# Isomorphism

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.

knows-req­ui­site(Group):

## 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 $$fg = \mathrm {id}_B$$.

Children:

• 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”.

Parents:

• Mathematics

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

• “iden­tity” is prob­a­bly not a suffi­ciently spe­cific link; I’d go for math_iden­tity, prob­a­bly.

• @267 Yeah I’ve been won­der­ing about the con­ven­tion of things like this. I’ve been call­ing my pages things like cat­e­gory_math­e­mat­ics.

• This is a great page! I think the in­tro/​sum­mary could be made a lit­tle more ac­cessible though? The use case I’m think­ing of is a per­son who wants a brief overview in rel­a­tively non-tech­ni­cal lan­guage, which is valuable for the pop­ups from links to here.

• @1yq Thank you very much! just to be clear, are you talk­ing about the ‘click­bait’, the in­tro para­graph in the text it­self, or both?

Feel free to sug­gest /​ make your own changes if you have any­thing spe­cific in mind by the way.

• The in­tro para­graph, the click­bait seems fine.

• I’d like to add some pic­tures to this page at some point, but due to cur­rent cir­cum­stances I can’t for now. If any­one wants to add pics (say differ­ent sta­tion maps with the same con­nec­tions, two ‘boxes’ with ran­dom items) please feel wel­come.

I also think I’ll change the names of the sta­tions from a, b etc. to funny made up sta­tion names.

The ma­jor­ity of this page will prob­a­bly end up in the least tech­ni­cal lens.