CatDat

representable

A functor $F : \mathcal{C} \to \mathcal{D}$ is representable if $\mathcal{C}$ is locally small, $\mathcal{D} = \mathbf{Set}$, and there is an object $A \in \mathcal{C}$ with $F \cong \mathrm{Hom}(A,-)$.

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Relevant implications

• representable implies continuous

*Those implications also require assumptions on the source or target category.

Examples

There are 3 functors with this property.

• contravariant power set functor
• forgetful functor for vector spaces
• identity functor on the category of sets

Counterexamples

There are 3 functors without this property.

• abelianization functor for groups
• covariant power set functor
• free group functor

Unknown

There are 0 functors for which the database has no information on whether they satisfy this property.

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