CatDat

essentially surjective

A functor $F : \mathcal{C} \to \mathcal{D}$ is essentially surjective when every object $Y \in \mathcal{D}$ is isomorphic to $F(X)$ for some $X \in \mathcal{C}$.

• Dual property: essentially surjective (self-dual)
• nLab Link

Relevant implications

• essentially surjective andfaithful andfull is equivalent to equivalence

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

Examples

There are 2 functors with this property.

• abelianization functor for groups
• identity functor on the category of sets

Counterexamples

There are 4 functors without this property.

• contravariant power set functor
• covariant power set functor
• forgetful functor for vector spaces
• 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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