monomorphism-preserving

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

Dual property: epimorphism-preserving
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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.

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