Recall that the Schroeder-Bernstein theorem states that given two sets X and Y and injections f: X->Y and g:Y->X there exists a bijection h: X->Y. Probably most of our readers have proved this result at some point using the nifty ladder argument. If you haven’t seen the proof it’s covered nicely over on wikipedia (I find the “other proof” easier to follow).
My freshman year of college several of my classmates (me, Jared Weinstein, Haiwen Chu, and Mike Hill are the ones I remember) played a game of proving or disproving Schroeder-Bernstein in other categories. For example, if you have two vector spaces with linear injections both ways are they automatically isomorphic? If you have two groups with group injections both ways are they automatically isomorphic? (I don’t want to spoil your fun so I’ll put the answers to these two easy questions in comments.)
This is still a game I like to play when I run accross a new category to test my understanding. Plus it’s fun. Here are some more categories you might try (with *’s by the most interesting ones):
- Abelian groups
- Topological spaces
- *Finite topological spaces
- *Vector spaces without the axiom of choice
- *Free modules over a noncommutative ring
Put proofs/counterexamples/other suggestions in comments!