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.

## 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.

## 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.

## The question

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