CatDat

isomorphism

A functor $F : \C \to \D$ is an isomorphism when there is a functor $G : \D \to \C$ with $F \circ G = \id_{\D}$ and $G \circ F = \id_{\C}$. In contrast to an equivalence of categories, these compositions are required to be equal to the identities. Warning: This property is not invariant under equivalences.

• Dual property: isomorphism (self-dual)
• Related properties: equivalence
• nLab Link

Relevant implications

• isomorphism implies equivalence

Examples

There are 3 functors with this property.

• identity functor on the category of sets
• opposite category functor
• opposite monoid functor

Counterexamples

There are 33 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
• walking isomorphism object inclusion

Unknown

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

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