monomorphism-preserving

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: epimorphism-preserving
nLab Link

Relevant implications

left exact implies monomorphism-preserving

*Those implications also require assumptions on the source or target category.

Examples

There are 5 functors with this property.

contravariant power set functor
covariant power set functor
forgetful functor for vector spaces
free group functor
identity functor on the category of sets

Counterexamples

There is 1 functor without this property.

abelianization functor for groups

Unknown

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

—