nLab
isomorphism

Context

Category theory

Equality and Equivalence

Contents

Idea

An isomorphism is an invertible morphism.

Two objects of a category are isomorphic if they are essentially equal without necessarily actually being equal. An isomorphism is a specific way of translating one object to an isomorphic one. Note that it's often not enough to know that two objects are isomorphic; you may need to know how they are isomorphic, that is to know the specific isomorphism in question.

Definitions

An isomorphism, or iso for short, is an invertible morphism, i.e. a morphism with a 2-sided inverse.

A morphism could be called isic (following the more common ‘monic’ and ‘epic’) if it is an isomorphism, but it's more common to simply call it invertible. Two objects xx and yy are isomorphic if there exists an isomorphism from xx to yy (or equivalently, from yy to xx). An automorphism is an isomorphism from one object to itself.

Properties

Note that the inverse of an isomorphism is an isomorphism, as is any identity morphism or composite of isomorphisms. Thus, being isomorphic is an equivalence relation on objects. The equivalence classes form the fundamental 0-groupoid? of the category in question.

Every isomorphism is both a split monomorphism (and thus about any other kind of monomorphism) and a split epimorphism (and thus about any other kind of epimorphism). In a balanced category, every morphism that is both monic and epic (called a bimorphism) is invertible, but this does not hold in general. However, any monic regular epimorphism (or dually, any epic regular monomorphism) must be an isomorphism.

A groupoid is precisely a category in which every morphism is an isomorphism. More generally, the core of any category CC is the subcategory consisting of all objects but only the isomorphisms; it is a kind of underlying groupoid of CC. In a similar way, the automorphisms of any given object xx form a group, the automorphism group of xx.

In higher categories, isomorphisms generalise to equivalences, which we expect to have only weak inverses.

Examples

Revised on September 17, 2012 23:52:11 by Urs Schreiber (82.169.65.155)