$R$-Module objects in cartesian closed categories

Question

I am looking for a reference for the following statement.

Theorem. Let

• $C$ be a regular, well-powered, countably complete cartesian closed category,
• $R$ be a (commutative) ring object in $C$,
• $R\mathrm{Mod}(C)$ be the category of $R$-module objects in $C$.

Then $R\mathrm{Mod}(C)$ is a regular, well-powered, countably complete closed symmetric monoidal category (with the closed monoidal structure given in certain specific ways).

Proof sketch/references I found: I can only piece together the proof from a number of (partially non-citable) places: First, the statement for abelian group objects can essentially be found in Zhen Lin's answer to this question, who refers to some unpublished lecture notes of his. Now, a ring object in $C$ is a monoid in the monoidal category of abelian group objects in $C$, and the modules for this monoid are exactly $R$-module objects, which inherit the closed monoidal structure (see this paper by Seal).

However, this is a bit hand-wavy and the statement seems useful enough to probably have appeared somewhere in the published literature -- I just can't find it.

EDIT: To be extra clear: by an $R$-module object, I mean an abelian group object $M$ together with a morphism $R \times M \to M$ making a number of "obvious" diagrams commute. If we suppose that we already know the monoidal structure on $Ab(C)$ (category of abelian group objects in $C$), then this is the same as an $R$-action/module when viewing $R$ as a monoid in the monoidal category $Ab(C)$.

Examples/Motivation:

• if $C=\mathsf{Set}$, then $R\mathrm{Mod}(C)$ are ordinary $R$-modules,
• if $C$ is a convenient category of topological spaces, then $\mathbb{R}$-module objects are essentially (but not quite) topological vector spaces,
• if $C=\mathsf{Difflg}$ is the category of diffeological spaces, then $\mathbb{R}$-module objects are exactly diffeological vector spaces,
• if $C=\mathsf{Set}^{\Delta}$, then $\mathbb{Z}\mathrm{Mod}(C)$ is the category of simplicial abelian groups.
• if $C=\mathsf{Cond(Set)}$ is the category of condensed sets, then $\mathbb{R}$-module objects are condensed vector spaces (see e.g. here), and $\mathbb{Z}$-module objects are condensed abelian groups,
• further examples: bornological vector spaces, internal $R$-modules in a topos, etc.

I would mainly like a simultaneous construction of the closed symmetric monoidal structure for all of these cases.

The main problem: What I haven't found a good reference for is the fact that the forgetful functor $R\mathrm{Mod}(C) \to C$ is monadic and that the resulting monad is a symmetric monoidal (equivalently, commutative strong) monad on $C$ (with respect to the cartesian monoidal structure on $C$).

EDIT: The main problem, reformulated: The main problem reduces to the assertion: the forgetful functor $\mathrm{Ab}(C) \to C$ is monadic and that the resulting monad is a symmetric monoidal (equivalently, commutative strong) monad on $C$ (with respect to the cartesian monoidal structure on $C$). This was answered in this question, as noted above, with the caveat that the reference provided are unpublished lecture notes which I don't consider citable (for example, it doesn't say which of the results there are original, and the material seems like it should be "old").

Answer 1

References for all the properties of $RMod(C)$ that you ask for can be found in Borceux's excellent book Handbook of Categorical Algebra 2: Categories and Structures, which is very worth having a copy of. I will use the observation that the category $RMod(C)$ is the category of algebras over the monad $T_R(M) = R\otimes M$ on $C$. I think this is clear, but if you need a reference for it, you can cite Theorem 4.1 (page 15) of Algebras and modules in monoidal model categories. That theorem also gives the closed symmetric monoidal structure on $RMod(C)$.

Borceux's Proposition 4.3.1 (page 197) proves that $RMod(C)$ has whichever limits $C$ has. Then, Proposition 4.3.2 does the same for colimits (note the description of $T_R$ above). The category $RMod(C)$ has limits and colimits computed in $C$, which is not true for general categories of algebras over a monad. Next, Theorem 4.3.5 proves that $RMod(C)$ is regular. Lastly, that $RMod(C)$ is well-powered is 4.8.8 (page 253).

EDIT: The OP has edited their question yet again. The "reformulated question" was already answered by me, in a comment, four hours ago. I paste that comment here to make it more visible. Not planning to waste any more time engaging with this particular question.

In Hovey's paper "Monoidal model categories", the first line of the proof of Proposition 2.8 is that "It is well-known that A-mod is closed symmetric monoidal: see [HSS98, Section 2.2]." In that HSS paper Symmetric Spectra, Lemma 2.2.8 shows how to make $R$-mod closed symmetric monoidal (assuming $C$ is).

Answer 2

My apologies for the length and verbosity of this answer, but I still believe that there is a significant gap in David White's answer that needs to be addressed, and my impression is that I need to be quite detailed at this point. [If there just turns out to be a misunderstanding from my side, I apologise, and I will delete this answer.]

For all I know at the moment, the answer to the original question is that there might in fact be a small gap in the literature: I could not find a reference that would prove the theorem in the question without requiring a (small, but non-trivial) leap of faith.

First of all, David's answer does provide detailed references for the well-poweredness, (countable) (co-)completeness, and regularity of $\mathrm{RMod}(C)$, once we know that the forgetful functor $\mathrm{RMod}(C)\to C$ is monadic. The point that's missing is a construction of the closed symmetric monoidal structure on $\mathrm{RMod}(C)$. David makes the right suggestion to use the following theorem (at least I assume this is what he means; and this is what Hovey rightfully calls "well known" in his paper):

Theorem 1. Let $D$ be a closed symmetric monoidal category and let $M$ be a monoid in $D$. Then the category of $M$-actions (perhaps confusingly called $M$-modules by Hovey and others, cf. Morgan Roger's comment) is a closed symmetric monoidal category.

In the above theorem, take $D=\mathrm{CMon}(C)$ the category of commutative monoids in $C$, and suppose that we already know the ("canonical") symmetric monoidal structure on $\mathrm{CMon}(C)$ (this tensor product on $\mathrm{CMon}(C)$ is not given by the coproduct in $\mathrm{CMon}(C)$/the product in $C$!). Then, indeed, it will turn out that a commutative semiring $R$ in $C$ is the same thing as a commutative monoid in $\mathrm{CMon}(C)$. So applying the above theorem to $M:=R$ reduces the construction of the closed symmetric monoidal structure on $\mathrm{RMod}(C)$ to constructing one on $\mathrm{CMon}(C)$. So we need to show:

Theorem 2. [no reference found] Let $C$ be a finitely complete, countably cocomplete closed cartesian closed category. Then $\mathrm{CMon}(C)$ is a closed symmetric monoidal category.

To prove this, we cannot apply Theorem 1, since a) $\mathrm{CMon}(C)$ is not generally a category of $M$-actions for some monoid $M$ in a symmetric monoidal category, in a non-trivial way, and b) Theorem 1 takes a general closed symmetric monoidal category as input, but Theorem 2 is false for $C$ a general closed symmetric monoidal category.

For a), take $C=\mathsf{Set}$, then $\mathrm{CMon}(C)$ is the category of commutative monoids. This is indeed trivially the category of $\mathbb{N}$-actions/modules in the symmetric monoidal category $\mathrm{CMon}(C)$, with the tensor product given by the tensor product of commutative monoids in the usual sense -- but for this we already have to know the monoidal structure on $\mathrm{CMon}(C)$.

For b), take the closed symmetric monoidal category $\mathrm{C}=\mathsf{Ab}$ of abelian groups. Then $\mathrm{CMon}(C)=\mathrm{Ring}$ is the category of commutative rings. This is not a closed monoidal category in any useful way. In particular, $\mathrm{Ring}$ is not closed monoidal with respect to its coproduct.

In the language of Hovey's paper, we are interested in a closed monoidal structure on $I$-algebras (where $I$ is the tensor unit), not on modules over some monoid.

Instead of Theorem 1, we will at least implicitly have to use:

Theorem 3. [Corollary 6.5.4, Brandenburg ] Let $C$ be closed symmetric monoidal category and let $T$ be a symmetric monoidal monad on $C$. Then the category $C^T$ of $T$-algebras (also sometimes called $T$-modules, to add to the confusion) is a closed symmetric monoidal category.

Remark. The tensor product and internal hom are constructed in a way that is analogous to the classical case of $R$-modules over a commutative ring $R$. This is why using Theorem 3 at least implicitly is likely unavoidable, as it is already implicitly used in the case $C=\mathsf{Set}$.

Now, in order to be able to employ Theorem 3, we will have to show i) that the forgetful functor $\mathrm{CMon}(C)\to \mathsf{C}$ has a left adjoint $S$, ii) that this forgetful functor is morover monadic, iii) that the resulting monad, denoted also by $S$ (by a slight abuse of notation), admits the structure of symmetric monoidal monad.

For i), a reference can be found in [Lemma 4.4.5, Brandenburg]. Explicitly, $S$ is given by, $$ S(X) := \coprod_{n \in \mathbb{N}} X^n/S_n, $$ where $X^n/S_n$ is the coequaliser obtained from the action of the symmetric group $S_n$ by "permuting the factors". This works more generally for $C$ any countably cocomplete monoidal category in which the tensor product distributes over countable coproducts (no cartesian monoidal structure needed for this point).

For ii), it is probably easiest to write down down an explicit (quasi-)inverse to the comparison functor $\mathrm{CMon}(C)\to C^S$. I haven't found any citable reference where this is done. (Here, we still don't need the cartesian monoidal structure.)

For iii), we do need $C$ to be a cartesian monoidal category, in order to write down the symmetric monoidal structure on $S$, $$ S(X) \times S(Y) \to S(X\times Y), \;\; 1 \to S(1), $$ I think we can do this by using the explicit construction of $S$ above, but we would then still have to verify that the necessary diagrams commute (probably not hard, but tedious). I haven't found this done in detail anywhere.

Finally, I also couldn't find explicit constructions of the closed symmetric monoidal structure on $\mathrm{CMon}(C)$ that don't use the language of symmetric monoidal/commutative monads. In other words, as far as I can tell at the moment, a detailed proof of the Theorem 2 seems to be missing from the published literature.