Quotient inductive-inductive types

Thorsten Altenkirch, Paolo Capriotti, Gabe Dijkstra, Nicolai Kraus, Fredrik Nordvall Forsberg

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

29 Citations (Scopus)
19 Downloads (Pure)


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.
Original languageEnglish
Title of host publicationFoundations of Software Science and Computation Structures
Subtitle of host publication21st International Conference, FOSSACS 2018, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2018, Thessaloniki, Greece, April 14–20, 2018. Proceedings
EditorsChristel Baier, Ugo Dal Lago
Place of PublicationCham
Number of pages18
Publication statusPublished - 14 Apr 2018

Publication series

NameLecture Notes in Computer Science
PublisherSpringer Berlin Heidelberg
ISSN (Print)0302-9743


  • higher inductive types
  • qotient inductive-inductive types
  • homotopy type theory
  • algebra


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