Let $C$ be a cartesian category.
We can form the category $\text{Mon}(C)$ of monoid objects in $C$. If $C$ is $\text{Set}$, this category is the category of monoids. If $C$ is $\text{Mon}$ (the category of monoids, with product the monoidal operation), then I suspect $\text{Mon}(C)$ is the category of commutative monoids.
I am thinking about properties of $C$ which give corresponding properties of $\text{Mon}(C)$. Here are some questions:
- What are optimal conditions to put on $C$ to make coproducts distribute over products in $\text{Mon}(C)$?
For instance, I suspect that free monoid distributes over product of monoids, but I am not certain.
- What are optimal conditions to put on $C$ to make products isomorphic to coproducts in $\text{Mon}(C)$?
I call this condition linearity. I think the category of commutative monoids has this property.
My interest in this question lies in trying to get a better feel for how taking monoid objects (or commutative monoid objects, or group objects,or abelian group objects) changes the flavour of the category. So I would be greatly appreciative of any theorems in this regard.
Here are some similar examples:
Abelian group objects in $R$-alg / $S$ (monoidal category with products the monoidal operation) are $S$-modules. So we started with a category which was codistributive (and also extensive) and got a category which is linear (and in fact abelian).
Of course, rings are monoid objects in the monoidal linear category of abelian groups. Here we start with a monoidal linear category (but where the monoidal operation is not product this time) and get a codistributive category.
Group objects in the category of groups (a codistributive category, I think) are just abelian groups (linear category). This follows from the Eckmann-Hilton argument.
Abelian group objects and commutative monoid objects always form linear categories. Abelian group object categories are always enriched over $\text{Ab}$.