Unique Constant in Category of Monoids

Theorem

Let $\mathbf{Mon}$ be the category of monoids.

Then every object $M$ of $\mathbf{Mon}$ has precisely one constant.

Corollary

The category of monoids $\mathbf{Mon}$ does not have enough constants.

Work In Progress In particular: Subpage, put up the trivial proof You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code.

Proof

From Trivial Monoid is Terminal Element, we obtain that a constant of $M$ is a morphism $f: \left\{{e}\right\} \to M$.

By Trivial Monoid is Initial Element, there is precisely one such morphism.

Hence $M$ has one constant.

$\blacksquare$

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.3$: Example $2.12$: $2$