Up to isomorphism

“The prop­erty \(P\) holds up to iso­mor­phism” is a phrase which means “we might say an ob­ject \(X\) has prop­erty \(P\), but that’s an abuse of no­ta­tion. When we say that, we re­ally mean that there is an ob­ject iso­mor­phic to \(X\) which has prop­erty \(P\)”. Essen­tially, it means “the prop­erty might not hold as stated, but if we re­place the idea of equal­ity by the idea of iso­mor­phism, then the prop­erty holds”.

Re­lat­edly, “The ob­ject \(X\) is well-defined up to iso­mor­phism” means “if we re­place \(X\) by an ob­ject iso­mor­phic to \(X\), we still ob­tain some­thing which satis­fies the defi­ni­tion of \(X\).”


Groups of or­der \(2\)

There is only one group of or­der \(2\) up to iso­mor­phism. We can define the ob­ject “group of or­der \(2\)” as “the group with two el­e­ments”; this ob­ject is well-defined up to iso­mor­phism, in that while there are sev­eral differ­ent groups of or­der \(2\) noteTwo such groups are \(\{0,1\}\) with the op­er­a­tion “ad­di­tion mod­ulo \(2\)”, and \(\{e, x \}\) with iden­tity el­e­ment \(e\) and the op­er­a­tion \(x^2 = e\)., any two such groups are iso­mor­phic. If we don’t think of iso­mor­phic ob­jects as be­ing “differ­ent”, then there is only one dis­tinct group of or­der \(2\).


  • Isomorphism

    A mor­phism be­tween two ob­jects which de­scribes how they are “es­sen­tially equiv­a­lent” for the pur­poses of the the­ory un­der con­sid­er­a­tion.