What is the difference between being unique, unique up to isomorphism and unique up to unique isomorphism?

Could anyone explain the difference between the above mentioned terms "unique", "unique up to isomorphism" and "unique up to unique isomorphism", preferably with an example? Are there other characterisations of being unique "up to something"?

Solution 1:

All of these terms only make sense relative to some specification. A thing $X$ being unique relative to the specification means there does not exist any other thing satisfying the specification: if $Y$ satisfies the specification, then it follows that $Y=X$. Being unique is common for values or elements given by a specification, but it is almost never used for structures (such as groups, topological spaces, and such) for the simple reason that one can almost always make carbon copies of structures: change the identity (name, label) of all individual elements of the structure, and transfer all structure without change to the new set. Like defining $\Bbb N'=\{0',1',2',\ldots\}$ with all usual operations transported to the new element: $2'+2'=4'$, and such.

So for structures one needs a different kind of uniqueness, if it is to be useful at all. Most types of structures come with a notion of isomorphism; invariably if there is an isomorphism $X\to Y$ there is also an isomorphism $Y\to X$ (the inverse isomorphism), so admitting an isomorphism (being isomorphic) is a symmetric relation. It may happen that for some specification of such a structure, any two structures that satisfy the specification must be isomorphic: if $X$ and $Y$ satisfy the specification then there exists at least one isomorphism $X\to Y$; in this case the structure specified is unique up to isomorphism. In case the more strict requirement holds that under the stated conditions there exists exactly one isomorphism $X\to Y$, then the structure specified is unique up to unique isomorphism.

This is most often used in the setting of category theory, where the basic notion is that of homomorphism (an isomorphism then is a homomorphism that admits an inverse homomorphism, and the relation of being inverse homomorphisms is symmetric). The specifications that specify a structure up to isomorphism are often given in category theoretic terms; for instance a structure $Z$ is initial in its category if for every structure $X$ of the category there is a unique homomorphism $Z\to X$. For instance $\Bbb Z$ is initial in the category of (unital) rings, and a one-element group $\{e\}$ is initial in the category of groups. Initial structures do not exist in all categories, but if they exist they are unique up to isomorphism: if $Z_1$ and $Z_2$ are initial structures, then by definition there are unique homomorphisms $Z_1\to Z_2$ and $Z_2\to Z_1$, and these must be inverses of each other, and therefore isomorphisms. In fact by the "initial" property, these are the unique isomorphisms between $Z_1$ and $Z_2$, so initial structures, if they exist, are unique up to unique isomorphism.

Nonetheless, "unique up to unique isomorphism" is not a very useful notion. It just means the structure specified is unique up to isomorphism and has no non-trivial automorphism (in other words an isomorphism $X\to X$ has to be the identity homomorphism); if $X$ had a non-trivial automorphism, then that automorphism would defeat the requirement of a unique isomorphism. Many properties used to specify structures up to isomorphism say nothing about the existence of automorphisms, so they will usually not specify anything up to unique isomorphism. What is very useful however is specifying something up to canonical automorphism: if $X$ and $Y$ are structures satisfying the specification, then there is an isomorphism $X\to Y$, and in case there should be more than one such isomorphism, there is a way to single out one of them by a special property. In practice this special property is usually directly related to the specification that $X,Y$ have to satisfy, and the existence of a canonical isomorphism an immediate consequence of it.

It is not so easy to give a nice example of a structure specified so as to be unique up to canonical isomorphism, since I don't know which structures you are familiar with, and the easiest examples have an insipid category theoretic taste to them. Here is one possible specification of a polynomial ring $K[X]$ over a field$~K$. A commutative ring $R$ is a polynomial ring over$~K$ when it satisfies

• there exists a ring homomorphism $f:K\to R$ (interpreting scalars as constant polynomials; it allows in particular to view $R$ as a $K$-vector space)
• there exists an injective map $g:\Bbb N\to R$ (which we shall think of as $i\mapsto X^i$) whose image is a basis of $R$ as $K$-vector space
• one has $f(1)=g(0)$, and $g(i)g(j)=g(i+j)$ for all $i,j\in\Bbb N$.

(the maps $f,g$ should be considered to be part of giving $R$, so I should really have said $(R,f,g)$ is a polynomial ring over $K$ if ...)

Now suppose $(S,f',g')$ also satisfies these requirements (you may think of $S=K[Y]$ with $f'(a)=a\in K[Y]$ as constant polynomial, and $g'(i)=Y^i$). The basis property ensures there is a unique $K$-linear map $R\to S$ sending $g(i)\mapsto g'(i)$ for all $i\in\Bbb N$ (think of sending $X^i\mapsto Y^i$, and more generally $P[X]\to P[Y]$), and it is easily shown to be a ring isomorphism. This is the canonical isomorphism $R\to S$; it is however very far from being the unique isomorphism.

Solution 2:

Unique means that there is only one object satisfying a given definition. There is a unique set whose elements are precisely $1$ and $2$, namely $\{1, 2\}$.

Unique up to isomorphism means that all the objects satisfying a given definition are isomorphic, or have the same structure. Less formally, it means that they are the same object with different names for things. As an example, there is a unique cyclic group of order $3$, which could be realized as a subgroup of the permutation group on $3$ letters generated by $(1 2 3)$, or as the cyclic group of integers modulo 3.

Unique up to unique isomorphism means that there's only one isomorphism relating the two isomorphic objects. Returning to the previous example, there is not a unique isomorphism, since there are two distinct generators in the group - so there's more than one way to rename things.