Examples of rings in monoidal categories

Question

Ring objects are usually defined on Cartesian monoidal categories, but one can define them more generally on non-Cartesian symmetric monoidal categories as follows:

• Let $(\mathcal{C},\otimes,\mathbf{1})$ be a symmetric monoidal category.
• When equipped with the tensor product of $\mathcal{C}$, the category $\mathsf{CCoMon}(\mathcal{C})$ of cocommutative comonoids in $\mathcal{C}$ becomes Cartesian monoidal ― note that this requires $\mathcal{C}$ to be symmetric.
• A ring object in $\mathcal{C}$ is then a ring object in $\mathsf{CCoMon}(\mathcal{C})$.

Alternatively, if

• $\mathcal{C}$ and the category $\mathsf{HopfMon}^{\mathrm{bicomm}}(\mathcal{C})$ of bicommutative Hopf monoids in $\mathcal{C}$ have all co/limits, and
• there is a "free bicommutative Hopf monoid functor" $\mathcal{C}\to\mathsf{HopfMon}^{\mathrm{bicomm}}(\mathcal{C})$,

then we can mimic the construction of tensor products of abelian groups in $\mathcal{C}$, obtaining a symmetric monoidal category $(\mathsf{Ab}(\mathcal{C}),\boxtimes)$, the monoids in which are then defined to be ring objects in $\mathcal{C}$.

Note that

• Replacing $\mathsf{HopfMon}^{\mathrm{bicomm}}(\mathcal{C})$ by $\mathsf{BiMon}^{\mathrm{bicomm}}(\mathcal{C})$ and carrying out the second approach, one obtains a notion of a rig object in $\mathcal{C}$.
• The latter approach is the one developed in Part II of Goerss's Hopf Rings, Dieudonné Modules, and $E_*\Omega^2S^3$. See there for more details and arXiv:1804.10153 for the example of Hopf algebras and affine and formal abelian group schemes.
• When both approaches can be carried out, they agree.

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Examples

Examples of ring and rig objects in monoidal categories are the following.

• When $\mathcal{C}=\mathsf{Sets}$, one recovers rings and rigs, as $\mathsf{CCoMon}(\mathsf{Sets})\cong\mathsf{Sets}$, or alternatively since \begin{align*} \mathsf{HopfMon}^{\mathsf{bicomm}}(\mathsf{Sets}) &\cong \mathsf{Ab},\\ \mathsf{BiMon}^{\mathsf{bicomm}}(\mathsf{Sets}) &\cong \mathsf{CMon}, \end{align*} with $\boxtimes$ recovering the tensor product of abelian groups or commutative monoids.
• More generally, rings in Cartesian monoidal categories coincide with the usual notion of a ring object in a category with finite limits.
• Rings in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$ and $(\mathsf{Rings},\otimes_{\mathbb{Z}},\mathbb{Z})$ are plethories.
• A categorification of this approach, where one replaces monoids by pseudomonoids, recovers $2$-rigs and $2$-rings.

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Background

Let $(\mathcal{C},\otimes,\mathbf{1})$ be a monoidal category. A monoid in $\mathcal{C}$ consists of an object $A$ of $\mathcal{C}$ together with maps $\mu\colon A\otimes A\to A$ and $\eta\colon\mathbf{1}\to A$ making the diagrams

commute. For example, monoids in the Cartesian monoidal category of sets recover ordinary monoids, while monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$ recover (non-commutative) rings. Moreover, if $\mathcal{C}$ has a braided monoidal structure, we say that a monoid $(A,\mu,\eta)$ in $\mathcal{C}$ is commutative if the diagram

commutes. Again, this recovers commutative monoids and commutative rings when applied to $\mathsf{Sets}$ and $\mathsf{Ab}$.

Dually, a comonoid in $\mathcal{C}$ is a monoid in $\mathcal{C}^{\mathsf{op}}$: it is a triple $(C,\Delta,\epsilon)$ consisting of an object $C$ of $\mathcal{C}$ equipped with maps $\Delta\colon C\to C\otimes C$ and $\epsilon\colon C\to\mathbf{1}$ making the diagrams

commute. Cocommutative comonoids are defined dually to commutative monoids.

Any object of a Cartesian monoidal category is canonically a comonoid when equipped with the diagonal and projection to the unit maps. They are quite more interesting if the category in question is non-Cartesian, however: in $\mathsf{Mod}_{R}$, for instance, they give rise to $R$-coalgebras.

Now, we can also consider bimonoids in (a braided monoidal category) $\mathcal{C}$. These are objects of $\mathcal{C}$ equipped with both a monoid and a comonoid structure in a compatible way (image from the bialgebra page in Wikipedia):

A bimonoid in $\mathcal{C}$ is bicommutative if it is commutative and cocommutative.

A Hopf monoid in $\mathcal{C}$ is a bimonoid $H$ in $\mathcal{C}$ together with a morphism $\sigma\colon H\to H$, called the antipode of $H$, making the diagrams

commute. In a sense, Hopf monoids are bimonoids with inverses: bimonoids in $\mathsf{Sets}$ are monoids, but Hopf monoids in $\mathsf{Sets}$ are groups.

The classical examples of bimonoids and Hopf monoids are bialgebras and Hopf algebras.

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The question

What are some other examples of rings and rigs in monoidal categories, in particular non-Cartesian ones?

Answer 1

Below is a natural example of a ring object in the monoidal category of affine schemes. If $p$ is a prime and $R$ is a commutative ring, then the ring $W(R)$ of $p$-typical Witt vectors is also a commutative ring. The functor $W: CommRings \rightarrow CommRings$ is in fact co-representable by some commutative ring $\mathbb{W}$ which has two coproducts inducing the addition and multiplication, respectively, on $W(R)$ for each $R$. That is, the two coproducts give $hom_{CommRings}(\mathbb{W},R)$ a natural ring structure for all commutative rings $R$, and with that ring structure it is isomorphic to $W(R)$. Consequently, Spec of that co-representing ring is a ring object $Spec\ \mathbb{W}$ in the category of affine schemes.

There is a nice, but unpublished, note about this affine ring scheme by Paul Pearson from the mid-2000s, which has a few calculations of the structure maps.

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Another class of examples is provided by Hopf rings, which are ring objects in the category of cocommutative coalgebras. These things arise naturally in topology from studying unstable operations on generalized homology theories. The point is that, if you have a suitably nice spectrum $E$, then when you evaluate $E$ on the spaces in the $\Omega$-spectrum for $E$ and take the direct sum over all those spaces in the $\Omega$-spectrum, the resulting bi-graded gadget inherits a lot of structure from the structure of $E$: in fact, it picks up the structure of a Hopf ring. This is pretty well-documented in various resources on the Web, e.g. here: https://encyclopediaofmath.org/wiki/Hopf_ring

Answer 2

Maybe not an answer that gives examples directly, but a general observation that for any symmetric monoidal category $C$, the category of cocommutative comonoid objects $\mathsf{CCoMon}(C)$ is cartesian monoidal under the tensor product of $C$, and then one can define varieties of algebras in $\mathsf{CCoMon}(C)$, such as ring objects.

So for example, if we start with the category of vector spaces over some field, we obtain cocommutative coalgebras as the cocommutative comonoid objects. Abelian group objects in $\mathsf{CCoMon}(Vect)$ are the same as bicommutative Hopf algebras. One can internalize the construction of tensor products of abelian groups to abelian group objects in this category, but in the end, the resulting (commutative) $\boxtimes$-monoids gives (commutative) ring objects defined in the ordinary way for the cartesian monoidal $\mathsf{CCoMon}(Vect)$.

Also, it could be interesting to apply this observation to the symmetric monoidal category you get by applying the center construction (or Drinfeld center) once to some given monoidal category $C$ (to get a braided monoidal category) and once again to that braided monoidal category to get a symmetric monoidal category. I've not thought much about these.