Definition:Representable Functor

Definition

Let $\mathbf C$ be a locally small category.

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

Let $F: \mathbf C \to \mathbf{Set}$ be a covariant functor.

Then $F$ is representable if and only if there exists an object $C \in \mathbf C$ such that $F$ is naturally isomorphic to the covariant hom functor $\map {\operatorname{Hom} } {C, \cdot}$.

That is, $F$ is representable if and only if $F$ has a representation.

Also see

• Uniqueness of Representing Objects
• Definition:Hom Functor
• Definition:Representation of Functor

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• Definition:Contravariant