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