isomorphism
A morphism f:A⟶B in a category is an isomorphism
if there exists a morphism f-1:B⟶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 Aut(A).
Examples:
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In the category of sets and functions, a function f:A⟶B 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
ϕ:G⟶H is an isomorphism if it has an inverse map ϕ-1:H⟶G 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 | 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 |