CatDat

conservative

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

• Dual property: conservative (self-dual)
• Related properties: essentially injective, full on isomorphisms, fully faithful
• nLab Link

Relevant implications

• comonadic implies conservative
• conservative andfull implies full on isomorphisms
• conservative andleft adjoint andpreserves coreflexive equalizers () implies comonadic
• conservative andpreserves coequalizers () implies faithful
• conservative andpreserves equalizers () implies faithful
• conservative andpreserves reflexive coequalizers andright adjoint () implies monadic
• fully faithful implies conservative
• left-invertible implies conservative
• monadic implies conservative

Examples

There are 23 functors with this property.

• binary coproduct functor on sets
• binary diagonal functor on the category of 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 vector spaces
• forgetful functor from abelian groups to groups
• forgetful functor from groups to monoids
• forgetful functor from groups to pointed sets
• forgetful functor from rings to monoids
• free group functor
• identity functor on the category of sets
• indiscrete topology functor
• monoid ring functor
• opposite category functor
• opposite monoid functor
• sequences functor on sets
• squaring functor on sets
• walking isomorphism object inclusion

Counterexamples

There are 13 functors without this property.

• abelianization functor for groups
• binary product functor on sets
• enveloping group functor
• forgetful functor for topological spaces
• functor of continuous functions
• fundamental group functor
• group of units functor
• modulo p functor
• p-torsion functor
• path components functor
• torsion functor
• trivial functor from the category of groups
• trivial functor from the category of sets

Unknown

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

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