessing $(C \otimes C') \cdot M \cong C \cdot (C' \cdot M)$, etc.

Next, given a symmetric monoidal functor $f : (\mathcal{C},\otimes) \to (\mathcal{D},\otimes')$ we get a "restriction of scalars" from a $\mathcal{D}$-module category to a $\mathcal{C}$-module category, and I would love for this functor to have left and right adjoints, given respectively by tensoring and homming with $\mathcal{D}$. (To make sense of homming I might want to be viewing $\mathrm{Cat}$ as an $(\infty,2)$-category).

I know there are some results in this vein written in the language of bicategories, but I was wondering if this can be reduced to a special case of something in Lurie's Higher Algebra. After all, (as I understand it), a symmetric monoidal category is "just" an $E_\infty$-algebra in $\mathrm{Cat}$, and it seems believable that there should already be theorems available about categories of modules over an $E_\infty$-algebra, with results about base change like I'm looking for. The main issue is that trying to look up results of this kind give things that are either general enough to be hard to read (with what I currently know) or applications to ring spectra (which are interesting, but not what I'm currently looking for).

Thanks in advance!

Answer 1

This is indeed the case, and follows from various things in Higher Algebra. The $\infty$-category of $\infty$-categories $\mathrm{Cat}_\infty$ is presentable, and moreover the cartesian product on it commutes with with colimits in both coordinates. In other words, it is presentably symmetric monoidal. From this, Lurie constructs a functor $$ \mathrm{Mod}_{(-)}(\mathrm{Cat}_\infty): \mathrm{CMon}(\mathrm{Cat}_\infty) \to \mathrm{Pr}^{\mathrm{L}} $$ where the source is the $\infty$-category of commutative monoids (i.e. $\mathbb{E}_\infty$-monoids) in $\mathrm{Cat}_\infty$, and $\mathrm{Pr}^{\mathrm{L}}$ is the $\infty$-category of presentable $\infty$-categories. This functor sends a morphism to a left adjoint functor (as it it lands in $\mathrm{Pr}^\mathrm{L}$), and it indeed sends it to the left adjoint to the restriction of scalars you mentioned, given by relative tensor product. Furthermore, the restriction of scalars admits a further right adjoint.

As for references, the fact that $\mathrm{Cat}_\infty$ is presentably symmetric monoidal is Lemma 4.8.4.2 (for the case $\mathcal{K}=\emptyset$). The parts on modules is in Section 4.8.5, which admittedly is not the easiest to read. The relevant part is Notation 4.8.5.10 and Theorem 4.8.5.16 (building on Construction 4.8.3.24), which construct a symmetric monoidal functor sending an $\mathbb{E}_1$-algebra in a presentable category to its $\infty$-category of right modules. Then, applying $\mathrm{CAlg}$ to get $\mathbb{E}_\infty$-version, and restricting to commutative algebras in $\mathrm{Cat}_\infty$ (as in Remark 4.8.5.17), gives the functor I mentioned above. The fact that the restriction of scalars admits adjoints from both sides follows from Corollary 4.2.3.7(2) and mentioned more explicitly below it in Remark 4.2.3.8. The description of the left adjoint appears in Proposition 4.6.2.17.