  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.
