isomorphism

A morphism $f: A \longrightarrow B$ in a category is an isomorphism if there exists a morphism $f^{-1}: B \longrightarrow A$ which is its inverse. The objects $A$ and $B$ are isomorphic if there is an isomorphism between them.

A morphism which is both an isomorphism and an endomorphism is called an automorphism. The set of automorphisms of an object $A$ is denoted $\operatorname{Aut}(A)$ .

Examples:

• In the category of sets and functions, a function $f: A \longrightarrow B$ is an isomorphism if and only if it is bijective.
• In the category of groups and group homomorphisms (or rings and ring homomorphisms), a homomorphism $\phi: G \longrightarrow H$ is an isomorphism if it has an inverse map $\phi^{-1}: H \longrightarrow G$ which is also a homomorphism.
• In the category of vector spaces and linear transformations, a linear transformation is an isomorphism if and only if it is an invertible linear transformation.
• In the category of topological spaces and continuous maps, a continuous map is an isomorphism if and only if it is a homeomorphism.