A computable structure is given by a computable domain, and then a set of computable relations and functions defined on that domain. The study of computable structures, going back as far as the work of Frohlich and Shepherdson, Rabin, and Malcev is part of a long-term programme to understand the algorithmic content of mathematics.
In mathematics generally, the notion of isomorphism is used to determine structures which are essentially the same. Within the context of effective (algorithmic) mathematics, however, one is presented with the possibility that pairs of computable structures may exist which, while isomorphic, fail to have a computable isomorphism between them. Thus the notion of computable categoricity has become of central importance: a computable structure S is computably categorical if any two computable presentations A and B of S are computably isomorphic. In this paper, my co-authors Downey, Kach, Lempp, Montalban, Turetsky and I, answer one of the longstanding questions in computable structure theory, showing the class of computably categorical structures has no simple structural or syntactic characterisation.
The complexity of computable categoricity, with Downey, Kach, Lempp, Montalban, and Turetsky, Advances in Mathematics 268 (2015), 423–466, pdf.