CatDat

faithful

A functor is faithful when it is injective on Hom-sets: If $F(f)=F(g)$ for two morphisms $f,g : A \rightrightarrows B$, then $f=g$.

• Dual property: faithful (self-dual)
• Related properties: equivalence, essentially injective, full, left-invertible
• nLab Link

Relevant implications

• comonadic implies faithful
• conservative andpreserves coequalizers () implies faithful
• conservative andpreserves equalizers () implies faithful
• essentially surjective andfaithful andfull is equivalent to equivalence
• faithful andfull implies conservative
• left-invertible implies faithful
• monadic implies faithful

Examples

There are 19 functors with this property.

• binary coproduct functor on sets
• binary diagonal functor on the category of sets
• contravariant power set functor
• countable copower functor on sets
• covariant power set functor
• discrete topology functor
• doubling functor on sets
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for topological spaces
• forgetful functor for vector spaces
• forgetful functor from abelian groups to groups
• forgetful functor from groups to monoids
• forgetful functor from rings to monoids
• free group functor
• identity functor on the category of sets
• monoid ring functor
• sequences functor on sets
• squaring functor on sets

Counterexamples

There are 8 functors without this property.

• abelianization functor for groups
• binary product functor on sets
• enveloping group functor
• functor of continuous functions
• group of units functor
• modulo p functor
• p-torsion functor
• torsion functor

Unknown

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

—