7.7. Monoids, Groups, in Symmetric Monoidal Categories

Recall from section ? that we were able to construct monoid and groups which were internal to some category $\mathcal{C}$. The philosophy behind the construction is one we've seen before: we of course think of monoids and groups by their elements, but we resist the temptation and instead present an object-free, diagrammatic set of axioms for monoids and rings. We utilized the cartesian product in the category $\mathcal{C}$ to demonstrate this. However, we now know that the cartesian product in any category is a small example of a category with a symmetric monoidal structure. Hence we revisit the concepts of a monoid and group, and expand their generality by demonstrating that they can be defined in a symmetric monoidal category.

Let $(\mathcal{M},\otimes ,I,\alpha ,\rho ,\lambda )$ be a monoidal category and let $M$ be an object of $\mathcal{M}$. We say $M$ is if there exist maps

$$\begin{array}{rl}\mu & :M\otimes M\to M\\ \eta & :I\to M\end{array}$$

referred to as the multiplication and identity maps, such that the diagrams below commute.

One of the most useful examples of this concept arises from the notion of an algebra $A$ over some field $k$, where $A$ is a vector space over the field $k$.