CatDat

fully faithful

A functor $F : \C \to \D$ is fully faithful when it is full and faithful. This means that for all objects $A,B \in \C$ the map $\Hom(A,B) \to \Hom(F(A),F(B))$, $f \mapsto F(f)$ is a bijection.

• Dual property: fully faithful (self-dual)
• Related properties: equivalence, faithful, full, pseudomonic
• nLab Link

Relevant implications

• essentially surjective andfully faithful is equivalent to equivalence
• faithful andfull is equivalent to fully faithful
• fully faithful implies conservative

Examples

There are 8 functors with this property.

• discrete topology functor
• forgetful functor from abelian groups to groups
• forgetful functor from groups to monoids
• identity functor on the category of sets
• indiscrete topology functor
• opposite category functor
• opposite monoid functor
• walking isomorphism object inclusion

Counterexamples

There are 28 functors without this property.

• abelianization functor for groups
• binary coproduct functor on sets
• binary diagonal functor on the category of sets
• binary product functor on sets
• contravariant power set functor
• countable copower functor on sets
• covariant power set functor
• doubling functor on sets
• enveloping group functor
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for topological spaces
• forgetful functor for vector spaces
• forgetful functor from groups to pointed sets
• forgetful functor from rings to monoids
• free group functor
• functor of continuous functions
• fundamental group functor
• group of units functor
• modulo p functor
• monoid ring functor
• p-torsion functor
• path components functor
• sequences functor on sets
• squaring functor on sets
• torsion functor
• trivial functor from the category of groups
• trivial functor from the category of sets

Unknown

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

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