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 nontrivial 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 welltyped syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductiveinductive definitions. We call those HITs quotient inductiveinductive 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 inductioninduction, despite many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initialalgebra 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 

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  293310 
Number of pages  18 
DOIs  
Publication status  Published  14 Apr 2018 
Publication series
Name  Lecture Notes in Computer Science 

Publisher  Springer Berlin Heidelberg 
Volume  10803 
ISSN (Print)  03029743 
Keywords
 higher inductive types
 qotient inductiveinductive types
 homotopy type theory
 algebra
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Fredrik Nordvall Forsberg
Person: Academic, Research Only