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I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state the question for the category of simplicial vector spaces $\mathsf{SVec}$.

As far as I understand, the structure of simplicial category for this kind of abelian categories of modules should be the same as the one for simplicial abelian groups from Goerss, Jardine, Theorem III.2.13. For $\mathsf{SVec}$:

The copowering functor $\cdot \otimes \cdot: \mathsf{SVec} \otimes \mathsf{SSet} \to \mathsf{SVec}$ is defined as $V \otimes K = V \otimes \mathbb{R} K$, the tensor product with the free simplicial vector space generated by $K$.

Then this fits in the two-variable adjunction

$$ \mathsf{SVec}(V \otimes K, W) \cong \mathsf{SSet} (K, \mathbf{Hom}_{\mathsf{SVec}}(V,W)) \cong \mathsf{SVec}(V, \mathbf{Hom}_{\mathsf{SSet}}(K, W)), \tag{1} $$

for any $K \in \mathsf{SSet}$, and $V,W \in \mathsf{SVec}$, where $\mathbf{Hom_{\mathsf{SVec}}}(V,W)$ is the mapping space between $V$ and $W$ (it consists of linear homotopies), and $\mathbf{Hom}_{\mathsf{SSet}}(K,W)$ is the mapping space between $K$ and the underlying simplicial set of $W$, which is actually the same as $\mathbf{Hom_{\mathsf{SVec}}}(\mathbb{R}K, W)$. Both of these also happen to be simplicial vector spaces and not just simplicial sets, but note that in the case $K=V$, they do not coincide. Now I want to know is if $\mathsf{SVec}$ is closed monoidal with respect to $\otimes$ and $\mathbf{Hom_{\mathsf{SVec}}}$, that is: Is there a two-variable tensor-hom adjunction $$ \mathsf{SVec}(V \otimes U, W) \cong \mathsf{SVec} (U, \mathbf{Hom}_{\mathsf{SVec}}(V,W)) \cong \mathsf{SVec}(V, \mathbf{Hom}_{\mathsf{SVec}}(U, W)), \tag{2} $$ for any $U,V,W \in \mathsf{SVec}$?

I managed to show that assuming that one can always choose a basis for a simplicial vector space, and write $U = \mathbb{R} B$, for some simplicial set $B$, there is a choice-dependent isomorphism as in (2) following from (1) since $\mathbf{Hom}_{\mathsf{SSet}}(B,W) = \mathbf{Hom}_{\mathsf{SVec}}(U,W)$. But is there a way to get a natural isomorphism?

In particular, is this adjunction general enough that it holds for all categories of simplicial $R$-modules over a ring $R$? If not, what are the minimal hypotheses for this to hold?

It seems to me this should be an example/exercise in books such as Hovey or Goerss, Jardine, but I could not find it there, so either it is more difficult than it looks or it is simply out of their main focus.

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The category of simplicial $R$-modules is the category of $R$-modules in the closed symmetric monoidal category of simplicial sets. I wrote an answer last week with references explaining why such categories of $R$-modules are monoidal. You were right to think this kind of thing was of interest to Hovey, but indeed was "out of his main focus" in the book. Instead of putting it in his book, he wrote it down in a paper written around the same time as the book, when Schwede and Shipley turned to the homotopy theory of monoids. Specifically, 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 cited HSS paper Symmetric Spectra, Lemma 2.2.8 shows how to make $R$-mod closed symmetric monoidal (assuming $C$ is). The proof goes along the lines you started in your post here, by writing down the internal hom and the tensoring, then checking that they are adjoint. Note that the tensor product you need is $-\otimes_R -$. The "function $R$-module" that Hovey mentions is the collection of morphisms $f: M\to N$ with the $R$-action you'd expect: $(r\cdot f)(m) = r\cdot (f(m))$. None of this requires $R$ to be a field. It's enough to be a commutative ring with unity.

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  • $\begingroup$ Not to be pedantic, but it seems to me that a simplicial $R$-module should be an $R$-module object in the category of simplicial abelian groups, not sets. The definition Hovey gives in the paper on Symmetric Spectra is simply: "A left $R$-module is an object M with an associative multiplication $\alpha: R \otimes M \to M$ that respects the unit." Is the abelian group structure implicit in this or is it extra data? If this is the case, then by lemma 2.2.8, my question reduces to "Is the category of simplicial abelian groups $\mathsf{SAb}$ closed monoidal?", right? $\endgroup$
    – SetR
    Commented Feb 29 at 14:15
  • $\begingroup$ In the end, I actually managed to find out the explicit adjunction, which is the same as that for simplicial sets found in Prop. I.5.1 in Goerss, Jardine, and if one defines the same bijection in $\mathsf{SAb}$, this ends up being linear, because everything else is. Is this interpretation correct? Do you think i should add it again as an answer? In any case thank you for your answer, it was helpful to see it from a different perspective even though I ended up preferring a less abstract approach. $\endgroup$
    – SetR
    Commented Feb 29 at 14:15
  • $\begingroup$ No, I mean simplicial sets. You tagged "at.algebraic-topology", and referenced Hovey and Goerss-Jardine. Your post emphasized the tensoring of SVect over sSet. The category of simplicial R-modules is a category of algebras over an operad in sSet. See Example 4 on page 387 of Goerss-Jardine (free-forgetful adjunction with sSet). See definition on bottom of page 8 of Schwede-Shipley "Algebras and Modules" paper. Separately, they discuss R-modules in sAb (which is closed symmetric monoidal, see page 13). You don't need to pass thru sAb to get from sSet to simplicial R-modules. $\endgroup$ Commented Feb 29 at 15:58
  • $\begingroup$ Should you write your own answer? I don't think it's necessary. The question was already borderline too basic for MO, and has been answered. I have no idea what you mean by "a less abstract approach." I literally spelled out the internal hom, tensor, and adjunction. Lastly, I'm growing slightly concerned that you are in fact the same user as BP from the other thread, due to writing style, font choices, and insistence on going thru sAb. It is a violation of the terms of MO to create "sock puppet" accounts. I hope I'm wrong. $\endgroup$ Commented Feb 29 at 16:01
  • $\begingroup$ I am still convinced there is a confusion between what you (and all references you cited) refer to as $R$-module (an object in $\mathsf{C}$ with an action of a monoid object $R$), and what I meant as $R$-module (in the most basic algebraic sense, an abelian group with an action of a ring $R$). I was not aware of this language overlap and I forgot to clarify that I meant $R$-module as a module over a ring $R$, which, according also to This example on nLab, should be an $R$-module in $\mathsf{Ab}$. $\endgroup$
    – SetR
    Commented Feb 29 at 21:22

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