PlanetPhysics/Isomorphism

Definition 0.1 \bigbreak A [[../TrivialGroupoid/|morphism]] $f:A\to B$ in a [[../Cod/|category]] $C$ is an isomorphism when there exists an inverse morphism of $f$ in $C$ , denoted by Failed to parse (unknown function "\inv"): {\displaystyle \inv f: B \to A} , such that Failed to parse (unknown function "\inv"): {\displaystyle f \circ \inv f =id_A = 1_A: A \to A} .

One also writes: $A\cong B$, expressing the fact that the [[../TrivialGroupoid/|object]] A is isomorphic with object B under the isomorphism $f$.

Note also that an isomorphism is both a [[../InjectiveMap/|monomorphism]] and an epimorphism; moreover, an isomorphism is both a section and a retraction. However, an isomorphism is not the same as an [[../TrivialGroupoid/|equivalence relation]].