PlanetMath: isomorphism

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

isomorphism

(Definition)

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 and 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 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.

"isomorphism" is owned by djao.

(view preamble)

Also defines:

isomorphic, automorphism

Attachments:: linear isomorphism (Definition) by matte

Log in to rate this entry.
(view current ratings)

Cross-references: homeomorphism, continuous maps, topological spaces, invertible linear transformation, linear transformations, vector spaces, map, ring homomorphisms, rings, group homomorphisms, groups, bijective, functions, category of sets, objects, inverse, category, morphism
There are 223 references to this entry.

This is version 3 of isomorphism, born on 2002-02-13, modified 2005-11-30.
Object id is 1936, canonical name is Isomorphism2.
Accessed 26504 times total.

Classification:

AMS MSC:	18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
	20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
	13A99 (Commutative rings and algebras :: General commutative ring theory :: Miscellaneous)
	15A04 (Linear and multilinear algebra; matrix theory :: Linear transformations, semilinear transformations)
	54A05 (General topology :: Generalities :: Topological spaces and generalizations )

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact