Results 1  10
of
19
A 2Categorical Approach To Change Of Base And Geometric Morphisms II
, 1998
"... We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibi ..."
Abstract

Cited by 45 (7 self)
 Add to MetaCart
We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2category, in such a way that arbitrary functors F: L ✲ K induce equipment arrows relF: relL ✲ relK, spnF: spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locallyfullyfaithful 2functor to the 2category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.
Presheaf Models for Concurrency
, 1999
"... In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their wo ..."
Abstract

Cited by 45 (19 self)
 Add to MetaCart
In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a builtin notion of bisimulation. We show how
Restriction categories I: Categories of partial maps
 Theoretical Computer Science
, 2001
"... ..."
A Relational Model of NonDeterministic Dataflow
 In CONCUR'98, volume 1466 of LNCS
, 1998
"... . We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits ..."
Abstract

Cited by 28 (13 self)
 Add to MetaCart
. We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higherorder dataflow as a biproduct, essentially by following the geometry of interaction programme. 1 Introduction A fundament...
A Theory of Recursive Domains with Applications to Concurrency
 In Proc. of LICS ’98
, 1997
"... Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains. ..."
Abstract

Cited by 23 (14 self)
 Add to MetaCart
Marcelo Fiore , Glynn Winskel (1) BRICS , University of Aarhus, Denmark (2) LFCS, University of Edinburgh, Scotland December 1997 Abstract We develop a 2categorical theory for recursively defined domains.
A Categorical Axiomatics for Bisimulation
 In Proc. of CONCUR’98, LNCS 1466
, 1998
"... We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in the Kleisli category for the 2monad T .
Doctrines Whose Structure Forms A Fully Faithful Adjoint String
 Theory Appl. Categ
, 1997
"... . We pursue the definition of a KZdoctrine in terms of a fully faithful adjoint string Dd a m a dD. We give the definition in any Graycategory. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The alge ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
. We pursue the definition of a KZdoctrine in terms of a fully faithful adjoint string Dd a m a dD. We give the definition in any Graycategory. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defined as adjunctions when we start with a KZdoctrine. 1. Introduction Free cocompletions of categories under suitable classes of colimits were the motivating examples for the definition of KZdoctrines. We introduce in this paper a notstrict version of such doctrines defined via a fully faithful adjoint string. Thus, a nonstrict KZdoctrine on a 2category K consists of a normal endo homomorphism D : K \Gamma! K, and strong transformations d : 1K \Gamma! D, and m : DD \Gamma! D in such a way that Dd a m a dD forms a fully faithful adjoint string, satisfying one equation involving the unit of...
Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to
Locating Reaction with 2categories
, 2004
"... Groupoidal relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner's approach to deriving labelled bisimulation congruences from reduction systems. In this paper, we develop the theory of GRPOs further, proving that wellknown equivalences, othe ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
Groupoidal relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner's approach to deriving labelled bisimulation congruences from reduction systems. In this paper, we develop the theory of GRPOs further, proving that wellknown equivalences, other than bisimulation, are congruences. To demonstrate the type of category theoretic arguments which are inherent in the 2categorical approach, we construct GRPOs in a category of `bunches and wirings.' Finally, we prove that the 2categorical theory of GRPOs is a generalisation of the approaches based on Milner's precategories and Leifer's functorial reactive systems.
On PropertyLike Structures
, 1997
"... A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precise mathemat ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of "essentially unique" and investigating its consequences. We call such 2monads propertylike. We further consider the more restricted class of fully propertylike 2monads, consisting of those propertylike 2monads for which all 2cells between (even lax) algebra morphisms are algebra 2cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which "structure is adjoint to unit", and which we now call laxidempotent 2monads: both these and their colaxidempotent duals are fully propertylike. We end by showing that (at least for finitary 2monads) the classes of propertylikes, fully propertylike...