PlanetPhysics/Isomorphism

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

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

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.