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 $$X$$.”

Examples

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 $$2$$.

Parents:

• Isomorphism

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