Essentially unique

In mathematics, the term essentially unique is used to indicate that while some object is not the only one that satisfies certain properties, all such objects are "the same" in some sense appropriate to the circumstances. This notion of "sameness" is often formalized using an equivalence relation.

A related notion is a universal property, where an object is not only essentially unique, but unique up to a unique isomorphism (meaning that it has trivial automorphism group). In general given two isomorphic examples of an essentially unique object, there is no natural (unique) isomorphism between them.

Examples

Set theory

Most basically, there is an essentially unique set of any given cardinality, whether one labels the elements $\{1,2,3\}$ or $\{a,b,c\}$ . In this case the non-uniqueness of the isomorphism (does one match 1 to a or to c?) is reflected in the symmetric group.

On the other hand, there is an essentially unique ordered set of any given finite cardinality: if one writes $\{1 < 2 < 3\}$ and $\{a< b< c\}$ , then the only order-preserving isomorphism maps 1 to a, 2 to b, and 3 to c.

Group theory

Suppose that we seek to classify all possible groups. We would find that there is an essentially unique group containing exactly 3 elements, the cyclic group of order three. No matter how we choose to write those three elements and denote the group operation, all such groups are isomorphic, hence, "the same".

On the other hand, there is not an essentially unique group with exactly 4 elements, as there are two non-isomorphic examples: the cyclic group of order 4 and the Klein four group.

Measure theory

Suppose that we seek a translation-invariant, strictly positive, locally finite measure on the real line. The solution to this problem is essentially unique: any such measure must be a constant multiple of Lebesgue measure. Specifying that the measure of the unit interval should be 1 then determines the solution uniquely.

Topology

Suppose that we seek to classify all two-dimensional, compact, simply connected manifolds. We would find an essentially unique solution to this problem: the 2-sphere. In this case, the solution is unique up to homeomorphism.

Lie theory

A maximal compact subgroup of a semisimple Lie group may not be unique, but is unique up to conjugation.