PlanetPhysics/Isomorphism

Definition 0.1 \bigbreak A morphism ${\displaystyle f:A\to B}$ in a category ${\displaystyle C}$ is an isomorphism when there exists an inverse morphism of ${\displaystyle f}$ in ${\displaystyle C}$ , denoted by $\displaystyle \inv f: B \to A$ , such that $\displaystyle f \circ \inv f =id_A = 1_A: A \to A$ .

One also writes: ${\displaystyle A\cong B}$, expressing the fact that the object A is isomorphic with object B under the isomorphism ${\displaystyle f}$.

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