[Ring objects](https://ncatlab.org/nlab/show/ring+object) 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](https://ncatlab.org/nlab/show/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 $E_*\Omega^2S^3$](https://sites.math.northwestern.edu/~pgoerss/papers/amshopf.pdf). See there for more details and [arXiv:1804.10153](https://arxiv.org/abs/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](https://ncatlab.org/nlab/show/plethory).
 - A categorification of this approach, where one replaces monoids by pseudomonoids, recovers $2$-rigs and $2$-rings.

---
Background
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Let $(\mathcal{C},\otimes,\mathbf{1})$ be a monoidal category. A [**monoid**](https://ncatlab.org/nlab/show/monoid+in+a+monoidal+category) 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

[![enter image description here][1]][1]

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**](https://ncatlab.org/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category) if the diagram

[![enter image description here][2]][2]

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

Dually, a [**comonoid**](https://ncatlab.org/nlab/show/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

[![enter image description here][3]][3]

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](https://en.wikipedia.org/wiki/Coalgebra).

Now, we can also consider [**bimonoids**](https://en.wikipedia.org/wiki/Bialgebra) 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](https://en.wikipedia.org/wiki/Bialgebra) in Wikipedia):

[![enter image description here][4]][4]

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

A [Hopf monoid](https://ncatlab.org/nlab/show/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

[![enter image description here][5]][5]

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](https://en.wikipedia.org/wiki/Bialgebra) and [Hopf algebras](https://en.wikipedia.org/wiki/Hopf_algebra).

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

  [1]: https://i.sstatic.net/NxTHI.png
  [2]: https://i.sstatic.net/z4rg3.png
  [3]: https://i.sstatic.net/VFEeM.png
  [4]: https://i.sstatic.net/75MGd.png
  [5]: https://i.sstatic.net/edBHr.png