The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological spaces of finite dimension, and well-ordered sets.

However, it fails in other categories: topological spaces, groups, rings, fields, graphs, posets, etc.

Can we caracterize Cantor-Bernsteiness in terms of other categorical properties?

[Edit: Corrected misspelling of Bernstein]

Whenever the objects in your category can be classified by a bounded collection of cardinal invariants, then you should expect to have the Schroeder-Bernstein property.

For example, vector spaces (over some fixed field $K$) or algebraically closed fields (of some fixed characteristic) can each be classified by a single cardinal invariant: the dimension of the vector space, or the transcendence degree of the field.

More interesting example: countable abelian torsion groups.  Suppose A and B are two such groups, $A$ is a direct summand of $B$, and vice-versa; are they isomorphic?  By Ulm's Theorem, $A$ and $B$ are determined up to isomorphism by countable sequences of cardinal numbers -- namely, the number of summands of $\mathbb{Z}_p^\infty$ and the "Ulm invariants," which are dimensions of some vector spaces associated with $A$ and $B$.  All of these invariants behave nicely with respect to direct sum decompositions, so it follows that $A$ and $B$ are isomorphic.  (See Kaplansky's Infinite Abelian Groups for a very nice, and elementary, proof of all this.)

If you like model theory, I could tell you a lot about when the categories of models of a complete theory have the Schroeder-Bernstein property (under elementary embeddings).  If not, at least I can tell you this:


Categories of structures with "definable" partial orderings with infinite chains (e.g. real-closed fields, atomless Boolean algebras) will NOT have the S-B property.  Again, I need some model theory to make this statement precise...
Let $C$ be a first-order axiomatizable class of structures (in a countable language) which is "categorical in $2^{\aleph_0}$" -- i.e. any two structures in $C$ of size continuum are isomorphic.  Then $C$ has the S-B property with respect to elementary embeddings.  (This generalizes the cases of vector spaces and algebraically closed fields.)


Addendum: A completely different way that a category $C$ might be Schroeder-Bernstein is if every object is "surjunctive" (i.e. any injective self-morphism of an object is necessarily surjective).  This covers Justin's example of the category of well-orderings.
