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Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:

In many cases, the category $\cal{M}od_A\left(\cal{C}\right)$ of $A$-modules in $\cal{C}$ inherits the structure of a symmetric monoidal category with respect to the relative tensor product over $A$.

Where can I find conditions, details and proofs of this (seemingly) elementary fact? (I didn't find it in the book I searched - MacLane, Barr & Wells, or in books about operads etc.)

I also need the facts that "extension of scalars" $-\otimes A$ is the left adjoint of the forgetful functor and it commutes with the tensor product. And the case when $\otimes$ distriute over the coproduct (or the bi-product).

I think I can prove most of these facts, but I want to be sure about them, and it would also be much easier to refer to them.

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  • $\begingroup$ This sort of stuff is shown in a lot of places that I know of for topological categories, and these things follow as degenerate cases of that, but that's probably overkill. $\endgroup$ – Jonathan Beardsley Sep 11 '14 at 23:14
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    $\begingroup$ I don't even see how can one check associativity of $\otimes_A$ if $\otimes$ doesn't preserve colimits, which is automatic in the closed case, of course. Maybe you can do it with weaker hypothesis but, do you really have a non-closed example in mind? $\endgroup$ – Fernando Muro Sep 11 '14 at 23:22
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    $\begingroup$ I believe that having $\otimes$ distribute over reflexive coequalizers should suffice for the first paragraph, and so having $\otimes$ distribute over finite colimits should suffice for the first and second paragraphs. But I'd need some time to track down suitable references. $\endgroup$ – Todd Trimble Sep 11 '14 at 23:29
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    $\begingroup$ As an $A$-module is an algebra for the monad $A \otimes - \colon \mathcal{C} \to \mathcal{C}$, you could use the references in this answer: mathoverflow.net/a/75929/10368. The sufficient condition there is that the module category has reflexive coequalizers; I'm not sure if it is also necessary. $\endgroup$ – Chris Heunen Sep 12 '14 at 9:17
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    $\begingroup$ One reference is "Dualizations and antipodes, Applied Categorical Structures 11 (2003) 229-260". See Section 5 there. It defines Comod rather than Mod, but it is essentially the same. Here's a pdf maths.mq.edu.au/~street/Antipode.pdf $\endgroup$ – Dimitri Chikhladze Sep 12 '14 at 12:02
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One needs that $C$ is cocomplete and that $\otimes$ preserves colimits in each variable (and then $\mathrm{Mod}_C(A)$ will have the corresponding properties). More precisely, you only need that $C$ has reflexive coequalizers and that $\otimes$ preserves them in each variable. This has been known for decades, but the first clear write-up of this, at least I know of, is Florian Marty's thesis. You can also find a discussion on this in my thesis, Section 4.1 (and Chapter 6 for the issue on reflexive coequalizers). See also MO/114457 for a discussion of the internal homs by Todd Trimble.

Edit. Here are some other references, which even discuss the case where $- \otimes A$ is replaced by a (suitable) symmetric monoidal monad:

H. Lindner. Commutative monads. In Deuxiéme colloque sur l'algébre des catégories. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

R. Guitart. Tenseurs et machines. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(1):5-62, 1980.

A. Kock. Closed categories generated by commutative monads. Journal of the Australian Mathematical Society, 12(04):405-424, 1971.

G. J. Seal. Tensors, monads and actions. Theory Appl. Categ., 28:No. 15, 403-433, 2013.

If I recall correctly, some of these references restrict the monads in such a way that they are of the form $- \otimes A$ anyway.

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    $\begingroup$ Well you do know of a slightly earlier write up: the very DAG III that the OP mentioned is from 2007 and contains the more general statement for $\infty$-categories. :) $\endgroup$ – Omar Antolín-Camarena Oct 8 '14 at 13:12
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    $\begingroup$ I would never cite a $\infty$-categorical result for a basic result about categories. Because the development is vice versa ... $\endgroup$ – Martin Brandenburg Oct 8 '14 at 13:53
  • $\begingroup$ I wouldn't cite an $\infty$-categorical result for a fact about ordinary categories either. (The ":)" indicated my remark was tongue in cheek.) I do find it funny that the earliest reference found so far seems to be DAG III when the result must have been known much earlier. $\endgroup$ – Omar Antolín-Camarena Oct 8 '14 at 14:17

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