### 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.

Original language | English |
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Title of host publication | Foundations of Software Science and Computation Structures |

Subtitle of host publication | 21st 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 |

Editors | Christel Baier, Ugo Dal Lago |

Place of Publication | Cham |

Pages | 293-310 |

Number of pages | 18 |

DOIs | |

Publication status | Published - 14 Apr 2018 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer Berlin Heidelberg |

Volume | 10803 |

ISSN (Print) | 0302-9743 |

### Keywords

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

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## Profiles

## Fredrik Nordvall Forsberg

Person: Academic, Research Only

## Cite this

Altenkirch, T., Capriotti, P., Dijkstra, G., Kraus, N., & Nordvall Forsberg, F. (2018). Quotient inductive-inductive types. In C. Baier, & U. Dal Lago (Eds.),

*Foundations of Software Science and Computation Structures: 21st 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*(pp. 293-310). (Lecture Notes in Computer Science; Vol. 10803). Cham. https://doi.org/10.1007/978-3-319-89366-2