# 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.

 Title isomorphism Canonical name Isomorphism Date of creation 2013-03-22 12:19:20 Last modified on 2013-03-22 12:19:20 Owner djao (24) Last modified by djao (24) Numerical id 6 Author djao (24) Entry type Definition Classification msc 54A05 Classification msc 15A04 Classification msc 13A99 Classification msc 20A05 Classification msc 18A05 Defines isomorphic Defines automorphism