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isomorphism


A morphismMathworldPlanetmath f:AB in a category is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if there exists a morphism f-1:BA which is its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. 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 Aut(A).

Examples:

  • In the category of sets and functions, a function f:AB is an isomorphism if and only if it is bijective.

  • In the category of groups and group homomorphisms (or rings and ring homomorphisms), a homomorphismMathworldPlanetmathPlanetmath ϕ:GH is an isomorphism if it has an inverse map ϕ-1:HG 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