On generalized algebraic theories and categories with families

We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.publishedVersio

Modal dependent type theory and dependent right adjoints

In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory, and spatial and cohesive type theory. In this paper we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K-axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any finite limit category with an adjunction of endofunctors gives rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal lambda-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes

Modal dependent type theory and dependent right adjoints

In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory, and spatial and cohesive type theory. In this paper we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K-axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any finite limit category with an adjunction of endofunctors gives rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal lambda-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes

Bicategories in univalent foundations

We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of those, we develop the notion of "displayed bicategories", an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. Displayed bicategories allow us to construct univalent bicategories in a modular fashion. To demonstrate the applicability of this notion, we prove several bicategories are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Our work is formalized in the UniMath library of univalent mathematics

Semantics for Homotopy Type Theory

The main aim of my PhD thesis is to define a semantics for Homotopy type theory based on elementary categorical tools. This led us to extend the study of this system in other directions: we proved a Normalisation theorem, and defined a generic syntax. All those results are obtained for a subset of the whole Homotopy type theory, which we called 1-HoTT theories. A 1-HoTT theory is composed by Martin-L\uf6f type theory with generic inductive types, the axioms of function extensionality and univalence, truncation and generic 1-higher inductive types, which are a subset of the higher inductive types in which the higher constructor of a type T is limited to the type =T . For those theories we obtained some proof theoretic results; the main one is a Normalisation theorem, following Girard's reducibility candidates technique. The semantics is sound and complete, with the completeness result following from the existence of a canonical model, which is also classifying. Our conjecture is that our proof theory and semantics can be extended to every single higher inductive type. The dissertation shows that a very large amount of higher inductive types can be analysed inside our framework: what prevents to extend the results is the lack of a systematic treatment of the syntax of the higher inductive types, which is still an open issue in Homotopy type theory

A Category Theoretic View of Contextual Types: from Simple Types to Dependent Types

We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann's work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad $\flat$ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et. al.

Semantics for Homotopy Type Theory

The main aim of my PhD thesis is to define a semantics for Homotopy type theory based on elementary categorical tools. This led us to extend the study of this system in other directions: we proved a Normalisation theorem, and defined a generic syntax. All those results are obtained for a subset of the whole Homotopy type theory, which we called 1-HoTT theories. A 1-HoTT theory is composed by Martin-Löf type theory with generic inductive types, the axioms of function extensionality and univalence, truncation and generic 1-higher inductive types, which are a subset of the higher inductive types in which the higher constructor of a type T is limited to the type =T . For those theories we obtained some proof theoretic results; the main one is a Normalisation theorem, following Girard's reducibility candidates technique. The semantics is sound and complete, with the completeness result following from the existence of a canonical model, which is also classifying. Our conjecture is that our proof theory and semantics can be extended to every single higher inductive type. The dissertation shows that a very large amount of higher inductive types can be analysed inside our framework: what prevents to extend the results is the lack of a systematic treatment of the syntax of the higher inductive types, which is still an open issue in Homotopy type theory

The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories

Seely's paper Locally cartesian closed categories and type theory contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Π, ∑, and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the Bénabou-Hofmann interpretation of Martin-Löf type theory in locally cartesian closed categories yields a biequivalence of 2-categories. To facilitate the technical development we employ categories with families as a substitute for syntactic Martin-Löf type theories. As a second result we prove that if we remove Π-types the resulting categories with families are biequivalent to left exact categories