isomorphism

Definition 0.1 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 $ f^{-1}: B \to A$ , such that $ f \circ f^{-1} =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.



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