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 $Y \hookrightarrow F(X)$.

Relevant implications

• dominant () implies essentially surjective
• essentially surjective implies dominant

Examples

There are 24 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
• morphism endpoints inclusion
• opposite category functor
• opposite monoid functor
• path components functor
• rational product functor
• sequences functor on sets
• squaring functor on sets
• Stone-Čech compactification 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
• walking isomorphism object inclusion

Counterexamples

There are 32 functors without this property.

• binary diagonal functor on the category of sets
• Brauer group functor
• contravariant power set functor
• covariant power set functor
• discrete topology functor
• empty functor to the category of sets
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for vector spaces
• 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 rings to monoids
• forgetful functor from torsion abelian groups to abelian groups
• forgetful functor from torsion-free abelian groups to abelian groups
• free group functor
• functor of continuous functions
• inclusion functor from extended natural numbers to ordinal numbers
• indiscrete topology functor
• modulo p functor
• monoid ring functor
• nerve functor
• p-torsion functor
• ring idempotents functor
• simple-group probing functor
• span endpoints inclusion
• torsion functor
• walking morphism representation

Unknown

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

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