# 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.