Consider the following notion of a [ring object](https://ncatlab.org/nlab/show/ring+object), which works also for (some) non-Cartesian monoidal categories:
> Let $(\mathcal{C},\otimes,\mathbf{1})$ be a monoidal category having Sweedler $\mathrm{Hom}$s and Sweedler products (for instance, we can take $\mathcal{C}$ to be a locally presentable braided monoidal category; see [arXiv:1509.07632, Theorem 4.1](https://arxiv.org/abs/1509.07632)).
>
> Taking categories of bicommutative Hopf monoids gives us a category
> $\mathsf{HopfMon}^{\mathrm{bicomm}}(\mathcal{C})$, which by assumption
> admits a Sweedler product $\boxtimes$ making it into a monoidal
> category.
>
> Then, define a (**commutative**) **ring object** in $\mathcal{C}$ to be a (commutative) monoid in $(\mathsf{HopfMon}^{\mathrm{bicomm}}(\mathcal{C}),\boxtimes,1)$.
Examples:
- This notion recovers (commutative) rings when applied to $\mathcal{C}=\mathsf{Sets}$, as $\mathsf{BiHopf}^{\mathsf{bicomm}}(\mathsf{Sets})\cong\mathsf{Ab}$ and the Sweedler product in $\mathsf{Ab}$ is the tensor product of abelian groups.
- 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).
**Question.** What are some other examples of rings in monoidal categories? Has this non-Cartesian variant been studied before?
---
**Edit:** Here's some background on Hopf monoids in monoidal categories, as requested in the comments by Paul Taylor.
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).
Lastly, here are some comments on Sweedler theory. Given a comonoid $C$ and a monoid $A$ in a _closed_ monoidal category $\mathcal{C}$, one can form a new monoid $[C,A]$, called the [**convolution monoid**](https://ncatlab.org/nlab/show/convolution+algebra#ConvolutionOfMapsFromACoalgebraToAnAlgebra) of $C$ and $A$. When $\mathcal{C}=\mathsf{Sets}$, this means we can endow a set of the form $\mathrm{Hom}_{\mathsf{Sets}}(X,M)$ with $X$ a set and $M$ a monoid with a monoid structure consisting of
- The multiplication map sending a pair $f,g\colon X\rightrightarrows M$ of maps of sets to the map $f*g$ defined by
$$
(f\ast g)(x)\overset{\mathrm{def}}{=}f(x)g(x)
$$
- The unit map $\Delta_{1_M}$ defined by
$$
\Delta_{1_{M}}(x)
\overset{\mathrm{def}}{=}
1_{M}.
$$
Taking convolution monoids in $\mathcal{C}$ defines a functor
$$
[-_{1},-_{2}]
\colon
\mathsf{CoMon}(\mathcal{C})^{\mathsf{op}}
\times
\mathsf{Mon}(\mathcal{C})
\longrightarrow
\mathsf{Mon}(\mathcal{C}).
$$
When $\mathcal{C}$ is sufficiently nice, this functor becomes part of a [two-variable adjunction](https://ncatlab.org/nlab/show/two-variable+adjunction) involving two new functors
$$
\begin{align*}
-_{1}\triangleright-_{2} &\colon \mathsf{CoMon}(\mathcal{C})\times\mathsf{Mon}(\mathcal{C}) \longrightarrow \mathsf{Mon}(\mathcal{C}),\\
\{-_{1},-_{2}\} &\colon \mathsf{Mon}(\mathcal{C})^{\mathsf{op}}\times\mathsf{Mon}(\mathcal{C}) \longrightarrow \mathsf{CoMon}(\mathcal{C}),
\end{align*}
$$
called the Sweedler product and the [Sweedler $\mathrm{Hom}$](https://ncatlab.org/nlab/show/measuring+coalgebra) (or the [measuring comonoid](https://en.wikipedia.org/wiki/Measuring_coalgebra)), respectively.
These are harder to describe, but for example the Sweedler $\mathrm{Hom}$ in $\mathsf{Sets}$ is given by the functor
$$
\mathrm{Hom}_{\mathsf{Mon}}
\colon
\mathsf{Mon}^{\mathsf{op}}\times\mathsf{Mon}
\to
\mathsf{Sets}
$$
taking a pair of monoids $A$ and $B$ to the set $\mathrm{Hom}_{\mathsf{Mon}}(A,B)$ of monoid maps from $A$ to $B$.
When $\mathcal{C}=\mathsf{Mod}_{R}$, Sweedler $\mathrm{Hom}$s are given by [measuring $R$-coalgebras](https://en.wikipedia.org/wiki/Measuring_coalgebra). These make sense also in the ($\infty$-)category of spectra; see [arXiv:2006.09408](https://arxiv.org/abs/2006.09408) and [Péroux's PhD thesis](https://indigo.uic.edu/articles/thesis/Highly_Structured_Coalgebras_and_Comodules/13475667/1).
A fundamental example of the Sweedler product is given when $\mathcal{C}$ is the category of bicommutative Hopf monoids in $\mathsf{Sets}$, i.e. the category of abelian groups, where we recover the tensor product of abelian groups.
Sweedler theory has also been carefully worked out by Anel and Joyal in [arXiv:1309.6952](https://arxiv.org/abs/1309.6952) in the setting of dg-co/algebras.
**TL;DR.** A comonoid in a monoidal category $\mathcal{C}$ is the dual notion of a monoid in $\mathcal{C}$. A bimonoid in $\mathcal{C}$ is a monoid and a comonoid in $\mathcal{C}$ in a compatible way. A Hopf monoid in $\mathcal{C}$ is roughly a bimonoid in $\mathcal{C}$ with inverses. We can convolve a comonoid $C$ with a monoid $A$, giving a kind of Hom set of maps from $C$ to $A$. This convolution monoid admits adjoints in nice situations, allowing us to tensor monoids with comonoids and to enrich $\mathsf{Mon}(\mathcal{C})$ in $\mathsf{CoMon}(\mathcal{C})$.
[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