A pair of math­e­mat­i­cal struc­tures are iso­mor­phic to each other if they are “es­sen­tially the same”, even if they aren’t nec­es­sar­ily equal.

An iso­mor­phism is a mor­phism be­tween iso­mor­phic struc­tures which trans­lates one to the other in a way that pre­serves all the rele­vant struc­ture. An im­por­tant prop­erty of an iso­mor­phism is that it can be ‘un­done’ by its in­verse iso­mor­phism.

An iso­mor­phism from an ob­ject to it­self is called an au­to­mor­phism. They can be thought of as sym­me­tries: differ­ent ways in which an ob­ject can be mapped onto it­self with­out chang­ing it.

Equal­ity and Identity

The sim­plest iso­mor­phism is equal­ity: if two things are equal then they are ac­tu­ally the same thing (and so not ac­tu­ally two things at all). Any­thing is ob­vi­ously in­dis­t­in­guish­able from it­self un­der what­ever mea­sure you might use (it has any prop­erty in com­mon with it­self) and so re­gard­less of the the­ory or lan­guage, any­thing is iso­mor­phic to it­self. This is rep­re­sented by the iden­tity (iso)mor­phism.


Group Isomorphisms

For a more tech­ni­cal ex­am­ple, the the­ory of groups only talks about the way that el­e­ments are com­bined via group op­er­a­tion. The the­ory does not care in what or­der el­e­ments are put, or what they are la­bel­led or even what they are. Hence, if you are us­ing the lan­guage and the­ory of groups, you want to say two groups are es­sen­tially in­dis­t­in­guish­able if you can pair up the el­e­ments such that their group op­er­a­tions act the same way. <div>

Iso­mor­phisms in Cat­e­gory Theory

In cat­e­gory the­ory, an iso­mor­phism is a mor­phism which has a two-sided in­verse func­tion. That is to say, \(f:A \to B\) is an iso­mor­phism if there is a mor­phism \(g: B \to A\) where \(f\) and \(g\) can­cel each other out.

For­mally, this means that both com­pos­ites \(fg\) and \(gf\) are equal to iden­tity mor­phisms (mor­phisms which ‘do noth­ing’ or de­clare an ob­ject equal to it­self). That is, \(gf = \mathrm {id}_A\) and \(fg = \mathrm {id}_B\).


  • Group isomorphism

    “Iso­mor­phism” is the proper no­tion of “same­ness” or “equal­ity” among groups.

  • Isomorphism: Intro (Math 0)

    Things which are ba­si­cally the same, ex­cept for some stuff you don’t care about.

  • Up to isomorphism

    A phrase math­e­mat­i­ci­ans use when say­ing “we only care about the struc­ture of an ob­ject, not about spe­cific im­ple­men­ta­tion de­tails of the ob­ject”.


  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.