Giuseppe Rosolini

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.

Type theory via exact categories

Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a category-theoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework.

Locally cartesian closed exact completions

AbstractWedetermineexplicitconditionsonacategoryPwhichareequivalenttotherequestthatitsexactcompletionP ex belocallycartesianclosed,orsimplycartesianclosed.InlinewiththeideathatweaklimitsinPgiverisetoactuallimitsintheexactcompletion,theconditionrequirestheexistenceofweakevaluationmaps.WeapplythecharacterizationtothecaseofthecategoryoftopologicalspacesandtoarelatedcategoryproposedbyDanaScott. c 2000ElsevierScienceB.V.Allrightsreserved.MSC:18B99;18A35;18B30;03B40;03B15 1.WeakconditionsofclosureWhenpresentingthee ectivetoposasanexactcompletion(see[1,4,10]),itiscertainlyasurprisetoseethatthefreeexactcategoryobtainedintheprocessturnsouttohaveasubobjectclassi er.Itislessofasurprisetorecognizeithasexponentials.Inthepresentworkweinvestigategeneralconditionstoexplainwhythishappens.Itisaneverendingrigmarolethatthenotionofcartesianclosurerequiresonly niteproducts,whilethenotionofexactcategoryrequiresonlypullbacks.Sothe rstapproximationtothereasonablecontextinwhichtostudyclosureinanexactcategoryisthatofacategorywith nitelimitsandclosed.Notethat,incaseacategoryChas nitelimitsandisclosed,then

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 ′.

ABOUT MODEST SETS

We present a new proof of the existence of a small full subcategory of the effective topos which is closed under very many limits, sufficient to give an intuitive and direct interpretation of polymorphism. We describe also how to construct many new categories with enough limits to give direct interpretation of polymorphisms and other strong programming languages.

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 Category Theoretic Formulation for Engeler-style Models of the Untyped λ-Calculus

We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset monad. We exploit the self-duality of the category Rel, i.e., the Kleisli category for the powerset monad, and the category theoretic structures on it that allow us to build models of the untyped λ-calculus, yielding a variant of the Engeler model. We replace the monad for commutative monoids by that for idempotent commutative monoids, which, on Set, is the finite powerset monad. This does not quite yield a distributive law, so requires a little more subtlety, but, subject to that subtlety, it yields exactly the original Engeler construction.

Two models of synthetic domain theory

Abstract Two models of synthetic domain theory encompassing traditional categories of domains are introduced. First, we present a Grothendieck topos embedding the category ω- Cpo of ω-complete posets and ω-continuous functions as a reflective exponential ideal. Second, we obtain analogous results with respect to a category of domains and stable functions.

A Categorial View of Process Refinement

A very general notion of refinement of event structures is presented that refines both the events and the relations of causality and conflict. It is based on a purely semantic construction based on sections of a functor between domain-like categories. The present construction is compared to others in the literature.

Comparing models of higher type computation

Models of higher type computation appear basically in two avours as real izability models partial equivalence relations over some appropriate weak al gebraic model of function application and as some kind of limit preserving functionals for some suitably weakened notion of topology The aim of this presentation is to show that the two approaches are no di erent in the sense that provided a certain theorem can be proved about the limit structures the higher order structures de ned via the limit preserving functionals coincides with one de ned in terms of PER s Glimpses of this fact can be found in various guises in the literature see