CatDat

essentially surjective

A functor $F : \C \to \D$ is essentially surjective when every object $Y \in \D$ is isomorphic to $F(X)$ for some $X \in \C$.

• Dual property: essentially surjective (self-dual)
• Related properties: equivalence, essentially injective
• nLab Link

Relevant implications

• essentially surjective andfaithful andfull is equivalent to equivalence

Examples

There are 7 functors with this property.

• abelianization functor for groups
• binary coproduct functor on sets
• binary product functor on sets
• enveloping group functor
• forgetful functor for topological spaces
• group of units functor
• identity functor on the category of sets

Counterexamples

There are 20 functors without this property.

• 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 vector spaces
• forgetful functor from abelian groups to groups
• forgetful functor from groups to monoids
• forgetful functor from rings to monoids
• free group functor
• functor of continuous functions
• 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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