Quotient inductiveinductive types
Altenkirch, Thorsten and Capriotti, Paolo and Dijkstra, Gabe and Kraus, Nicolai and Nordvall Forsberg, Fredrik; Baier, Christel and Dal Lago, Ugo, eds. (2018) Quotient inductiveinductive types. In: Foundations of Software Science and Computation Structures. Lecture Notes in Computer Science . Springer Berlin/Heidelberg, Cham, pp. 293310. ISBN 9783319893655 (https://doi.org/10.1007/9783319893662)
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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 nontrivial 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 welltyped syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductiveinductive definitions. We call those HITs quotient inductiveinductive 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 inductioninduction, despite many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initialalgebra 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.
ORCID iDs
Altenkirch, Thorsten, Capriotti, Paolo, Dijkstra, Gabe, Kraus, Nicolai and Nordvall Forsberg, Fredrik ORCID: https://orcid.org/0000000161579288; Baier, Christel and Dal Lago, Ugo

Item type: Book Section ID code: 63612 Dates: DateEvent14 April 2018Published17 December 2017AcceptedKeywords: higher inductive types, qotient inductiveinductive types, homotopy type theory, algebra, Electronic computers. Computer science, Computer Science(all) Subjects: Science > Mathematics > Electronic computers. Computer science Department: Faculty of Science > Computer and Information Sciences Depositing user: Pure Administrator Date deposited: 05 Apr 2018 09:04 Last modified: 10 Jan 2022 09:42 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/63612