CatDat

faithful

A functor is faithful when it is injective on Hom-sets: If $F(f)=F(g)$, then $f=g$.

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

Relevant implications

• coequalizer-preserving andconservative implies faithful *
• comonadic implies conservative andfaithful andleft adjoint
• conservative andequalizer-preserving implies faithful *
• essentially surjective andfaithful andfull is equivalent to equivalence
• monadic implies conservative andfaithful andright adjoint

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

Examples

There are 5 functors with this property.

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

Counterexamples

There is 1 functor without this property.

• abelianization functor for groups

Unknown

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

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