Edmund Robinson

Premonoidal categories and notions of computation

We introduce the notions of premonoidal category and premonoidal functor, and show how these can be used in the denotational semantics of programming languages. We characterize the semantic definitions of Eugenio Moggis monads as notions of computation, exhibit a representation theorem for our premonoidal setting in terms of monads, and give a fibrational setting for the structure.

Categories of partial maps

Abstract This paper attempts to reconcile the various abstract notions of “category of partial maps” which appear in the literature. First a particular algebraic theory ( p -categories) is introduced and a representation theorem proved. This gives the authors a coherent framework in which to place the various other definitions. Both algebraic theories and theories which make essential use of the poset-enriched structure of partial maps are discussed. Proofs of equivalence are given where possible and counterexamples where known. The paper concludes with brief sections on the representation of partial maps and on partial algebras.

Proof Nets for Classical Logic

This paper introduces a notion of proof net for classical logic, provides a static correctness condition for these nets, and analyses the connection between nets and conventional sequent calculus. The main surprise of the paper is that there are no surprises at the static level. Subsequent work reveals that there are few at the dynamic either.

Variations on Algebra: Monadicity and Generalisations of Equational Therories

Abstract. This is a largely tutorial paper about the categorical notion of monad and the ways in which monads on different categories correspond to variations on the standard notion of algebraic theory.

Colimit completions and the effective topos

The family of readability toposes, of which the effective topos is the best known, was discovered by Martin Hyland in the late 1970s. Since then these toposes have been used for several purposes. The effective topos itself was originally intended as a category in which various recursion-theoretic or effective constructions would live as natural parts of the higher-order type structure. For example the hereditary effective operators become the higher types over N (Hyland [1982]), and effective domains become the countably-based domains in the topos (McCarty [1984], Rosolini [1986]). However, following the discovery by Moggi and Hyland that it contained nontrivial small complete categories, the effective topos has also been used to provide natural models of polymorphic type theories, up to and including the theory of constructions (Hyland [1987], Hyland, Robinson and Rosolini [1987], Scedrov [1987], Bainbridge et al. [1987]). Over the years there have also been several different constructions of the topos. The original approach, as in Hyland [1982], was to construct the topos by first giving a notion of Pω-valued set. A Pω-valued set is a set X together with a function = x : X × X → Pω. The elements of X are to be thought of as codes, or as expressions denoting elements of some “real underlying” set in the topos. Given a pair ( x , x ′) of elements of X , the set = x ( x , x ′) (generally written ) is the set of codes of proofs that the element denoted by x is equal to the element denoted by x ′.

How complete is PER

The category of partial equivalence relations (PER) on the natural numbers has been used extensively in recent years to model various forms of higher-order type theory. It is known that PER can be viewed as a category of sets in a nonstandard model of intuitionistic Zermelo-Fraenkel set theory. The use of PER as a vehicle for modeling-type theory then arises from completeness properties of this category. The paper demonstrates these completeness properties, and shows that, constructively, some complete categories are more complete than others.<<ETX>>

Categorical proof theory of classical propositional calculus

We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite well-behaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. Finally we explain the consequences of insisting on more familiar categorical behaviour.

Algebraic types in PER models

Huet has conjectured that the interpretations of a class of types (the “algebraic types”) in the PER model on the natural numbers for the second-order lambda calculus are in a certain sense the initial algebras. In this paper we examine several different PER models, and show that Huets conjecture holds in each.

A characterization of pie limits

It is well-known that limits in 2-categories are more complex than limits in ordinary categories. Most readers will at least be familiar with terms such as ‘lax limit’ and ‘pseudo-limit’. In the more modern treatments, these become special cases of a more general class of ‘weighted’ or ‘indexed’ limits (see Kelly [7] and Section 1 of this paper).

The p-adic spectrum

Abstract An account of the theory of the p-adic analogue of the real spectrum, up to and including the proof that it is spatial, together with some remarks on possible extensions of the theory to more general complete local fields.