All possible total orderings of a finite set are isomorphic. I find these kinds of results remarkable. Here's a few more.
Assume that $S$ is a finite set. Then:
- All possible field structures on $S$ are isomorphic (and there exists a field structure on $S$ iff the cardinality of $S$ is $p^n$ for some prime $p$ and positive integer $n$.)
- All possible Boolean algebra structures on $S$ are isomorphic (and there exists a Boolean algebra structure on $S$ iff the cardinality of $S$ is $2^n$ for some natural number $n$.)
- All possible cyclic group structures on $S$ are isomorphic.
- If the cardinality of $S$ is prime, then all possible ways of making $S$ into a group yield isomorphic groups.
- All possible ways of making $S$ into a cycle (in the sense of graph theory) are isomorphic.
Question. Not necessarily assuming that $S$ is a set (e.g. it can be an abelian group, or a ring, or whatever) what are some other examples of this phenomenon?
We can make things a little more precise using the language of categories.
Given a functor $U : \mathbf{S} \leftarrow \mathbf{C}$ and an object $S$ of $\mathbf{S}$, define that:
- $U^{-1}(S)$ is the full subcategory of $\mathbf{C}$ consisting of precisely those objects whose image under $U$ is $S$.
- $U^{-1}(\mathrm{id}_S)$ is the wide subcategory of $U^{-1}(S)$ consisting of precisely those morphisms of $U^{-1}(S)$ whose image under $U$ is $\mathrm{id}_S$.
We're interested in results of the form: the category $U^{-1}(\mathrm{id}_S)$ has multiple non-isomorphic objects; nonetheless, all objects of $U^{-1}(S)$ are isomorphic.