“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 \(2\).