We introduce the notion of a Martin-Löf category-a locally cartesian closed category with disjoint coproducts and initial algebras of container functors (the categorical analogue of W-types)-and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any Martin-Löf category.Central to our development are the notions of containers and container functors. These provide a new conceptual analysis of data structures and polymorphic functions by exploiting dependent type theory as a convenient way to define constructions in Martin-Löf categories. We also show that morphisms between containers can be full and faithfully interpreted as polymorphic functions (i.e. natural transformations) and that, in the presence of W-types, all strictly positive types (including nested inductive and coinductive types) give rise to containers.

Abstract. We introduce the notion of containers as a mathematical formalisation of the idea that many important datatypes consist of templates where data is stored. We show that containers have good closure properties under a variety of constructions including the formation of initial algebras and final coalgebras. We also show that containers include strictly positive types and shapely types but that there are containers which do not correspond to either of these. Further, we derive a representation result classifying the nature of polymorphic functions between containers. We finish this paper with an application to the theory of shapely types and refer to a forthcoming paper which applies this theory to differentiable types.

Abstract. We study the interplay between dependent types and general computational e↵ects. We define a language with both value types and terms, and computation types and terms, where types depend only on value terms. We use computational ⌃-types to account for typedependency in the sequential composition of computations. Our language design is justified by a natural class of categorical models. We account for both algebraic and non-algebraic e↵ects. We also show how to extend the language with general recursion, using continuous families of cpos.

We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses.

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