# Up to isomorphism

https://arbital.com/p/up_to_isomorphism

by Patrick Stevens Sep 24 2016 updated Sep 24 2016

A phrase mathematicians use when saying "we only care about the structure of an object, not about specific implementation details of the object".

[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".

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 $2$ %%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 $2$.