We present a framework for compositional program verification based on polynomial functors in dependent type theory. In this framework, polynomial functors serve as program interfaces, Kleisli morphisms for the free monad monad serve as implementations, and dependent polynomials encode pre/postcondition specifications. We show that implementations and their verifications compose via wiring diagrams, and that Mealy machines provide a compositional coalgebraic operational semantics. We identify the abstract categorical structure underlying this compositionality as a monoidal functor from specifications to interfaces with a compatible monoidal natural transformation of lax monoidal presheaves; this opens the door to generalizations to other categories, monoidal products, etc., including settings for concurrency and relational verification, which we sketch. As a proof-of-concept, the entire framework has been formalized in Agda.

Awodey, later with Newstead, showed how polynomial functors with extra structure (termed “natural models”) hold within them the categorical semantics for dependent type theory. Their work presented these ideas clearly but ultimately led them outside of the usual category of polynomial functors to a particular tricategory of polynomials in order to explain all of the structure possessed by such models. This paper builds off that work – explicating the categorical semantics of dependent type theory by axiomatizing them entirely in terms of the usual category of polynomial functors. In order to handle the higher-categorical coherences required for such an explanation, we work with polynomial functors in the language of Homotopy Type Theory (HoTT), which allows for higher-dimensional structures to be expressed purely within this category. The move to HoTT moreover enables us to express a key additional condition on polynomial functors – univalence – which is sufficient to guarantee that models of type theory expressed as univalent polynomials satisfy all higher coherences of their corresponding algebraic structures, purely in virtue of being closed under the usual constructors of dependent type theory. We call polynomial functors satisfying this condition polynomial universes. As an example of the simplification to the theory of natural models this enables, we highlight the fact that a polynomial universe being closed under dependent product types implies the existence of a distributive law of monads, which witnesses the usual distributivity of dependent products over dependent sums.

We define strict and lax orthogonal factorization systems on double categories. These consist of an orthogonal factorization system on arrows and one on double cells that are compatible with each other. Our definitions are motivated by several explicit examples, including factorization systems on double categories of spans, relations and bimodules. We then prove monadicity results for orthogonal factorization systems on double categories in order to justify our definitions. For fibrant double categories we discuss the structure of the double orthogonal factorization systems that have a given orthogonal factorization system on the arrows in common. Finally, we study the interaction of orthogonal factorization systems on double categories with double fibrations.

Ordered, linear, and other substructural type systems allow us to expose deep properties of programs at the syntactic level of types. In this paper, we develop a family of unary logical relations that allow us to prove consequences of parametricity for a range of substructural type systems. A key idea is to parameterize the relation by an algebra, which we exemplify with a monoid and commutative monoid to interpret ordered and linear type systems, respectively. We prove the fundamental theorem of logical relations and apply it to deduce extensional properties of inhabitants of certain types. Examples include demonstrating that the ordered types for list append and reversal are inhabited by exactly one function, as are types of some tree traversals. Similarly, the linear type of the identity function on lists is inhabited only by permutations of the input. Our most advanced example shows that the ordered type of the list fold function is inhabited only by the fold function.

Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have emerged with the aim of expressing such parametric reasoning in their internal logic, toward the end of solving various problems arising from the complexity of higher-dimensional coherence conditions in type theory. This paper presents a first step toward the unification, simplification, and extension of these various methods for internalizing parametricity. Specifically, I argue that there is an essentially modal aspect of parametricity, which is intimately connected with the category-theoretic concept of cohesion. On this basis, I describe a general categorical semantics for modal parametricity, develop a corresponding framework of axioms (with computational interpretations) in dependent type theory that can be used to internally represent and reason about such parametricity, and show this in practice by implementing these axioms in Agda and using them to verify parametricity theorems therein. I then demonstrate the utility of these axioms in managing the complexity of higher-dimensional coherence by deriving induction principles for higher inductive types, and in closing, I sketch the outlines of a more general synthetic theory of parametricity, with applications in domains ranging from homotopy type theory to the analysis of program modules.

This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems possessing either syntax or semantics inclusive of certain practical applications, but has struggled to combine these all in one and the same system. Toward resolving this difficulty, I propose a novel categorical interpretation of substructural dependent types, analogous to the use of monoidal categories as models of linear and ordered logic, that encompasses a wide class of mathematical and computational examples. On this basis, I develop a general framework for substructural dependent type theories, and proceed to prove some essential metatheoretic properties thereof. As an application of this framework, I show how it can be used to construct a type theory that satisfactorily addresses the problem of effectively representing cut admissibility for linear sequent calculus in a logical framework.