Elementary embeddings, elementary substructures,category of sets I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.
 A: Since a set is a structure in the empty language - there's no additional structure besides the underlying carrier set itself - things are quite simple to describe.
Specifically, we have full quantifier elimination, and the quantifier-free formulas can basically only say finitely many facts about size. This analysis - which I've left a bit vague, since it's a good exercise - shows the following:


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*If $A,B$ are infinite sets, then the elementary embeddings from $A$ to $B$ are precisely the injections.

*If $A,B$ are finite sets, then the elementary embeddings from $A$ to $B$ are precisely the bijections.
Since an elementary substructure is just a substructure for which the inclusion map is an elementary embedding, this also characterizes the elementary substructures: $A$ is an elementary substructure of $B$ iff $A\subseteq B$ and either $A=B$ or $A$ (and hence $B$ as well) is infinite.
Finally, note that we also get a characterization of elementary equivalence: $A\equiv B$ iff $A,B$ either have the same cardinality or are each infinite. In particular, two sets are elementarily equivalent iff there is an elementary embedding of one of them into the other (since cardinalities are comparable - note that this uses a weak form of the axiom of choice!).
So what we see in the "pure set situation" is a near-complete collapse of logical notions: elementary equivalence/embedding/substructure mean almost the bare minimum of what they have to mean. The situation is far more interesting with even a little bit of structure involved, and this situation should really be thought of as pathological.
