Type-Theoretic Signatures for Algebraic Theories and Inductive Types

Abstract

Summary of Thesis This thesis develops the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated algebraic theories. In the thesis, we describe three type theories where each typing context can be viewed as an algebraic signature, specifying sorts, operations and equations. These signatures are useful in broader mathematical contexts, but we are also concerned with potential implementation in proof assistants. In Chapter 3, we describe a way to use two-level type theory as a metalanguage for developing semantics of algebraic signatures. This makes it possible to work in a concise internal notation of a type theory, and at the same time build semantics internally to arbitrary structured categories. For example, the signature for natural number objects can be interpreted in any category with finite products. In Chapter 4, we describe finitary quotient inductive-inductive (FQII) signatures. Most type theories themselves can be specified with FQII signatures. We build a structured category of algebras for each signature, where equivalence of initiality and induction can be shown. We additionally present term algebra constructions, constructions of left adjoint functors of signature morphisms, and we describe a way to use self-describing signatures to minimize necessary metatheoretic assumptions. In Chapter 5, we describe infinitary quotient inductive-inductive signatures. These allow specification of infinitely branching trees as initial algebras. We adapt the semantics from the previous chapter. We also revisit term models, left adjoints of signature morphisms and self-description of signatures. We also describe how to build semantics of signatures internally to the theory of signatures itself, which yields numerous ways to build new signatures from existing ones. In Chapter 6, we describe higher inductive-inductive signatures. These differ from previous semantics mostly in that their intended semantics is in homotopy type theory, and allows higher-dimensional equalities. In this more general setting we only consider enough semantics to compute notions of initiality and induction for each signature.