CatDat

preserves monomorphisms

A functor $F : \C \to \D$ preserves monomorphisms if every monomorphism in $\C$ is mapped to a monomorphism in $\D$.
This property is useful to rule out some adjunctions.

• Dual property: preserves epimorphisms
• Related properties: left exact
• nLab Link

Relevant implications

• left exact implies preserves monomorphisms
• preserves equalizers () implies preserves monomorphisms

Examples

There are 24 functors with this property.

• 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
• discrete topology functor
• doubling functor on sets
• forgetful functor for groups
• forgetful functor for rings
• forgetful functor for topological spaces
• 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
• group of units functor
• identity functor on the category of sets
• monoid ring functor
• p-torsion functor
• sequences functor on sets
• squaring functor on sets
• torsion functor

Counterexamples

There are 3 functors without this property.

• abelianization functor for groups
• enveloping group functor
• modulo p functor

Unknown

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

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