Quotient inductive-inductive types

Abstract

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total 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 the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having 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 and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules.

References in corpus (5)

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• Computational Higher Type Theory IV: Inductive Types
• W-Types with Reductions and the Small Object Argument
• Large and Infinitary Quotient Inductive-Inductive Types
• Synthetic topology in Homotopy Type Theory for probabilistic programming
• Signatures and Induction Principles for Higher Inductive-Inductive Types
• Free Higher Groups in Homotopy Type Theory
• Constructing Higher Inductive Types as Groupoid Quotients
• Synthetic Spectra via a Monadic and Comonadic Modality
• Connecting Constructive Notions of Ordinals in Homotopy Type Theory
• Constructing Initial Algebras Using Inflationary Iteration
• An Initial Algebra Theorem Without Iteration
• A class of higher inductive types in Zermelo-Fraenkel set theory