Abstract not found

In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ of this pseudo-distributive law are precisely symmetric polycategories. 1

Abstract. This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2-categories, and Cat-categories. The latter two are exactly the same (except that strictly speaking a Cat-category should have small hom-categories, but that need not concern us here). The first two are nominally different — the 2-categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2-categories. Nonetheless, the theories of bicategories, 2-categories, and Catcategories have rather different flavours.

Abstract. I describe the main results obtained during the EPSRC postdoctoral fellowship that I held at the University of Cambridge. The fellowship focused on the interplay between category theory and mathematical logic. 1. Wellfounded trees W-types in categories. Types of wellfounded trees, or W-types, are one of the most important components of Martin-Löf’s dependent type theories. They allow us to define a wide class of inductive types, play an essential role in the ‘sets-astrees’ interpretation of constructive set theories, and contribute considerably to the proof-theoretic strength of dependent type theories. A categorical counterpart of W-types was introduced in [18] by defining W-types in a locally cartesian closed category to be initial algebras for endofunctors of a special kind, generally referred to as polynomial functors. In collaboration with Martin Hyland, I set out to investigate the consequences of the assumption that a locally cartesian closed category has W-types. To explore these consequences we introduced the notion of a dependent polynomial functor, a

This paper studies fundamental connections between profunctors (i.e., distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, "lifting", and indexed families.

A category of event structures with symmetry is introduced and its categorical properties investigated. Applications to the event-structure semantics of higher order processes, nondeterministic dataflow and the unfolding of Petri nets with multiple tokens are sketched.