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nLab article on tensor product says:

"Given two objects in a monoidal category $(C,\otimes)$ with a right and left action, respectively, of some monoid $A$, their tensor product over $A$ is the quotient of their tensor product in $C$ by this action. If $A$ is commutative, then this is a special case of the tensor product in a multicategory."

Where can I find a reference introducing this tensor product between actions? I couldn't find any.

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  • $\begingroup$ No particular reference comes to mind, although this type of construction is a basic tool in category theory. Suitably abstracted, for example, this construction yields relatively free algebras (see e.g. the discussion here: ncatlab.org/nlab/show/…), and appears as a canonical augmentation in two-sided bar constructions. You can find some related discussion here: mathoverflow.net/questions/180673/… $\endgroup$ – Todd Trimble Dec 22 '14 at 4:36
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Let's observe what this statement is saying. Let $(\mathcal{C},\otimes)$ by a monoidal category, and $A$ be a monoid in $(\mathcal{C},\otimes)$. Let $M,N$ be objects of $(\mathcal{C},\otimes)$ with a left and right action, respectively, of $A$. Then this statement is saying that $M\otimes_{A}N$ is $M\otimes N$ quotiented by the $A$-action. But this is by construction of $M\otimes_{A}N$; see the subheading in the nlab page on the tensor product. Now if $A$ is a commutative monoid (now we're assuming $(\mathcal{C},\otimes)$ is symmetric monoidal), then $M\otimes_{A}N$ is an object of $(\mathcal{C},\otimes)$ with an action of $A$.This agrees with the monoidal structure thanks to the fact that this tensor product over $A$ can be constructed as a coequalizer $M\otimes A \otimes N \;\rightrightarrows\; M\otimes N$ given by the action of $A$ on $M$ and on $N$. Then, as the nlab says, "there is a multicategory of $A$-modules whose tensor product agrees with the coequalizer defined above".

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A motivating example: if you consider the monoidal category $\mathbf{Ab}$ of abelian groups with tensor product $\otimes_{\mathbb Z}$, a monoid $A$ in $\mathbf{Ab}$ is a unital ring. If $M$ resp. $N$ are abelian groups with a right $R$-module, resp. left $R$-module structure, then the tensor product $M\otimes_R N$ (which is just an abelian group as $R$ is in general not commutative) is defined as the quotient of the abelian group $M\otimes_{\mathbb Z}N$ by the subgroup generated by the elements of the form $(m\cdot r)\otimes_{\mathbb Z} n - m\otimes_{\mathbb Z} (r\cdot n)$ for all $m\in M$, $n\in N$ and $r\in R$; by its very definition this subgroup takes into account the action of the ring on both modules.

These lecture notes contain more details on the topic.

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In the case when $A$ is not commutative, this result appear as exercise 6 in section VII.4 of MacLane "Categories". For the commutative case, see the references to my question.

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