Investigating the computable Friedman-Stanley jump

The Friedman-Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on ω, written , recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman-Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility In particular, we show that this jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from.