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Let $(C, \otimes)$ be a symmetric monoidal category (maybe braided is also okay). Then the category $\text{Mon}(C)$ of monoid objects in $C$ is also a symmetric monoidal category with the same monoidal product. The category $\text{Mon}(\text{Mon}(C))$ is, by the Eckmann-Hilton argument, just the category of commutative monoids in $C$; by a second application of Eckmann-Hilton, the process of constructing monoid categories stabilizes in the sense that the corresponding sequence of forgetful functors

$$... \to \text{Mon}(\text{Mon}(C)) \to \text{Mon}(C) \to C$$

stabilizes to a sequence of equivalences by the third functor at worst. However, the same argument shows that if $C$ is itself the category of monoids in some other symmetric monoidal category, then this sequence in fact stabilizes by the second functor, and if $C$ is itself the category of commutative monoids in some other symmetric monoidal category, then this sequence in fact stabilizes by the first functor.

So symmetric monoidal categories are divided up into three classes based on how soon the process of repeatedly constructing monoid categories stabilizes. Are there names for these three classes? Are there theorems characterizing them, for example theorems to the effect that if $C$ stabilizes early then it is nontrivially a category of monoids (not just $\text{Mon}(C)$ in the case that it stabilizes by the first functor)? (Vaguer question: is there some kind of homotopy-theoretic interpretation of these three classes?)

A somewhat nontrivial example of a category $C$ such that $\text{Mon}(C) \to C$ is already an equivalence is $C = \text{Set}^{op}$ with categorical coproduct (product in $\text{Set}$) as the monoidal product. This is due to the fact that every set has a unique comonoid structure given by the diagonal $X \ni x \mapsto (x, x) \in X \times X$. The corresponding monoid interpretation is that $\text{Set}^{op}$ can be realized as a suitable category of Boolean rings, which are commutative monoid objects in the category of abelian groups (with tensor product).

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Do you know any example of a category $C$ where the process stabilizes after the first step, i.e. such that $Mon(C) \to C$ is not an equivalece, but $CMon(C) \to Mon(C)$ is an equivalence? – Thomas Nikolaus Sep 4 '12 at 9:40
Thomas: here's one. Let $M$ be a nontrivial monoid, regarded as a discrete monoidal category $C$. (So the objects of $C$ are the elements of $M$.) Then $Mon(C) = 1$, so $Mon(C) \to C$ is not an equivalence but $CMon(C) \to Mon(C)$ is. – Tom Leinster Sep 4 '12 at 10:42
(Here $M$ must be commutative, in order that $C$ is symmetric.) – Tom Leinster Sep 4 '12 at 10:44
@Thomas: the category of monoids in $(\text{Set}, \times)$ or $(\text{Ab}, \otimes)$, etc. is an example. – Qiaochu Yuan Sep 4 '12 at 17:41
Sub-question: A symmetric monoidal category $C$ stabilizes after one step if and only if every monoid in $C$ is commutative. However, in all examples people have given thus far, every monoid in $C$ is commutative for a trivial reason: every object in $C$ has at most one monoidal structure. Moreover, every morphism between objects with a monoidal structure preserves that monoidal structure, so that the map $\operatorname{Mon}(C) \to C$ is an inclusion of a full subcategory. Are there any examples of such $C$ where the monoidal structure is not unique, or at least where the map is not full? – Evan Jenkins Sep 4 '12 at 21:47
up vote 11 down vote accepted

Your example suggests the characterization of those $C$ which are already stabilized: they are exactly the monoidal categories whose monoidal structure is cocartesian. To see this, it suffices to observe that the monoidal structure on $\mathrm{CMon}(C)=\mathrm{Mon}(\mathrm{Mon}(C))$ is always cocartesian, and that cocartesianness makes every object into a commutative monoid in a unique way.

I don't know a similar characterization of monoidal categories that stablize after one step.

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If you take commutative monoids in the first step, there is another natural choice of monoidal product on $\mathrm{CMon}(C)$. Namely, the universal "bilinear" construction, due in this generality to Anders Kock: in case $(C,\otimes) = (\mathrm{Set},\times)$, morphisms $M \otimes N \to K$ correspond to functions $M \times N \to K$ that are monoid morphisms in each argument separately when the other is fixed. In a sense this is more interesting, because no collapse of the kind you describe occurs. In fact, $\mathrm{Mon}(\mathrm{CMon}(C))$ is precisely the category of semirings in $C$. So heightening the "tower" of monoid categories then adds structure that occurs naturally and often in algebra.

See also and references therein. Incidentally, that paper also has a precise proof of the result Mike Shulman's answers refers to.

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What's an explicit description of this monoidal product for, say, commutative rings (that is, taking $C = (\text{Ab}, \otimes)$)? – Qiaochu Yuan Sep 4 '12 at 17:41

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