Logic Seminar

Please note that the time is PST

It is a folklore principle of locally countable Borel combinatorics that most arguments amount to doing countable combinatorics "in a uniform Borel way". We prove a result making this precise, by showing that there is a dual equivalence of categories between countable Borel equivalence relations (CBERs) and countable Lω1ωLω1ω theories admitting a (one-sorted) interpretation of the distinguished theory of Lusin–Novikov uniformizing functions together with a separating family of unary predicates. This allows all combinatorial concepts on CBERs to be characterized in terms of definability in countable structures, modulo said distinguished theory which describes structure available "for free" on every CBER. If time permits, I will also discuss a generalization of this correspondence to locally countable Borel groupoids. This is joint work with Rishi Banerjee.