Definition:Underlying Set Functor

Definition

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

Let $\mathbf C$ be a metacategory with:

A collection of sets as objects;

A collection of mappings as morphisms.

The underlying set functor $\size {\, \cdot \,}: \mathbf C \to \mathbf {Set}$ is defined in the following cases.

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Category of Monoids

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:		\(\ds \size f := f \)

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