An algebraic view of structural induction
 Claudio Hermida,
 Bart Jacobs
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Abstract
We propose a uniform, categorytheoretic 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.
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 Title
 An algebraic view of structural induction
 Book Title
 Computer Science Logic
 Book Subtitle
 8th Workshop, CSL '94 Kazimierz, Poland, September 25–30, 1994 Selected Papers
 Pages
 pp 412426
 Copyright
 1995
 DOI
 10.1007/BFb0022272
 Print ISBN
 9783540600176
 Online ISBN
 9783540494041
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 933
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
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 Editors
 Authors

 Claudio Hermida ^{(1)}
 Bart Jacobs ^{(2)}
 Author Affiliations

 1. Computer Science Department, Aarhus University, DK8000, Denmark
 2. CWI, Kruislaan 413, 1098, SJ Amsterdam, The Netherlands
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