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isomorphism (Definition)

Definition 0.1 A morphism $f: A \to B$ in a category $C$ is an isomorphism when there exists an inverse morphism of $f$ in $C$, denoted by $f^{-1}: B \to A$, such that $f \circ f^{-1} =id_A = 1_A: A \to A$.

One also writes: $A \cong B$, expressing the fact that the object A is isomorphic with object B under the isomorphism $f$.

Note also that an isomorphism is both a monomorphism and an epimorphism; moreover, an isomorphism is both a section and a retraction. However, an isomorphism is not the same as an equivalence relation.



"isomorphism" is owned by bci1.
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Also defines:  isomorphism, inverse morphism, section, retraction, mobomorphism, epimorphism, isomorphic objects under an isomorphism
Keywords:  isomorphism, mono and epimorhism, analogous systems, isomorphic objects, inverse morphism, identity morphism

Cross-references: equivalence relation, monomorphism, object, category, morphism
There are 105 references to this object.

This is version 5 of isomorphism, born on 2010-11-01, modified 2010-11-01.
Object id is 889, canonical name is Isomorphism.
Accessed 2555 times total.

Classification:
Physics Classification: 00. (GENERAL)
Pending Errata and Addenda
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