CatDat

full

A functor is full when it is surjective on Hom-sets: Every morphism $F(A) \to F(B)$ is induced by a morphism $A \to B$.

• Dual property: full (self-dual)
• nLab Link

Relevant implications

• comonadic andfull implies equivalence
• essentially surjective andfaithful andfull is equivalent to equivalence
• full andmonadic implies equivalence

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

Examples

There is 1 functor with this property.

• identity functor on the category of sets

Counterexamples

There are 5 functors without this property.

• abelianization functor for groups
• 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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