Abstract and Concrete Type Theories
Taichi Uemura
Abstract:
In this thesis, we study abstract and concrete type theories. We introduce an abstract notion of a type theory to obtain general results in the semantics of type theories, but we also provide a syntactic way of presenting a type theory to allow us a further investigation into a concrete type theory to obtain consistency and independence results.
In Chapter 3, we introduce an abstract notion of a type theory. After reviewing the theory of natural models of a type theory introduced by Awodey where models of a type theory are described in terms of representable maps of discrete fibrations, we introduce a notion of a category with representable maps as an abstract definition of a type theory. A model of a type theory is defined to be a structure-preserving functor to a category of discrete fibrations.
In Chapter 4, we introduce a syntactic presentation of a type theory called a second-order generalized algebraic theory, inspired by the definition of general type theories given by Bauer, Haselwarter, and Lumsdaine. This presentation coincides with the traditional presentation of a type theory by a grammar and inference rules. We show that every second-order generalized algebraic theory generates a type theory with an appropriate universal property and that, conversely, every type theory is generated by some second-order generalized algebraic theory.
In Chapter 5, we develop the semantics of type theories based on the definitions introduced in Chapter 3. Remarkably, the results in this chapter are theorems in pure category theory and do not depend on the syntactic presentations introduced in Chapter 4. The main result is the correspondence between theories and models for any type theory. This formally justifies the use of the internal language of a model of a type theory.
Once a type theory is defined as a category with certain structure, its higher dimensional generalization makes sense. We study an (∞, 1)-categorical generalization of a type theory called an ∞-type theory in Chapter 6. It turns out that ∞-type theories are a useful tool for understanding and tackling coherence problems which arise in the categorical and higher categorical semantics of type theories. We see that so-called non-split models of a type theory are naturally regarded as models of an ∞-type theory and that coherence problems in both categorical and higher categorical semantics of type theories can be formulated in the language of ∞-type theories and related concepts. As an application, we sketch a solution to Kapulkin and Lumsdaine’s conjecture that the dependent type theory with intensional identity types provides internal languages for finitely complete (∞, 1)-categories.
From Chapter 7, we turn our attention to a concrete type theory, univalent type theory. This type theory can be presented by a second-order generalized algebraic theory and thus defined as a type theory in the sense introduced in Chapter 3. Chapter 7 is mostly devoted to rephrasing the internal construction of models of univalent type theory given by Orton and Pitts and Licata et al. in terms of our notion of a model of a type theory.
Finally, in Chapter 8, we study concrete models of univalent type theory. Applying the construction explained in Chapter 7 to the assembly model of the extensional type theory, we have a model of univalent type theory called the cubical assembly model. We use the cubical assembly model to obtain consistency and independence results. We show that the cubical assembly model has a univalent and impredicative universe, and thus univalence is consistent with an impredicative universe. This impredicative universe does not satisfy the propositional resizing axiom in that we can construct a proposition in the cubical assembly model that is not equivalent to any proposition in the impredicative universe. Hence, the propositional resizing axiom is not provable over univalent type theory. We then study the cubical assembly model in relation to Constructive Recursive Mathematics. We show that the cubical assembly model satisfies Markov’s Principle but does not satisfy Church’s Thesis. Using the theory of modalities in homotopy type theory developed by Rijke, Shulman, and Spitters, we construct a reflective subuniverse of the cubical assembly model in which both Markov’s Principle and Church’s Thesis hold. Thus, these principles of Constructive Recursive Mathematics are consistent with univalent type theory.