Date: 15 Jun 2005

An algebraic view of structural induction

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Abstract

We propose a uniform, category-theoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial algebras for certain kind of endofunctors T: \(\mathbb{B} \to \mathbb{B}\) on a bicartesian/distributive category \(\mathbb{B}\) . Regarding a predicate logic as a fibration p: \(\mathbb{P} \to \mathbb{B}\) over \(\mathbb{B}\) , we consider a logical predicate lifting of T to the total category \(\mathbb{P}\) . 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 the ‘truth’ predicate functor ⊤: \(\mathbb{B} \to \mathbb{P}\) preserve initial algebras. We then show that when the fibration admits a comprehension principle, analogous to the one in set theory, it satisfies the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. the ‘stability’ property of the induction principle.