## Profunctors, open maps and bisimulation (2000)

Venue: | Mathematical Structures in Computer Science, To appear. Available from the Glynn Winskel’s web |

Citations: | 12 - 4 self |

### BibTeX

@TECHREPORT{Cattani00profunctors,open,

author = {Gian Luca Cattani and Glynn Winskel},

title = {Profunctors, open maps and bisimulation},

institution = {Mathematical Structures in Computer Science, To appear. Available from the Glynn Winskel’s web},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

3202 |
Communication and Concurrency
- Milner
(Show Context)
Citation Context ...s . . . . . . . . . . . . . . . . . . . . . . . . . . 49s1 Introduction At first sight, it is perhaps surprising that profunctors,1 a categorical generalisation of relations [4, 31], and bisimulation =-=[37, 38]-=-, a central equivalence in the study of processes are intimately related. Briefly, the chain of connections runs: - Nondeterministic processes can be represented as presheaves. A presheaf over a categ... |

377 | Basic concepts of enriched category theory, volume 64
- Kelly
- 1982
(Show Context)
Citation Context ...gory !P of bP is closed under all finite colimits. The category !P with y!P : P !!P is a free finite colimit completion of P. Proof: The closure of !P under finite colimits is shown in Theorem 5.8 of =-=[29]-=-. The proof of freeness is straightforward. 2 4The ! in !-Acc refers to the fact that filtered colimits are specified in terms of finite subdiagrams. For more on the notion of ^-accessible category (f... |

228 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ...his line is very fruitful in a range of categories of process models and domain theories, and often furnishes useful proof principles of coinduction, echoing the technique promoted by Milner and Park =-=[25]-=-. The other approach is based on open maps. Open maps have a prehistory in pure mathematics [27], but first appeared in computer science in [28]. Their initial role was in giving a unified approach to... |

215 |
Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics
- Lane
- 1998
(Show Context)
Citation Context ...given by F *(E) = E(F (-), E). In applications to the semantics of concurrent processes, the category P is to be thought of as consisting of path objects, or computation-path shapes. The Yoneda Lemma =-=[33]-=-, by providing a natural bijection between bP(yP(P ), X) and X(P ), justifies the intuition that a presheaf X : Pop ! Set can be thought of as specifying for a typical path object P the set X(P ) of c... |

116 | Bisimulation from open maps
- Joyal, Nielsen, et al.
- 1996
(Show Context)
Citation Context .... Familiar models of processes such as known categories of synchronisation trees and event structures, and many others, can be realised as presheaf categories P for some suitable choice of category P =-=[28]-=-. - Bisimulation between processes is caught via spans of open maps. An open map between presheaves is a generalisation of a functional bismulation between transition systems (i.e., a bisimulation who... |

115 |
Autonomous Categories
- Barr
- 1979
(Show Context)
Citation Context ...inition of dualiser is straightforward and direct in contrast to the definition of the corresponding pseudo-functor on Cocont. The bicategory Prof might reasonably be called a *-autonomous bicategory =-=[3]-=-. 5.4 Function space Combining tensor and dualiser, yields a "linear function space". Define the pseudo functor( : Prof op * Prof ! Prof as (= \Omegasffi ((-)? * 1), so P ( Q = Pop * Q, for any small ... |

107 |
Coherence for compact closed categories
- Kelly, Laplaza
- 1980
(Show Context)
Citation Context ...s "categories" of domains of nondeterministic processes, the techniques required to solve recursive domain equations are explored in [11]. 5 The structure of Prof It has been remarked, for example in =-=[30]-=-, that Prof has enough structure to be, what might be called, a compact closed bicategory. To see this, we first need to define certain bicategorical limits explicitly. 5.1 Pseudo-products and -coprod... |

90 | Full abstraction for a simple parallel programming language
- HENNESSY, PLOTKIN
- 1979
(Show Context)
Citation Context ... sets (see e.g. [5]). In this section we pursue another analogy relating presheaf categories to non-deterministic domains, in which the presheaf construction corresponds to a powerdomain construction =-=[20, 43]-=-. With presheaf categories as analogues of powerdomains, Prof can be regarded as a bicategory of non-deterministic domains [20]. 7.1 !-Accessible categories The operation of ideal completion, familiar... |

79 |
Foncteurs analytiques et espèces de structures
- Joyal
- 1985
(Show Context)
Citation Context ... 5This view is amplified in [40, 53]. For now, note that special profunctors of this form, viz., FB(1) + 1 where B is the category of finite sets and bijections, are used in Joyal's theory of species =-=[26]-=-. A profunctor FB(1) + 1 corresponds to a functor F : B ! Set; such a functor in turn corresponds to an analytic functor from Set to Set, taking a set X to R n2B F n * Xn. See Example 9.9. 33sfor hXii... |

67 |
Semantics of weakening and contraction
- Jacobs
- 1994
(Show Context)
Citation Context ...ap bisimulation. 8 Lifting and connected colimits Our next example of a pseudo-comonad is provided by the lifting operation on Prof . Its co-Kleisli bicategory provides a model of affine linear logic =-=[24]-=-. Arrows in the co-Kleisli bicategory will correspond to connected colimit preserving functors between presheaf categories. Such functors do not have to send the empty presheaf to the empty presheaf, ... |

64 |
Accessible categories: the foundations of categorical model theory, volume 104 of Contemporary Mathematics
- Makkai, Paré
- 1989
(Show Context)
Citation Context ...isation to categories, in which a category is completed with all filtered colimits (see [33] for a discussion of filtered categories and colimits). 18sDefinition 7.1 (Completion by filtered colimits) =-=[35, 2]-=- Let P be a small category. We write eP for the full subcategory of bP consisting of presheaves whose categories of elements (see Definition A.13) are filtered. As the category of elements of each rep... |

45 | Presheaf models for concurrency
- Cattani, Winskel
- 1996
(Show Context)
Citation Context ...F, F 0 : P + Q are open map bisimilar and that profunctors G, G0 : Q + R are open map bisimilar. Then, the compositions GF, G0F 0 : P + R are open map bisimilar. Proof: A direct proof can be found in =-=[10]-=-. In fact, both these results follow from the seemingly weaker Theorem 3.3 and Corollary 3.4, once we observe that the composition of profunctors preserves colimits in each argument. 16s(i) This can b... |

45 |
A fully abstract denotational model for higher-order processes
- Hennessy
- 1993
(Show Context)
Citation Context ...We refer the reader to that work and the recent work of Power and Tanaka [46, 49] for the definitions and results of pseudo-monads and pseudo-comonads 2Another place is in the work of Matthew Hennessy=-=[19]-=-, who in developing a domain theory for concurrency used a direct analogue of Prof, essentially one based on relations F : P * Qop ! 2 where the role of the category Set in defining a profunctor has b... |

41 |
A completeness theorem for open maps
- Joyal, Moerdijk
- 1994
(Show Context)
Citation Context ...ten furnishes useful proof principles of coinduction, echoing the technique promoted by Milner and Park [25]. The other approach is based on open maps. Open maps have a prehistory in pure mathematics =-=[27]-=-, but first appeared in computer science in [28]. Their initial role was in giving a unified approach to a range of models for concurrent computation, from interleaving models like transition systems ... |

35 |
Some free constructions in realizability and proof theory
- Carboni
- 1995
(Show Context)
Citation Context ...tion. Because the functors JF are full and faithful, F-polynomials correspond to within isomorphism to special functors between presheaf categories (under suitable conditions, they are exact functors =-=[9]-=-). 37sExample 9.8 The category I consists of finite sets and injections. (Alternatively we can work with the equivalent category with objects natural numbers understood as sets with injections.) There... |

34 |
Lane and Ieke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext
- Mac
- 1992
(Show Context)
Citation Context ...tween presheaves over P. Let f : X ! Y be a surjective IP-open map. By definition it is an epimorphism. To show that f is an isomorphism it is now enough to show that f is a monomorphism as well (see =-=[34]-=-). Since f is a natural transformation between presheaves, f is a monomorphism iff for every object P of P, the function fP : X(P ) ! Y (P ) is injective. Suppose then that x, x0 2 X(P ) are such that... |

27 | A relational model of non-deterministic dataflow
- Hildebrandt, Panangaden, et al.
- 2004
(Show Context)
Citation Context ...onn preserve open map bisimulation leading automatically to congruence results [52, 10]. The category Conn also supports a trace operation associated with a feedback loop in nondeterministic dataflow =-=[23]-=-. 9 Pseudo comonads via families 9.1 Motivation According to the discipline of linear logic, nonlinear maps from P to Q are introduced as linear maps from !P to Q--the exponential ! applied to P allow... |

26 |
On Semantic Foundations for Applicative Multiprogramming
- Abramsky
- 1983
(Show Context)
Citation Context ...sets, as in \Omegasabove, but this time allowing all functions as maps.) In this case FF(P) is the finite coproduct completion of a small categoryP (a construction dual to the categorical powerdomain =-=[32, 1]-=-). Clearly F has singletons. It has a dependent sum given by disjoint union. There is an isomorphism FF(P) \OmegasFF(Q) ,= FF(P&Q) expressing how a family in FF(P&Q) can be broken down into a pair of ... |

23 | A theory of recursive domains with applications to concurrency
- Cattani, Fiore, et al.
- 1998
(Show Context)
Citation Context ... Q, the type of higher order processes which take a process of type P as argument and deliver a process of type Q as result. Recursive domain equations can also be treated in this generalised setting =-=[11]-=-. It is sensible to view a profunctor F : P + Q as a linear map which on input of a process of type P yields a process of type Q. Linearity is about how to manage without a presumed ability to copy or... |

19 | A Fully Abstract Presheaf Semantics of SCCS with Finite Delay
- Hildebrandt
- 1999
(Show Context)
Citation Context ...in applying the mathematics, in connecting with process languages and 38soperational semantics [40, 41], the semantics of nondeterministic dataflow [23], independence/causal models [22, 39], fairness =-=[21]-=-, pi-Calculus and name generation for higher order processes [12, 57], and weak bisimulation [16]. These are all examples of how we can bring categorical reasoning to bear on issues of concurrent comp... |

19 | Bisimulation proof methods for mobile ambients
- Merro, Nardelli
- 2003
(Show Context)
Citation Context ... obtained from an operational semantics. Though a pattern has emerged, bisimulation is most often defined in an ad hoc manner for the language at hand, and sometimes can be a matter of great subtlety =-=[36]-=-. Broadly speaking, there are two lines of development in making the definition of bisimulation more systematic; so that the variety of bisimulation is determined by the denotational semantics given t... |

16 |
Metric spaces, generalized logic, and closed
- Lawvere
- 1974
(Show Context)
Citation Context ....4.6 Extensions of functors . . . . . . . . . . . . . . . . . . . . . . . . . . 49s1 Introduction At first sight, it is perhaps surprising that profunctors,1 a categorical generalisation of relations =-=[4, 31]-=-, and bisimulation [37, 38], a central equivalence in the study of processes are intimately related. Briefly, the chain of connections runs: - Nondeterministic processes can be represented as presheav... |

11 |
Cattani and Glynn Winskel. Presheaf models for concurrency
- Luca
- 1996
(Show Context)
Citation Context ... tree. That, for example F * preserves open map bisimulation implies that two hereditary history preserving bisimilar event structures are sent to strongly bisimilar synchronisation trees. The papers =-=[13, 14]-=- contain several examples directly using this result, including a characterisation of a well known refinement operation on event structures [18] as an instance of F!. 4 The bicategory Prof and the 2-c... |

11 | Pseudo-distributive laws
- Cheng, Hyland, et al.
(Show Context)
Citation Context ...cal issues. The use in this paper of pseudo-comonads predated and to some extent motivated Cheng, Hyland and Power's systematic definition and study of pseudo monads and their attendant constructions =-=[15]-=-. We refer the reader to that work and the recent work of Power and Tanaka [46, 49] for the definitions and results of pseudo-monads and pseudo-comonads 2Another place is in the work of Matthew Hennes... |

10 |
Locally Presentable and Accessible
- Adámek, Rosick´y
- 1994
(Show Context)
Citation Context ...isation to categories, in which a category is completed with all filtered colimits (see [33] for a discussion of filtered categories and colimits). 18sDefinition 7.1 (Completion by filtered colimits) =-=[35, 2]-=- Let P be a small category. We write eP for the full subcategory of bP consisting of presheaves whose categories of elements (see Definition A.13) are filtered. As the category of elements of each rep... |

9 |
Categories for Fixpoint Semantics
- Lehmann
- 1976
(Show Context)
Citation Context ...sets, as in \Omegasabove, but this time allowing all functions as maps.) In this case FF(P) is the finite coproduct completion of a small categoryP (a construction dual to the categorical powerdomain =-=[32, 1]-=-). Clearly F has singletons. It has a dependent sum given by disjoint union. There is an isomorphism FF(P) \OmegasFF(Q) ,= FF(P&Q) expressing how a family in FF(P&Q) can be broken down into a pair of ... |

6 | Categorical Models for Concurrency: Independence, Fairness and Dataflow
- Hildebrandt
- 1999
(Show Context)
Citation Context ...ave been successes in applying the mathematics, in connecting with process languages and 38soperational semantics [40, 41], the semantics of nondeterministic dataflow [23], independence/causal models =-=[22, 39]-=-, fairness [21], pi-Calculus and name generation for higher order processes [12, 57], and weak bisimulation [16]. These are all examples of how we can bring categorical reasoning to bear on issues of ... |

5 |
Handbook of categorical algebra I, volume 50 of Encyclopedia of Mathematics and its Applications
- Borceux
- 1994
(Show Context)
Citation Context ...ng functors bP ! bQ correspond to functorsP ! bQ , which correspond by "uncurrying" to functors P * Qop ! Set. Functors of this latter kind are often called profunctors (or bimodules or distributors) =-=[5, 31, 4]-=-. For a functor F : P*Qop ! Set, we write F : P + Q to signify that F is a profunctor from P to Q. Often operations are best described on profunctors, which provide an alternative (bicategorical) pres... |

5 |
Presheaf models for the ss-calculus
- Cattani, Stark, et al.
- 1997
(Show Context)
Citation Context ...ng constitutes the basic prefix operation in the presheaf semantics of affine HOPLA, the higher order affine language in [41], and underlies the semantics of many essentially affine process languages =-=[51, 12, 52, 53]-=-.) They also play a key role in harnessing open map preservation in Prof to connected colimit preserving functors. Proposition 8.16 The functor b-c : bP ! cP? preserves surjective open maps. Proof: In... |

4 |
G.: A Higher-Order Calculus for Categories
- Cáccamo, Winskel
- 2001
(Show Context)
Citation Context ...; a correct typing judgement will ensure the functoriality of a term in its free variables. Such judgements can be accompanied by a useful catalogue of natural isomorphisms of the kind appearing here =-=[7, 6]-=-. 40sDefinition A.1 (Dinatural transformations) Let F, G : Cop * C ! D be two functors. A dinatural transformation ff : F ..-! G from F to G consists of a family of arrows (ffC : F (C, C) ! G(C, C))C2... |

4 |
Limit preservation from naturality
- Caccamo, Winskel
- 2005
(Show Context)
Citation Context ...nt to ensure that G preserves the colimit. However, with minor side conditions, naturality of the isomorphism in d does ensure the colimit is preserved. Proofs of the following lemmas may be found in =-=[8, 6]-=-. Lemma A.8 Suppose the category I is small and connected. Suppose categories C, D have initial objects and all I-colimits. A functor G : C ! D preserves I-colimits iff there are isomorphisms `d : G(c... |

4 | Weak bisimulation and open maps (extended abstract
- Fiore, Cattani, et al.
- 1999
(Show Context)
Citation Context ... the compositions with the projections R ,! X * Y ss1! X and R ,! X * Y ss2! Y are surjective open. The following preservation property of open maps along adjunctions will be useful in Section 9 (see =-=[16, 28]-=- for other applications and a related result): Lemma 2.5 If P H A L? B, R are three functors with L left adjoint to R, we have for every arrow g in B, that Rg is H-open iff g is LH-open. Proof: "only ... |

2 |
Les distributeurs. Rapport nffi 33. Seminaires de Math'ematiques Pure, Institut de Math'ematiques, Universit'e Catholique de Louvain
- B'enabou
- 1973
(Show Context)
Citation Context ....4.6 Extensions of functors . . . . . . . . . . . . . . . . . . . . . . . . . . 49s1 Introduction At first sight, it is perhaps surprising that profunctors,1 a categorical generalisation of relations =-=[4, 31]-=-, and bisimulation [37, 38], a central equivalence in the study of processes are intimately related. Briefly, the chain of connections runs: - Nondeterministic processes can be represented as presheav... |

2 | Lecture notes in category theory. Notes for a course given by Glynn Winskel based - Caccamo, Winskel - 1995 |

1 |
18] Rob Van Glabeek and Ursula Goltz. Equivalence notions for concurrent systems and refinement of actions
- logic
- 1987
(Show Context)
Citation Context ...5 Linear logic We might summarise, informally and imprecisely, by saying that Prof is a compact closed bicategory. From a logical point of view, Prof forms an interpretation of classical linear logic =-=[17]-=- once it is equipped with a suitable exponential, and so provides a basis for a rich linear type discipline. Though, as a model of classical linear logic, Prof is somewhat degenerate; the operations &... |