Definition:Representable Functor
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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
- Definition:Contravariant
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