CatDat

conservative

A functor $F : \mathcal{C} \to \mathcal{D}$ is conservative when it is isomorphic-reflecting: If $f$ is a morphism in $\mathcal{C}$ such that $F(f)$ is an isomorphism, then $f$ is an isomorphism.

• Dual property: conservative (self-dual)
• nLab Link

Relevant implications

• coequalizer-preserving andconservative implies faithful *
• comonadic implies conservative andfaithful andleft adjoint
• conservative andequalizer-preserving implies faithful *
• monadic implies conservative andfaithful andright adjoint

*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.

—