Type-Theoretic Signatures for Algebraic Theories and Inductive Types

Abstract

I would like to thank my supervisor Ambrus Kaposi for his support and the remarkably low-stress experience throughout my studies, and also for the research collaboration which provided ample content for this thesis. Many thanks to Tamás Kozsik for being a likewise supportive supervisor in my first year of studies, and to Zoltán Horváth for doing a great job providing financial and administrative support, and I also want to thank the secretaries at department and at the faculty.I would like to thank my opponents Fredrik Forsberg and Christian Sattler for their careful reading and comments on my thesis.I would like to thank attendees of the Budapest type theory seminar for the many discussions over the years, in particular Balázs Kőműves, Péter Diviánszky, Gábor Lehel, Gergő Érdi and my fellow PhD students Rafaël Bocquet and István Donkó.I would like to thank everyone in the research community who educated and inspired me through various discussions and interactions. I would like to express special thanks to my coauthors Thorsten Altenkirch, Nicolai Kraus and Ambroise Lafont.Finally, I would like to thank my parents for their love and support. 2 1.1. OVERVIEW progress in this respect.Our signatures are useful in broader mathematical contexts, but we are also concerned with potential implementation in proof assistants. Although it is unlikely that our signatures can be deployed in practice exactly as they are, they should be still helpful as formal bases of practical implementations. OverviewIn Chapter 2, we present a minimal example of a type theory of signatures. This allows specifying single-sorted signatures without equations. The purpose of the chapter is didactic. We develop just enough semantics to get notions of initiality and induction for algebras. We also present a term algebra construction: this shows that the initial algebra for each signature can be constructed from the syntax of signatures itself.In Chapter 3 we describe a metatheoretic setting which is often used in the thesis. This is two-level type theory [ACKS19]. It allows us to develop general semantics for signatures, while still working inside a convenient type theory. As a demonstration, we generalize the semantics from Chapter 2 so that it is given internally to arbitrary categories-with-families. As a special case, signatures can be interpreted in arbitrary categories with finite products.In Chapter 4 we describe finitary quotient inductive-inductive signatures.These are close to generalized algebraic theories [Car86] in expressive power. In particular, most type theories themselves can be specified with finitary quotient inductive-inductive signatures. We significantly expand the semantics of signatures, now for each signature we provide a category of algebras with certain extra structure, which is equivalent to having finite limits. This allows us to prove for each signature the equivalence of initiality and induction. Also, owing to two-level type theory, signatures can be interpreted internally to any category with finite limits. Additi...