[summary(brief):A phrase mathematicians use when saying "we only care about the [structure_mathematics structure] of an [object_mathematics object], not about specific implementation details of the object".]

[summary: "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$~$."]

"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".

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 $~$2$~$ %%note: Two 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 $~$2$~$.