Unifying Structured Recursion Schemes

Abstract

Folds over inductive datatypes are well understood and widely used. In their plain form, they are quite restricted; but many disparate generalisations have been proposed that enjoy similar calculational benefits. There have also been attempts to unify the various generalisations: two prominent such unifications are the `recursion schemes from comonads' of Uustalu, Vene and Pardo, and our own `adjoint folds'. Until now, these two unified schemes have appeared incompatible. We show that this appearance is illusory: in fact, adjoint folds subsume recursion schemes from comonads. The proof of this claim involves standard constructions in category theory that are nevertheless not well known in functional programming: Eilenberg-Moore categories and bialgebras.