TY - GEN

T1 - Quotient inductive-inductive types

AU - Altenkirch, Thorsten

AU - Capriotti, Paolo

AU - Dijkstra, Gabe

AU - Kraus, Nicolai

AU - Nordvall Forsberg, Fredrik

PY - 2018/4/14

Y1 - 2018/4/14

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85045667307&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-89366-2_16

DO - 10.1007/978-3-319-89366-2_16

M3 - Conference contribution

AN - SCOPUS:85045667307

SN - 9783319893655

T3 - Lecture Notes in Computer Science

SP - 293

EP - 310

BT - Foundations of Software Science and Computation Structures

A2 - Baier, Christel

A2 - Dal Lago, Ugo

PB - Springer Verlag

T2 - 21st International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2018 Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2018

Y2 - 14 April 2018 through 20 April 2018

ER -