Some years ago, Louveau and Rosendal showed that the relation of
bi-embeddability for countable graphs is complete for analytic equivalence
relations. This is in strong contrast to the case of the isomorphism relation,
which as an equivalence relation on graphs (or on any class of countable
structures consisting of the models of an L_{omega_1,omega}-sentence) is
far from complete. In this talk I will present a strenghtening of the
Louveau-Rosendal result by showing that not only does bi-embeddability give
rise to analytic equivalence relations which are complete under Borel
reducibility, but in fact any analytic equivalence relation is Borel equivalent
to such a relation. This proves that, quite surprisingly, the logic relation of
bi-embeddability is able to capture the great complexity of the whole structure
of analytic equivalence relations (up to Borel-equivalence). Similar results
can also be obtained looking at various notions of morphism naturally arising
e.g. in Model Theory and Descriptive Set Theory. Moreover, the techniques
introduced allow to answer to some questions posed by Louveau and Rosendal
about the possible relationships between isomorphism and bi-embeddability.
This is partially joint work with Sy-David Friedman and Riccardo Camerlo.