One can define abstract data types by using algebras with sorts, operations, and systems of equations; recursive, or “circular,” data types (such as stacks and trees) are fixed-point solutions to such equations. Algebraic theories are, roughly, categories of numbers and sequences; iteration theories are algebraic theories for which a fixed-point operation (iteration) that satisfies some identities is defined. It would therefore be reasonable to define recursive types in terms of iteration theories. The authors construct such definitions, but the main parts of their paper consider properties of what they call ‘enriched functors’ and the relationship between iteration theories and initial algebras.

The paper is written in a concise and lucid style. Unfortunately, although it contains a brief review of the basic definitions, a reader who has only a tenuous grasp of category theory will be hopelessly lost. This complexity does not reflect badly on the paper, which appears in a theoretical journal and cannot be expected to summarize the research of the past decade, but it does limit the potential readership.