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A categorical semantics for inductive-inductive definitions

Altenkirch, Thorsten and Morris, Peter and Nordvall Forsberg, Fredrik and Setzer, Anton (2011) A categorical semantics for inductive-inductive definitions. In: Algebra and Coalgebra in Computer Science. Lecture Notes in Computer Science . Springer Berlin/Heidelberg, Berlin, pp. 70-84. ISBN 9783642229435

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Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A → Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules.