CatDat

equivalence

A functor $F : \C \to \D$ is an equivalence when there is a functor $G : \D \to \C$ with $F \circ G \cong \id_{\D}$ and $G \circ F \cong \id_{\C}$. In contrast to an isomorphism of categories, these compositions are only required to be isomorphic to the identities.

• Dual property: equivalence (self-dual)
• Related properties: essentially surjective, faithful, full, isomorphism, left-invertible, right-invertible
• nLab Link

Relevant implications

• equivalence implies comonadic andcoreflector
• equivalence is equivalent to left-invertible andright-invertible
• equivalence implies monadic andreflector
• essentially surjective andfully faithful is equivalent to equivalence
• isomorphism implies equivalence

Examples

There are 4 functors with this property.

• identity functor on the category of sets
• opposite category functor
• opposite monoid functor
• walking isomorphism object inclusion

Counterexamples

There are 32 functors without this property.

• abelianization functor for groups
• 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
• enveloping group functor
• 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 groups to pointed sets
• forgetful functor from rings to monoids
• free group functor
• functor of continuous functions
• fundamental group functor
• group of units functor
• indiscrete topology functor
• modulo p functor
• monoid ring functor
• p-torsion functor
• path components functor
• sequences functor on sets
• squaring functor on sets
• 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.

—