Definition:Underlying Set Functor/Category of Monoids

Definition

Let $\mathbf {Set}$ be the category of sets.

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

The underlying set functor $\size {\, \cdot \,}: \mathbf {Mon} \to \mathbf {Set}$ is the functor defined by:

			Object functor:		\(\ds \size {\struct {M, \circ} } := M \)
			Morphism functor:		\(\ds \size f := f \)

The underlying set functor thus comes down to deleting the information that $\struct {M, \circ}$ is a monoid, and that $f$ is a monoid homomorphism.

It is thus seen to be an example of a forgetful functor.

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.7$