Commutative Monoid Object

Let $(\mathcal{V},\otimes,I)$ be a Braided Monoidal Category with braiding $\beta$, and $(M,\eta,\mu)$ be a Monoid Object in $\mathcal{V}$. $M$ is a commutative monoid object if $\mu$ commutes with the braiding.

In most cases, $\mathcal{V}$ is a Symmetric Monoidal Category.

Examples

• A commutative monoid object in $(\mathrm{Sets}{.},\times,1)$ is simply a Commutative Monoid.
• A commutative monoid object in $(\mathrm{CMon}{.},\otimes,\mathbb{N})$ is a Commutative Semiring.
• A commutative monoid object in $(\mathrm{Ab}{.},\otimes,\mathbb{Z})$ is a Commutative Ring.

References

• https://ncatlab.org/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category