Quotient inductive-inductive types

Altenkirch, Thorsten and Capriotti, Paolo and Dijkstra, Gabe and Kraus, Nicolai and Nordvall Forsberg, Fredrik (2018) Quotient inductive-inductive types. In: Foundations of Software Science and Computation Structures. Lecture Notes in Computer Science . Springer Berlin/Heidelberg, Cham, pp. 293-310. ISBN 978-3-319-89365-5

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    Abstract

    Higher inductive types (HITs) in Homotopy Type Theory allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types with non-trivial higher equality types, such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define types satisfying uniqueness of equality proofs, such as the Cauchy reals, the partiality monad, and the well-typed syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on a general theory of HITs, there is not yet a theoretical foundation for the combination of equality constructors and induction-induction, despite many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics. We further derive a section induction principle , stating that every algebra morphism into the algebra in question has a section, which is close to the intuitively expected elimination rules.