It can happen that two equivariant sets with atoms are related by a finitely supported bijection, but not by an equivariant bijection.
It is easy to see that, for equivariant sets with equality atoms and , a finitely supported function may exist even if an equivariant function from to does not. For example, let be a singleton and the set of atoms ; one can map the unique element to an atom, but one needs to choose the atom first.
It is less clear what happens if the finitely supported function is a bijection: does an equivariant bijection exist as well?
The answer is negative. For example, consider
choose an atom and define a function by:
- for .
It is easy to see that is a bijection supported by . However, no equivariant bijection (or even an equivariant function) from to exists, since contains an element with empty support and does not.
Interestingly though, if and are orbit-finite, then finitely supported bijections can be smoothed into equivariant ones. This was proved by Joanna, who in the near future will have written a post about it.