# An algebraic view of structural induction

## 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.

## Keywords

Logical Predicate Finite Product Structural Induction Induction Principle Distributive Category## Preview

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## References

- [CS91]J.R.B. Cockett and D. Spencer. Strong categorical datatypes I. In
*Proceedings Category Theory 1991*. Canadian Mathematical Society, 1991.Google Scholar - [Her93]C. Hermida.
*Fibrations, logical predicates and indeterminates*. PhD thesis, University of Edinburgh, 1993. Tech. Report ECS-LFCS-93-277. Also available as Aarhus Univ. DAIMI Tech. Report PB-462.Google Scholar - [HJ93]C. Hermida and B. Jacobs. Fibrations with indeterminates: Contextual and functional completeness for polymorphic lambda calculi. In Book of Abstracts of
*Category Theory in Computer Science 5*, September 1993. Extended version to appear in*Mathematical Structures in Computer Science*.Google Scholar - [HJ95]C. Hermida and B. Jacobs. Induction and coinduction via subset types and quotient types. presented at CLICS/TYPES workshop, Götenburg, January 1995.Google Scholar
- [Jac91]B. Jacobs.
*Categorical Type Theory*. PhD thesis, Nijmegen, 1991.Google Scholar - [Jac95]B. Jacobs. Parameters and parameterization in specification using distributive categories.
*Fundamenta Informaticae*, to appear, 1995.Google Scholar - [Kel89]G.M. Kelly. Elementary observations on 2-categorical limits.
*Bulletin Australian Mathematical Society*, 39:301–317, 1989.Google Scholar - [Law70]F.W. Lawvere. Equality in hyperdoctrines and comprehension scheme as an adjoint functor. In A. Heller, editor,
*Applications of Categorical Algebra*. AMS Providence, 1970.Google Scholar - [LS81]D. Lehmann and M. Smyth. Algebraic specification of data types: A synthetic approach.
*Math. Systems Theory*, 14:97–139, 1981.CrossRefGoogle Scholar - [LS86]J. Lambek and P.J. Scott.
*Introduction to Higher-Order Categorical Logic*, volume 7 of*Cambridge Studies in Advanced Mathematics*. Cambridge University Press, 1986.Google Scholar - [MR91]Q. Ma and J. C. Reynolds. Types, abstraction and parametric polymorphism 2. In S Brookes, editor,
*Math. Found. of Prog. Lang. Sem.*, volume 589 of*Lecture Notes in Computer Science*, pages 1–40. Springer Verlag, 1991.Google Scholar - [Pav90]D. Pavlović.
*Predicates and Fibrations*. PhD thesis, University of Utrecht, 1990.Google Scholar - [Pav93]D. Pavlović. Maps I: relative to a factorisation system. Draft, Dept. of Math. and Stat., McGill University, 1993.Google Scholar
- [Pit93]A. Pitts. Relational properties of recursively defined domains. Tech. Report TR321, Cambridge Computing Laboratory, 1993.Google Scholar
- [Str72]R. Street. The formal theory of monads.
*Journal of Pure and Applied Algebra*, 2:149–168, 1972.CrossRefGoogle Scholar - [Str73]R. Street. Fibrations and Yoneda's lemma in a 2-category. In
*Category Seminar*, volume 420 of*Lecture Notes in Mathematics*. Springer Verlag, 1973.Google Scholar