Elementary epimorphisms were introduced by Philipp Rothmaler. A surjective homomorphism f: M –> N between two model-theoretic structures is an elementary epimorphism if and only if every formula with parameters satisfied by N is satisfied in M using a preimage of those parameters.

Philipp asked me whether nontrivial elementary epimorphisms between models of set theory exist. We answer this question in the negative for fully elementary epimorphisms between models of ZFC, but in the positive under weaker assumptions.

In particular, we show that every Pi_1-elementary epimorphism between models of ZF is an isomorphism. On the other hand, nonisomorphic Sigma_1-elementary epimorphisms between models of ZF can be constructed, as can fully elementary epimorphisms between models of ZFC^-.