Dependent Inductive and Coinductive Types are Fibrational Dialgebras

Henning Basold
(Radboud University)

In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types. It is well-known that dependent type theories à la Martin-Löf can be interpreted using fibrations. Modern theorem provers, however, are based on more sophisticated type systems that allow the definition of powerful inductive dependent types (known as inductive families) and, somewhat limited, coinductive dependent types. I define a class of functors on fibrations and show how data type definitions correspond to initial and final dialgebras for these functors. This description is also a proposal of how coinductive types should be treated in type theories, as they appear here simply as dual of inductive types. Finally, I show how dependent data types correspond to algebras and coalgebras, and give the correspondence to dependent polynomial functors.

In Ralph Matthes and Matteo Mio: Proceedings Tenth International Workshop on Fixed Points in Computer Science (FICS 2015), Berlin, Germany, September 11-12, 2015, Electronic Proceedings in Theoretical Computer Science 191, pp. 3–17.
Published: 9th September 2015.

ArXived at: http://dx.doi.org/10.4204/EPTCS.191.3 bibtex PDF

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