The Construction of Set-Truncated Higher Inductive Types

Abstract

We construct finitary set-truncated higher inductive types (HITs) from quotients and the propositional truncation. For that, we first define signatures as a modification of the schema by Basold et al., and we show they give rise to univalent categories of algebras in both sets and setoids. To interpret HITs, we use the well-known method of initial algebra semantics. The desired algebra is obtained by lifting the quotient adjunction to the level of algebras and adapting Dybjer's and Moeneclaey's interpretation of HITs in setoids. From this construction, we conclude that the equality types of HITs are freely generated and that HITs are unique. The results are formalized in the UniMath library.