Group of Units of a Monoid

Let $M$ be a Monoid. The group of units of $M$ consists of all Invertible Elements of $M$.

Universal Property

This construction extends to a functor $\mathrm{Units}:\mathrm{Mon}\to\mathrm{Grp}$ that is Right Adjoint to the forgetful functor $U:\mathrm{Grp}\to\mathrm{Mon}$.

As a Decategorification

This is a decategorification of the Core of a Category.