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.

Relevant implications

• essentially surjective andfully faithful is equivalent to equivalence
• faithful andfull is equivalent to fully faithful
• fully faithful andleft adjoint implies comonadic andleft-invertible
• fully faithful andright adjoint implies left-invertible andmonadic
• fully faithful implies conservative

Examples

There are 20 functors with this property.

• discrete topology functor
• empty functor to the category of sets
• forgetful functor from abelian groups to groups
• forgetful functor from commutative rings to rings
• forgetful functor from finite abelian groups to abelian groups
• forgetful functor from finite groups to groups
• forgetful functor from finite sets to sets
• forgetful functor from groups to monoids
• forgetful functor from Hausdorff spaces to topological spaces
• forgetful functor from torsion abelian groups to abelian groups
• forgetful functor from torsion-free abelian groups to abelian groups
• identity functor on the category of sets
• inclusion functor from extended natural numbers to ordinal numbers
• indiscrete topology functor
• nerve functor
• opposite category functor
• opposite monoid functor
• span endpoints inclusion
• walking isomorphism object inclusion
• walking morphism representation

Counterexamples

There are 36 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
• Brauer group functor
• 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
• morphism endpoints inclusion
• p-torsion functor
• path components functor
• rational product functor
• ring idempotents functor
• sequences functor on sets
• simple-group probing functor
• squaring functor on sets
• Stone-Čech compactification functor
• torsion functor
• trivial functor from the category of groups
• trivial functor from the category of sets
• trivial functor from the delooping
• trivial functor from the walking idempotent

Unknown

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

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