An Algebraic View of Structural Induction

We propose a uniform, category-theoretic account of structural induction for inductively deened data types. The account is based on the understanding of inductively deened data types as initial algebras for certain kind of endofunctors T : B !B on a bicartesian/distributive category B. Regarding a predicate logic as a bration p : P!B over B , we consider a logical predicate lifting of T to the total category P. Then, a predicate is inductive precisely when it carries an algebra structure for such lifted endofunctor. The validity of the induction principle is formulated by requiring that thètruth' predicate functor > : B !Ppreserve initial algebras. We then show that when the bration admits a comprehension principle, analogous to the one in set theory, it satisses the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. thèstability' property of the induction principle.