CatDat

dominant

A functor $F : \C \to \D$ is dominant when every object $Y \in \D$ there is an object $X \in \C$ such that $Y$ is a retract of $F(X)$, i.e. there is a split monomorphism $Y \hookrightarrow F(X)$.

• Dual property: dominant (self-dual)
• Related properties: essentially surjective
• nLab Link

Relevant implications

• essentially surjective implies dominant

Examples

There are 19 functors with this property.

• abelianization functor for groups
• binary coproduct functor on sets
• binary product functor on sets
• countable copower functor on sets
• doubling functor on sets
• enveloping group functor
• forgetful functor for topological spaces
• forgetful functor from groups to pointed sets
• fundamental group functor
• group of units functor
• identity functor on the category of sets
• opposite category functor
• opposite monoid functor
• path components functor
• sequences functor on sets
• squaring functor on sets
• trivial functor from the category of groups
• trivial functor from the category of sets
• walking isomorphism object inclusion

Counterexamples

There are 17 functors without this property.

• binary diagonal functor on the category of sets
• contravariant power set functor
• covariant power set functor
• discrete topology functor
• 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
• indiscrete topology functor
• modulo p functor
• monoid ring functor
• p-torsion functor
• torsion functor

Unknown

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

—