isomorphism
A morphism^{} $f:A\u27f6B$ in a category is an isomorphism^{} if there exists a morphism ${f}^{1}:B\u27f6A$ 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 $\mathrm{Aut}(A)$.
Examples:

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In the category of sets and functions, a function $f:A\u27f6B$ is an isomorphism if and only if it is bijective.

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In the category of groups and group homomorphisms (or rings and ring homomorphisms), a homomorphism^{} $\varphi :G\u27f6H$ is an isomorphism if it has an inverse map ${\varphi}^{1}:H\u27f6G$ which is also a homomorphism.

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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.

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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  20130322 12:19:20 
Last modified on  20130322 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 