"The property 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 $2$ ^[1], 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 $2$ .

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Two such groups are $0, 1$ with the operation "addition modulo $2$ ", and $e, x$ with identity element $e$ and the operation $x^{2} = e$ .

Up to isomorphism