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)
• Related properties: equivalence, essentially injective, faithful
• nLab Link

Relevant implications

• conservative andfull implies essentially injective
• essentially surjective andfaithful andfull is equivalent to equivalence
• faithful andfull implies conservative

Examples

There are 4 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

Counterexamples

There are 23 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 rings to monoids
• free group functor
• functor of continuous functions
• group of units functor
• modulo p functor
• monoid ring functor
• p-torsion functor
• sequences functor on sets
• squaring functor on sets
• torsion functor

Unknown

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

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