A computable structure C is $\Delta^0_2$-categorical if for every computable structure B isomorphic to C there exists a 0'-computable isomorphism f: B --> C. There are also several uniform versions of this notion such as effective, uniform, and relative $\Delta^0_2$-categoricity (to be defined).
In this talk we address the following problem:

    Given a class K of computable structures, characterize all members of K which are (effectively\uniformly\relatively) $\Delta^0_2$-categorical.

The problem has already been studied for well-orderings (Ash), linear orders under some effective restrictions (McCoy), Boolean algebras (Harris),  algebraic fields (Miller), and completely decomposable groups (Downey and M.).

 We restrict ourselves to (algebraically) simple classes, namely to computable equivalence relations and multi-cyclic abelian groups (those are direct sums of cyclic and quasi-cyclic groups). Cenzer, Calvert, Harizanov, and Morozov were the first to
investigate categoricity notions in these classes.
We will see that, rather unexpectedly, the study of $\Delta^0_2$-categorical members in these classes requires new ideas and advanced recursion-theoretic techniques such as the 0''' priority method. Our machinery enables us to obtain several structural results and also answer questions which were left open in (Cenzer et al).
Joint work with Rod Downey and Keng Meng (Selwyn) Ng.