G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, volume 1258 of LNCS, pages 5875, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....components. This opens the way towards a possibility of verifying certain properties of complex systems in a modular way, thus increasing eciency. Among existing approaches to concurrency based on berings, sheaf and presheaf theory we mention [MP86] Pfa91] Gog92] Lil93] Mal94] Win96] [CW96]. The starting point of our research is the work of Pfalzgraf [Pfa91] and the ideas of Goguen [Gog92] and Monteiro and Pereira [MP86] The idea of modeling states, actions and transitions by sheaves with respect to a topological space, and of using geometric logic for studying the link between ....

G.L. Cattani and G. Winskel. Presheaf models for concurrency. In D. van Dalen and M. Bezem, editors, Proceedings of Computer Science Logic '96, LNCS 1258, pages 58-75. Springer Verlag, Berlin, 1996.

....leads to familiar notions of bisimulation. One may then ask how one identifies a suitable category of paths in a model for concurrency; or even, if it is possible to construct a model for concurrency, given a category of computation paths. An answer is given by presheaf models for concurrency [CW96]. A presheaf over a category P is a contravariant functor into the category of sets, P Set. The category of presheaves, P, has presheaves as objects, and natural transformations as morphisms. A category P is a subcategory of the presheaf category P via the Yoneda embedding. In fact, P ....

....relabelling, and fibrations model inverse relabelling and restriction. Furthermore, because a category P is embedded in the presheaf category P, the presheaf category comes with a notion of bisimulation [JNW96] and weak bisimulation [FCW99] Presheaf models have been given to CCS like languages [CW96], a value passing version of CCS [Win96] and the p calculus [CSW97] An approach towards a theory of domains for concurrency based on presheaf models is presented in [CFW98] Presheaf models differ from the models presented before in that they do not present processes as morphisms, and types as ....

[Article contains additional citation context not shown here]

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In Proc. CSL, 1996.

....have more structure than traditional traces, e.g. allowing path semantics to take nondeterministic branching into account in a limited way. For example, path equivalence is related to simulation equivalence in Sect. 3.5 below. The other path based approach is that of categories of presheaf models [14] in which processes denote mappings from computation paths to sets of realisers saying how each computation path may be realised. This extra structure allows the incorporation of complete branching information, and the corresponding notion of process equivalence is a form of bisimulation [26] ....

G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

....es for a typical path p the set X(p) of computation paths of shape p, and acts on e : p q to give a function X(e) saying how paths of shape q restrict to paths of shape p. In this way a presheaf can model the nondeterministic branching of a process. For more information of presheaf models, see [14, 4]. A presheaf category has all colimits and so in particular all sums (coproducts) for any set I , the sum i2I X i of presheaves X i over P has a contribution i2I X i (p) the disjoint union of sets, at p 2 P. The empty sum is the presheaf with empty contribution at each p. In process ....

....maps [13, 14] Open maps are a generalization of functional bisimulations, or zig zag morphisms, known from transition systems. Presheaves over P are openmap bisimilar i there is a span of surjective open maps between them. The maps of Lin and A preserve open maps and so open map bisimulation [7, 4], giving congruence results for free when a process language is interpreted in these categories. Interestingly, although the operational bisimulation of Park Milner is a congruence for the higher order language, maps of Cts need not preserve open maps. This suggests that one should look at other ....

G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

....to unify models of concurrency and their notions of bisimulation into one framework, a domain theory based on presheaf models and open map bisimulation. The endeavor was initiated by Joyal, Nielsen and Winskel in [15] and pursued further in two PhD theses [3, 11] and a number of papers including [6, 20, 5, 4, 13, 21, 9, 12, 7]. Presheaf models are instantiations of the following situation: Let P be a small category in which objects are viewed as computation path shapes with morphisms saying how paths can be extended. The path category P embeds via Yoneda in the category = P , Set] of presheaves over P in which ....

....with arrows colimit preserving functors between the associated presheaf categories, and with natural transformations as 2 cells. The 2 category is equivalent to a bicategory Prof with the same objects, but with profunctors (which are often easier to work with) as arrows [7, 1, 2] It was shown in [6] that open map bisimulation is preserved along the arrows of Cocont, giving rise to congruence results for free when one interprets terms of a process language as such arrows. In fact, the arrows need only preserve connected colimits to preserve open map bisimulation, and this is exploited in ....

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, number 1258 in Lecture Notes in Computer Science, pages 58--75. Springer-Verlag, 1997.

....context of this work is an ongoing effort to unify models of concurrency and their notions of bisimulation into one framework, a domain theory based on presheaf models and open map bisimulation. The endeavor was initiated by Joyal, Nielsen and Winskel in [15] and pursued further in two PhD theses [3, 11] and a number of papers including [6, 20, 5, 4, 13, 21, 9, 12, 7] Presheaf models are instantiations of the following situation: Let P be a small category in which objects are viewed as computation path shapes with morphisms saying how paths can be extended. The path category P embeds via Yoneda ....

Gian Luca Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999.

....(co)monads on a biequivalent 2 category. At present this theory, even the definitions, are not sufficiently developed (though remedial work has started with Martin Hyland and John Power) Acknowledgements A good deal of the background for this work was developed with Gian Luca Cattani for his PhD [6]. Discussions with Martin Hyland and John Power have played a crucial role in the work on nonlinearity. ....

G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

....allowing process passing, though there remains the problem of relating the notion of bisimulation obtained from open maps to a more traditional definition from the operational semantics. A tentative beginning is made of a domain theory supporting presheaf models. Introduction The papers [12, 4] explore presheaf models for concurrency. Here begins an investigation of the use of presheaves to model higher order features, most dramatic in the situation of process calculi where processes can be communicated as values. Something of higher order appears even in value passing process calculi ....

....path lifting property of h see [12] Say presheaves X;Y are bisimilar iff there is a span of surjective open maps between them, equivalently, iff there is R , X Theta Y such that the compositions with the projections R , X Theta Y X and R , X Theta Y are surjective open. In [12, 4] we defined bisimulation between rooted presheaves, presheaves X, over a category assumed to have an initial object I, for which X(I) is a 4 singleton. For rooted presheaves bisimulation is defined merely through the presence of a open maps (not requiring surjectivity) This is because open maps ....

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Cattani, G.L., and Winskel, G., Presheaf models for concurrency. Manuscript, 1996.

....This yields a form of domain theory but boosted to the level of using categories rather than partial orders as the appropriate domains. One argument for the definition of bisimulation based on open maps is the powerful preservation properties associated with it. Notable is the result of [8] that any colimit preserving functor between presheaf categories preserves bisimulation, Basic Research in Computer Science, Centre of the Danish National Research Foundation. which besides obvious uses in relating semantics in different models with different notions of bisimulation is, along ....

....of bisimulation is all well and good but it needs to be tested and its consequences understood for a range of process languages. Another argument in favour of the presheaf approach to bisimulation is that when tried against traditional process languages it yields persuasive results, as in [8, 40, 7]. But still these are just examples and it is hoped that a more satisfying and conclusive argument will come from an endeavour to ascertain the operational content of presheaf models more generally. One difficulty has been in understanding the operational significance of the bisimulation which ....

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CATTANI, G. L., AND WINSKEL, G. Presheaf models for concurrency. In CSL'96, vol. 1258 of LNCS, Springer, pp. 58--75. 17

....by event structures when the morphisms on event structures are strict in that they preserve the partial order of causal dependency. 1 Introduction Presheaves have been advanced as a model of nondeterministic processes which supports a notion of bisimulation and as well extends to higher order [7, 12, 5, 4, 3, 10, 13, 6, 2]. At the start of this work, the paper [7] showed that the category of (labelled) event structures embedded fully and faithfully in the category of presheaves over pomsets; the embedding arises canonically from the fact that pomsets can be regarded as event structures. The paper [7] gave several ....

G. L. Cattani. Presheaf models for concurrency. PhD thesis, University of Aarhus, 1999. Forthcoming.

....indeed distinguishes the two agents given above. The background of the present paper is the work on presenting models for concurrency categorically as initiated by Winskel and Nielsen [23] and developed further in the work on bisimulation from open maps [12] and presheaf models for concurrency [3, 6, 9, 22]. Our goal is twofold: We want to extend the categorical approach (in which the issue of infinite computations and fairness has been absent so far) to models for infinite computations and we want to give a denotational semantics to SCCS with finite delay which captures a behavioural equivalence ....

....of bisimulation from open maps introduced in [12] from which one gets an abstract behavioural equivalence by choosing a path category, 3 i.e. a subcategory of the model at issue identifying the observable computations. The open maps approach increased its worth through the further development [3, 6, 5, 22] of the presheaf models for concurrency proposed in [12] Here one starts with a path category P and then takes the category b P of presheaves over P as model, justified categorically by being the free colimit completion of P [6] Now any presheaf model b P comes with a canonical notion of ....

[Article contains additional citation context not shown here]

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In CSL'96, volume 1258 of LNCS, pages 58--75. Springer, 1997.

....indeed distinguishes the two agents given above. The background of the present paper is the work on presenting models for concurrency categorically as initiated by Winskel and Nielsen [23] and developed further in the work on bisimulation from open maps [12] and presheaf models for concurrency [3, 6, 9, 22]. Our goal is twofold: We want to extend the categorical approach (in which the issue of infinite computations and fairness has been absent so far) to models for infinite computations and we want to give a denotational semantics to SCCS with finite delay which captures a behavioural equivalence ....

....of bisimulation from open maps introduced in [12] from which one gets an abstract behavioural equivalence by choosing a path category, 3 i.e. a subcategory of the model at issue identifying the observable computations. The open maps approach increased its worth through the further development [3, 6, 5, 22] of the presheaf models for concurrency proposed in [12] Here one starts with a path category P and then takes the category b P of presheaves over P as model, justified categorically by being the free colimit completion of P [6] Now any presheaf model b P comes with a canonical notion of ....

[Article contains additional citation context not shown here]

G. L. Cattani. Presheaf models for concurrency. PhD thesis, Aarhus University, 1999.

....definitions and standard examples. The paper [71] also suggested a class of abstract models for concurrency, the presheaf models for concurrency, which came with a canonical notion of bisimulation from open maps. The general theory of presheaf models have been developed over the last 5 years [71, 22, 143, 147, 21, 144, 19, 24, 37], culminating in the recent thesis of Cattani. We summarise some of the most important results in Sec. 2.3. 2.1 Models for Concurrency The work reported in the chapter [146] was initiated by work of Winskel [140] followed by work of Nielsen, Rozenberg and Thiagarajan [99] and Sassone et al. ....

....key to understanding how respectively fairness and feedback could be interpreted operationally correct in the abstract presheaf models. Bisimulation congruence and a domain theory for concurrency. The work on presheaf models which have followed [71] in particular the recent thesis of Cattani [24], has redeemed many of the initial promises and intuitions. From the view of presheaf categories as a concrete representations of free colimit completions (or connected colimit completions) it follows that they can themselves can be naturally related in a category of cocontinuous, i.e. colimit ....

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Gina Luca Cattani. Presheaf models for concurrency. PhD thesis, Aarhus University, 1999.

....categories. It has been shown that interpretations of the usual operators of CCSlike languages: action prefix, sum, restriction, and parallel composition can be obtained quite elegantly from the left Kan extension as described in (2. 2) above, in the case where Q is it self a presheaf category [23, 24]. Another important fact is that for any two fixed path categories P and Q, the category of colimit (or connected colimit) preserving functors between and natural transformations between them is equivalent to the presheaf category # P (or # P Presheaves in # P are also ....

....This yields a form of domain theory but boosted to the level of using categories rather than partial orders as the appropriate domains. One argument for the definition of bisimulation based on open maps is the powerful preservation properties associated with it. Notable is the result of [23] that any colimit preserving functor between presheaf categories preserves bisimulation, which besides obvious uses in relating semantics in di#erent models with di#erent notions of bisimulation is, along with several other general results, useful in establishing congruence properties of process ....

[Article contains additional citation context not shown here]

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In Computer Science Logic: Tenth international Workshop, CSL'96, Annual Conference of the EACSL. Selected Papers, number 1258 in Lecture Notes in Computer Science, pages

....definitions and standard examples. The paper [71] also suggested a class of abstract models for concurrency, the presheaf models for concurrency, which came with a canonical notion of bisimulation from open maps. The general theory of presheaf models have been developed over the last 5 years [71, 22, 143, 147, 21, 144, 19, 24, 37], culminating in the recent thesis of Cattani. We summarise some of the most important results in Sec. 2.3. 2.1 Models for Concurrency The work reported in the chapter [146] was initiated by work of Winskel [140] followed by work of Nielsen, Rozenberg and Thiagarajan [99] and Sassone et al. ....

.... in # P are also referred to as profunctors between P and Q , which forms the arrows of a bicategory Prof and the equivalence above gives an equivalence between Prof and the category of colimit preserving functors[15, 24] This opens up for a domain theory for concurrency as explored in [22, 143, 18], replacing domains and Scott continuous functions with presheaf models and cocontinuous (or connected colimit preserving) functors. This line of work is followed in [144] proposing a meta language for (linear) higher order CCS like languages which can be given a categorical semantics [31] in ....

[Article contains additional citation context not shown here]

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In CSL'96,

.... by choosing a path category , or to be a bit more general [14] a functor from a category of path shapes to the model at issue, identifying the observable computations (in [9] assumed to be the inclusion of a subcategory) The open maps approach gained ground through the further development [10,11,15,13,16] of the presheaf models for concurrency also proposed in [9] Here one starts with a path category P (of non empty path shapes) and then takes the category b P of presheaves over P as model. The categorical justi cation [10,16] for doing this, is the fact that Y P : P b P, the strict ....

....P : P b P, the strict extension of the well known Yoneda embedding, is the free connected colimit completion of the category P obtained by adding a new initial object to P. By the embedding Y P any presheaf model b P comes with a canonical notion of bisimulation from open maps. In [11,13,10] it is shown that presheaf models themselves can be related within a category in which arrows are connectedcolimit preserving functors, that such functors in fact preserve the canonical bisimulation and general techniques for their construction are provided. 3 Perhaps the simplest example of a ....

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G. L. Cattani, G. Winskel, Presheaf models for concurrency, in: CSL'96, Vol. 1258 of LNCS, Springer, 1997, pp. 58-75.

.... and the material also appears as a chapter in the authors PhDthesis [7] Its background is the work on presenting models for concurrency categorically as initiated by Winskel and Nielsen [8] and developed further in the work on bisimulation from open maps [9] and presheaf models for concurrency [10 13]. Our goal is to extend the categorical approach (in which the issue of in nite computations and fairness has been absent so far) to models for in nite computations. As quality check, we want to apply the models to give both operational and denotational semantics for SCCS with nite delay, and ....

.... by choosing a path category , or to be a bit more general [14] a functor from a category of path shapes to the model at issue, identifying the observable computations (in [9] assumed to be the inclusion of a subcategory) The open maps approach gained ground through the further development [10,11,15,13,16] of the presheaf models for concurrency also proposed in [9] Here one starts with a path category P (of non empty path shapes) and then takes the category b P of presheaves over P as model. The categorical justi cation [10,16] for doing this, is the fact that Y P : P b P, the strict ....

[Article contains additional citation context not shown here]

G. L. Cattani, Presheaf models for concurrency, Ph.D. thesis, Aarhus University (1999).

....Models for Fairness: Completion vs Delay Thomas T. Hildebrandt The IT University of Copenhagen, Denmark, hilde it c. dk Abstract We study two different approaches to semantics for fairness within the categorical framework of presheaf models for concurrency [1, 2]. The first approach, used in e.g. dataflow models, is based on representing finite or infinite completed observations. A completed observation is the result of an infinite computation. A finite completed observation is then the result of an infinite computation that only produces finitely many ....

G. L. Cattani. Presheaf models for concurrency. PhD thesis, Aarhus University, 1999.

....leads to familiar notions of bisimulation. One may then ask how one identifies a suitable category of paths in a model for concurrency; or even, if it is possible to construct a model for concurrency, given a category of computation paths. An answer is given by presheaf models for concurrency [CW96]. A presheaf over a category P is a contra variant functor into the category of sets, P op Set. The category of presheaves, P, has presheaves as objects, and natural transformations as morphisms. A category P is a subcategory of the presheaf category P via the Yoneda embedding. In fact, P ....

....relabelling, and fibrations model inverse relabelling and restriction. Furthermore, because a category P is embedded in the presheaf category P, the presheaf category comes with a notion of bisimulation [JNW96] and weak bisimulation [FCW99] Presheaf models have been given to CCS like languages [CW96], a value passing version of CCS [Win96] and the p calculus [CSW97] An approach towards a theory of domains for concurrency based on presheaf models is presented in [CFW98] Presheaf models differ from the models presented before in that they do not present processes as morphisms, and types as ....

[Article contains additional citation context not shown here]

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In Proc. CSL, 1996.

.... paper is a revised version of chapter 7 in the authors PhD thesis [15] Its background is the work on presenting models for concurrency categorically as initiated by Winskel and Nielsen [30] and developed further in the work on bisimulation from open maps [19] and presheaf models for concurrency [4, 7, 16, 29]. Our goal is to extend the categorical approach (in which the issue of infinite computations and fairness has been absent so far) to models for infinite computations. As quality check, we 2 want to apply the models to give both operational and denotational semantics for SCCS with finite delay, ....

.... by choosing a path category, or to be a bit more general [12] a functor from a category of path shapes to the model at issue, identifying the observable computations (in [19] assumed to be the inclusion of a subcategory) The open maps approach gained ground through the further development [4, 7, 6, 29, 4, 8] of the presheaf models for concurrency also proposed in [19] Here one starts with a path category P (of non empty path shapes) and then takes the category b P of presheaves over P as model. The categorical justification [4, 8] for doing this, is the fact that Y P : P b P, the strict ....

[Article contains additional citation context not shown here]

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In CSL'96, volume 1258 of LNCS, pages 58--75. Springer, 1997.

.... paper is a revised version of chapter 7 in the authors PhD thesis [15] Its background is the work on presenting models for concurrency categorically as initiated by Winskel and Nielsen [30] and developed further in the work on bisimulation from open maps [19] and presheaf models for concurrency [4, 7, 16, 29]. Our goal is to extend the categorical approach (in which the issue of infinite computations and fairness has been absent so far) to models for infinite computations. As quality check, we 2 want to apply the models to give both operational and denotational semantics for SCCS with finite delay, ....

.... by choosing a path category, or to be a bit more general [12] a functor from a category of path shapes to the model at issue, identifying the observable computations (in [19] assumed to be the inclusion of a subcategory) The open maps approach gained ground through the further development [4, 7, 6, 29, 4, 8] of the presheaf models for concurrency also proposed in [19] Here one starts with a path category P (of non empty path shapes) and then takes the category b P of presheaves over P as model. The categorical justification [4, 8] for doing this, is the fact that Y P : P b P, the strict ....

[Article contains additional citation context not shown here]

G. L. Cattani. Presheaf models for concurrency. PhD thesis, Aarhus University, 1999.

.... property on the state category [Law86b, Law86c] as for every state X in X and every control path fl : f(X) C there should exist a unique evolution e : X Y in X along fl (i.e. such that f(e) fl) The second case corresponds essentially to Winskel s et al. presheaf models [JNW96, CW97, WN97]. It allows non determinism but, as we explain in Section 1, imposes a particular notion of state on the state category. The approach adopted here will be to consider a more general notion of control functor; viz. unique factorisation lifting (ufl) functors . Our motivation for doing so comes ....

....may be embedded (as a full reflective subcategory) in a presheaf topos; viz. c C # where C # is Mac Lane s twisted arrow category [ML71] On the one hand, this result establishes connections with presheaf models; thus opening the possibility of reusing the techniques developed in that setting [CW97]. On the other hand, it highlights important differences. First, the path category C # is obtained from the control category C and this passage equips the former with extra structure; notably notions of cover. Second, presheaf models can be over generous. For instance, by considering all ....

[Article contains additional citation context not shown here]

G.L. Cattani and G. Winskel. Presheaf models for concurrency. In D. van Dalen and M. Bezem, editors, Proceedings of the CSL'96 Conf., volume 1258 of Lecture Notes in Computer Science, pages 58--75. Springer-Verlag, 1997.

No context found.

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

....These categories are very rich in structure. They point towards more expressive languages than HOPLA and AL. In particular, the a#ne category accommodates the independence model of event structures to the extent of supporting the standard event structure semantics of CCS and related languages [12], as well as the trace of nondeterministic dataflow [23] In fact, AL can be given an event structure semantics which at first order provides a representation of the presheaf denotations of AL. Nevertheless, it is here we meet the limitations of AL, and HOPLA. They can be shown not to support ....

....of Lin. 28 There is a category analogous to A# based on presheaves rather than path sets; its maps preserve connected colimits in presheaf categories [39, 15] This a#ne category is host to the semantics of nondeterministic dataflow [23] event structure semantics of CCS and related languages [12] as well as a semantics for AL. It came as a recent surprise [38] that the presheaf denotations of first order processes in AL can be represented by event structures; the elements of definable presheaves can be understood as finite configurations of an event structure. In more detail, maps ....

[Article contains additional citation context not shown here]

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

....constructions arise in manipulating that structure. For concreteness, we restrict our attention for the bulk of the paper to the operations needed in modelling CCS processes by synchronisation trees. We occasionally refer to more involved examples that were treated using presheaf categories in [22, 6, 5]. We rst observe that, given a category of observations, P, the basic operations of CCS lead us to consider the free completion of the category Punder countable colimits: for our choice of P, that is equivalent to the category of countably branching trees, as we shall see in Section 2. The ....

....It also gives us a candidate for higher order structure (though outside Kl(T ) allowing a possible way to model a process passing extension of CCS. This analysis forms Sections 4 and 5. Finally, in Section 6, we compare, especially as far as higher order is concerned, this work with that of [22, 6, 5, 7] using presheaves and profunctors to model process calculi, and we suggest directions for future research. We do not address weak bisimulation at all here. In no way do we suggest that it is unimportant. But it is such a large issue that it requires a full paper devoted to it. There are delicate ....

[Article contains additional citation context not shown here]

G. L. Cattani and G. Winskel. Presheaf Models for Concurrency. In Proceedings of CSL' 96, LNCS 1258, pages 58-75, 1997.

....observational congruence on presheaf models. Presheaf models have been shown to include traditional models like synchronisation trees and event structures [13] along with their notion of bisim ulation, to be related by powerful preservation properties associated with colimit preserving functors [9], and to form a domain theory for bisimulation [23, 7, 6] in which a wide range of, possibly higher order, process languages can receive a denotational semantics. In [17] an approach to an open map account of weak bisimulation on transition systems is based on weak morphisms , a reconstruction ....

....canonical choice for open map bisimulation in a presheaf category [13] Presheaf categories subsume more traditional categories of models. It was suggested in [13] that they deserve to be studied as categories of models for concurrency in their own right. Subsequent work on this line appeared in [9, 23, 26, 8, 7, 11, 6]. 5 Synchronisation trees. A simple example of a presheaf model is given by synchronisation trees. A synchronisation tree is a transition system whose transition graph is a tree with root the initial state. Definition 2.1 For a set of actions A, let STA be the full subcategory of TSA of ....

[Article contains additional citation context not shown here]

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, volume 1258 of LNCS, pages 58--75. Springer-Verlag, 1997.

.... models have been shown to include traditional models like synchronisation trees and event structures [13] along with their notion of bisim ulation, to be related by powerful preservation properties associated with colimit preserving functors [9] and to form a domain theory for bisimulation [23, 7, 6] in which a wide range of, possibly higher order, process languages can receive a denotational semantics. In [17] an approach to an open map account of weak bisimulation on transition systems is based on weak morphisms , a reconstruction of the maps between transition systems to account for the ....

....s 0 t and f(t) t # . In other words, f not only preserves but also reflects reachable transitions. If f is a zig zag morphism, then its graph is a bisimulation between the two transition systems. In [13] the above situation was axiomatised in terms of the following path lifting property (see [13, 6] for more detailed explanations) Suppose that within a category of models, one can distinguish a subcategory of computation paths (or shapes) P. The objects of P account for runs (or history) of a process and their morphisms tell us how one path can be extended by another one. Given the ....

[Article contains additional citation context not shown here]

G. L. Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999. Forthcoming.

....by event structures when the morphisms on event structures are strict in that they preserve the partial order of causal dependency. 1 Introduction Presheaves have been advanced as a model of nondeterministic processes which supports a notion of bisimulation and as well extends to higher order [7, 12, 5, 4, 3, 10, 13, 6, 2]. At the start of this work, the paper [7] showed that the category of (labelled) event structures embedded fully and faithfully in the category of presheaves over pomsets; the embedding arises canonically from the fact that pomsets can be regarded as event structures. The paper [7] gave several ....

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL' 96, LNCS 1258, pages 58--75, 1997.

.... presheaves associated with event structures, in the same way as the refinement operation on event structures proposed by Goltz and van Glabbeek in [12] To highlight the gain of working at a more abstract level than is common in concurrency theory, we can often exploit an important general result [6, 3] that any colimit preserving functor between presheaf categories automatically preserves open maps. In particular, this result specialises to show that the refinement, obtained as a Kan extension, preserves open maps and so open map bisimulation. One point of approaching models for concurrency as ....

....in the following section the relevant semantic constructions are discussed and proved to preserve bisimulation. Section 5 and 6 present the two applications to hereditary history preserving bisimulation and event structures discussed above. An earlier extended abstract of this paper appeared as [6]. 1 Preliminaries This section recalls three fundamental models for concurrent computation and provides the categorical background. 1.1 Traditional models We focus on three traditional models for concurrency: transition systems, synchronisation trees and event structures (see [29] for more ....

[Article contains additional citation context not shown here]

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In D. van Dalen and M. Bezem, editors, Computer Science Logic. 10th International Workshop, CSL '96, Annual Conference of the European Association for Computer Science Logic. Selected Papers, volume 1258 of Lecture Notes in Computer Science, pages 58--75. Springer-Verlag, 1997.

.... presheaves associated with event structures, in the same way as the refinement operation on event structures proposed by Goltz and van Glabbeek in [12] To highlight the gain of working at a more abstract level than is common in concurrency theory, we can often exploit an important general result [6, 3] that any colimit preserving functor between presheaf categories automatically preserves open maps. In particular, this result specialises to show that the refinement, obtained as a Kan extension, preserves open maps and so open map bisimulation. One point of approaching models for concurrency as ....

....shows that any P factorisable functor preserves open maps and so bisimulation. In contrast we aim to take advantage of the preservation properties of universal constructions a strategy proposed in the conclusion of [19] to the constructions of presheaf models for more sophisticated languages [5, 3]. Other applications, to nondeterministic dataflow, fairness and weak bisimulation, can be found in [16, 15, 10] Specifically, this paper builds on the analysis of the handbook chapter [29] with open map bisimulation in mind, to axiomatise a presheaf based semantics of the general purpose ....

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G. L. Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999.

.... assembled together in the bicategory of profunctors [2] or its equivalent presentation as the 2 category Prof of colimit preserving functors between presheaf categories (with natural transformations as 2 cells) The key facts here are: open maps and so bisimulation are preserved by such functors [4]; the 2 category is rich in constructions which can be summarised as those we expect from a model of classical linear logic [32, 3] open maps are closed under a wide range of constructions [12, 13] We have the basics of a domain theory for concurrency with a compositional account of ....

....fK j Rg Theta fK j Rg fK j Rg. 13 7 Open map bisimulation We provide models of part of the type theory of Section 4 in the Kcats of relations fProf M j Eg and fProf j Ig. In these models, the denotation of a type provides a model for concurrency (in the form of a presheaf category [4]) equipped with a relation. The presheaf models so obtained coincide with those of Section 4, whilst the relations will be shown to be in accordance with open map bisimulation (viz. the relation holding between presheaves that are connected by a surjective open span) Thus, by the results of the ....

[Article contains additional citation context not shown here]

G. L. Cattani and G. Winskel. Presheaf Models for Concurrency. In CSL'96,

....obtained as a colimit preserving functor coincides on event structures with a refinement proposed in [41] and again this entails that such a strong history preserving bisimulation is preserved by this refinement. The results of this chapter were announced in a joint paper with Glynn Winskel [26] that appeared in the proceedings of CSL 96. Chapter 4 is devoted to the study of the bicategory Prof of profunctors or equivalently the 2 category Cocont of presheaf categories, colimit preserving functors and natural transformations. Here we make explicit the categorical folklore about Prof ....

....di#erent fibres along cocartesian liftings preserves bisimulation. But can we deduce more, i.e. can we deduce that bisimulation is a congruence with respect to the operations of the denotational semantics of processes The answer is yes and it relies mainly on the following result (Corollary 4 in [26]) Proposition 3.2.5 (Colimit preserving functors preserve open maps) If C and D are two small categories and F : D is a colimit preserving functor, then F preserves open maps: If # : X Y is a C open map, then F (#) F (X) F (Y ) is a D open Proof: The proof is postponed to ....

[Article contains additional citation context not shown here]

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In van Dalen and Bezem [133], pages 58--75.

....realised that presheaf models can themselves be usefully assembled together in a category in which the maps are colimit preserving functors. There are two main bene ts: one is a general result stating that colimit preserving functors between presheaf categories preserve open maps and bisimulation [6]; the other that the category of the presheaf models is a form of domain theory for concurrency, with a compositional account of bisimulation, though at the cost that domains are categories rather than special partial orders [20, 4] Invited paper AMAST 98, Brazil Basic Research in CS, ....

G. L. Cattani and G. Winskel. Presheaf Models for Concurrency. In Proc. of CSL' 96, LNCS 1258, 1997.

.... distinctions [49] Moreover, the categorical study of process calculi gives the possibility of obtaining general congruence results: in [58] categorical models of CCS like processes are axiomatised and in [31] an abstract model theoretic notion of bisimulation is introduced (via open maps) in [13, 14] these two are combined to give abstract congruence results for strong bisimulation over a wide range of models. It is our hope that the present work serves as a rst step towards similar results for calculus like process languages. In particular, we would like a categorical understanding of our ....

.... calculus early metatheory can be developed in a syntax free style, independently of the exact choice of calculus. In particular, one could show congruence results for operational equivalences with respect to the N LTS operations de ned in Section 4, perhaps developing the open map approach of [31, 13, 14], and could prove characterisation results relating operational equivalences with classes of formulae of suitable modal logics. Model Checking We believe our structures may form a useful basis for calculus interleaving and partial order model checking, via notions of nitely presentable N LTS ....

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In D. van Dalen and M. Bezem, editors, Computer Science Logic. 10th International Workshop, CSL '96, Annual Conference of the European Association for Computer Science Logic. Selected Papers, volume 1258 of Lecture Notes in Computer Science, pages 58-75. Springer-Verlag, 1997.

.... late bisimulation, our focus here is on early semantics, both to obtain a simpler notion of transition system, and because we have found the early style suits work on concurrent language semantics and on secure encapsulation [51, 53, 54, 52] Presheaf models exist for both early and late notions [10]. Moreover we should add that, in contrast to [56, 20] which have full abstraction results wrt. strong bisimulation) we focus on intensional models, over which a number of equivalences can be de ned (though we give results only for bisimulation) The literature contains also testing based models ....

G. L. Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999.

....are an appropriate generalisation. 11 The future To conclude we present some remarks on research directions in the semantics of concurrent computation. These views are admittedly biased, and quite possibly limited to some extent they represent natural continuations of our recent work. The paper (Cattani and Winskel 1996) parallels the work here, but for presheaf models, obtaining abstract results for a whole class of presheaf models, which can then be transferred to concrete models like event structures. In particular, it 12 Although for simplicity we have assumed that path categories are subcategories of the ....

Cattani, G. L. and G. Winskel (1996). Presheaf models for concurrency. To appear as a BRICS report.

.... [SNW96] Moreover, the categorical study of process calculi gives the possibility of obtaining general congruence results: in [WN95] categorical models of CCS like processes are axiomatised and in [JNW96] an abstract model theoretic notion of bisimulation is introduced (via open maps) in [CW97, CW99] these two are combined to give abstract congruence results for strong bisimulation over a wide range of models. It is our hope that the present work serves as a rst step towards similar results for calculus like process languages. In particular, we would like a categorical understanding of our ....

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, volume 1258 of LNCS, pages 5875, 1997.

.... our focus here is on early semantics, both to obtain a simpler notion of transition system, and because we have found the early style suits work on concurrent language semantics and on secure encapsulation [Sew97, SV99a, SV99b, Sew00] Presheaf models exist for both early and late notions [Cat99]. Moreover we should add that, in constrast to [Sta96, FMS96] which have full abstraction results wrt. strong bisimulation) we focus on intensional models, over which a number of equivalences can be de ned (though we give results only for bisimulation) The literature contains also ....

G. L. Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999.

No context found.

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proceedings of CSL'96, volume 1258 of LNCS, pages 5875, 1997.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. PhD thesis, University of Aarhus, 1999.

No context found.

Gian Luca Cattani. Presheaf Models for Concurrency. PhD thesis, BRICS - University of Aarhus, 1999.

No context found.

Gian Luca Cattani. Presheaf Models for Concurrency. PhD thesis, BRICS - University of Aarhus, 1999.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

No context found.

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. BRICS Dissertation Series DS-99-1, 1999.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

No context found.

G. L. Cattani. Presheaf Models for Concurrency. Ph.D. Dissertation, University of Aarhus, 1999. BRICS Dissertation Series DS-99-1.

No context found.

G. L. Cattani and G. Winskel. Presheaf models for concurrency. In Proc. CSL'96, LNCS 1258.

No context found.

Gian Luca Cattani and Glynn Winskel. Presheaf models for concurrency. In Proc. CSL, 1996.

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