# PlanetPhysics/Isomorphism

**Definition 0.1**
\bigbreak
A morphism in a category ** is an ***isomorphism* when there exists an *inverse morphism* of in ** , 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: , expressing the fact that the object A is isomorphic with object B under the isomorphism .

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*.