Effective categoricity of equivalence Structures
  An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure B isomorphic to A there is a $\Delta^0_\alpha$ function witnessing that the two are isomorphic.   The present paper gives an exact characterization of $\Delta^0_\alpha$ equivalence structures where $\alpha = 1$ or $\alpha \geq 3$. Extensive results for $\alpha = 2$ are also given, and open cases are exhaustively described. 
W. Calvert, D. Cenzer, V. S. Harizanov, and A. Morozov