Constructing quotient inductive-inductive types

Abstract

Quotient inductive-inductive types (QIITs) generalise inductive types in two ways: a QIIT can have more than one sort and the later sorts can be indexed over the previous ones. In addition, equality constructors are also allowed. We work in a setting with uniqueness of identity proofs, hence we use the term QIIT instead of higher inductive-inductive type. An example of a QIIT is the well-typed (intrinsic) syntax of type theory quotiented by conversion. In this paper first we specify finitary QIITs using a domain-specific type theory which we call the theory of signatures. The syntax of the theory of signatures is given by a QIIT as well. Then, using this syntax we show that all specified QIITs exist and they have a dependent elimination principle. We also show that algebras of a signature form a category with families (CwF) and use the internal language of this CwF to show that dependent elimination is equivalent to initiality.