Given a symmetric monoidal category and a monoid object A in it, one can form the category of modules over this monoid object, i.e. objects are $A \otimes M \rightarrow M$ satisfying the natural properties analogous to modules over a ring and morphisms respecting this. The following seems to be true and I would like to know why:


If the category of modules has a closed symmetric monoidal structure with A as unit object, then A is a *commutative* monoid.


This is how I read the statement right after Proposition 2.3.4 in Hovey/Shipley/Smith's paper "Symmetric Spectra" and it would give an excellent motivation for introducing symmetric spectra...



Why is a monoid with closed symmetric monoidal module category commutative?
(I have rewritten parts to respond to potential objections, since someone disliked this answer. I didn't change the thrust of my discussion, but I did alter some places where I might have been slightly too glib. If I missed something, please feel free to correct me in the comments!)


The situation is even better than that! Suppose we are given an $E\_1$-algebra $A$ of a presentable symmetric monoidal $\infty$-category $\mathcal{C}$. 


Call an $E\_n$-monoidal structures on the $\infty$-category $\mathbf{Mod}(A)$ of left $A$-modules *allowable* if $A$ is the unit and the right action of $\mathcal{C}$ on $\mathbf{Mod}(A)$ is compatible with the $E\_n$ monoidal structure, so that $\mathbf{Mod}(A)$ is an $E\_n$-$\mathcal{C}$-algebra. Then the space of allowable $E\_n$-monoidal structures is equivalent to the space of $E\_{n+1}$-algebra structures on $A$ itself, compatible with the extant $E\_1$ structure on $A$. (This is even true when $n=0$, if one takes an $E\_0$-monoidal category to mean a category with a distinguished object.) The object $A$, regarded as the unit $A$-module, admits an $E\_n$-algebra structure that is suitably compatible with the $E\_1$ structure an $A$. [Reference: Jacob Lurie, DAG VI, Corollary 2.3.15.]


Let's sketch a proof of this claim in the case Peter mentions. Suppose $A$ is a monoid in a presentable symmetric monoidal category $(\mathbf{C},\otimes)$. Suppose $\mathbf{Mod}(A)$ admits a monoidal structure (not even a priori symmetric!) in which $A$, regarded as a left $A$-module, is the unit. I claim that $A$ is a commutative monoid. Consider the monoid object $\mathrm{End}(A)$ of endomorphisms of $A$ as a left $A$-module; the Eckmann-Hilton argument described below applies to the operations of tensoring and composing to give $\mathrm{End}(A)$ the structure of a commutative monoid object. The multiplication on $A$ yields an isomorphism of monoids $A\simeq\mathrm{End}(A)$.


In the case you mention, the result amounts to the original Eckmann-Hilton, as follows. If $X$ admits magma structures $\circ$ and $\star$ with the same unit (Below, Tom Leinster points out that I only have to assume that each has *a* unit, and it will *follow* that the units are the same. He's right, of course.) with the property that


$$(a\circ b)\star(c\circ d)=(a\star c)\circ(b\star d)$$


for any $a,b,c,d\in X$, then (1) the magma structures $\circ$ and $\star$ coincide; (2) the product $\circ$ is associative; and (3) the product $\circ$ is commutative. That is, a unital magma in unital magmas is a commutative monoid.


