G.L. Cattani and G. Winskel. Presheaf models for concurrency. Presented at CSL'96 Conf., 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....be available. This might help in settling our first conjecture (see Section 3, x Complete cuboidal sets) which is important for our representation programme. At a more speculative level, we wonder what the relationship is between this work and presheaf or higherdimensional models of concurrency [23, 42, 6, 41, 19, 18, 5, 20]. ....

G.L. Cattani and G. Winskel. Presheaf models for concurrency. Presented at CSL'96 Conf., 1996.

....and 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 bisimulation, 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]. 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 ....

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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 bisimulation, 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 ....

....0 t and f(t) t 0 : 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, M, 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 ....

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

....Basic Research in Computer Science, Centre of the Danish National Research Foundation. 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 [5]; the 2 category is rich in constructions which can be summarised as those we expect from a model of classical linear logic [36, 4] open maps are closed under a wide range of constructions [13, 14] We have the basics of a domain theory for concurrency with a compositional account of ....

....j (R 0 ; R) is a T # bisimulationg : 7 Open map bisimulation We provide models of parts of the type theory of Section 4 in the Kcats of relations fProf M j Extg and fProf j Intg. In these models, the denotation of a type provides a model for concurrency (in the form of a presheaf category [5]) 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 ....

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G. L. Cattani and G. Winskel. Presheaf models for concurrency. In [35], pages 58--75, 1997.

....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 [34,10,9]. We rst observe in Section 1 that, given a category of observations, P, the basic operations of CCS lead us to consider the free completion of the category P under countable colimits: for our choice of P, that is equivalent to the category of countably branching trees. The construction of ....

....also gives us candidates for higher order structure (though some are outside Kl(T ) allowing a possible way to model a process passing extension of CCS. This analysis forms Sections 6 and 7. Finally, in Section 8, we compare, especially as far as higher order is concerned, this work with that of [34,10,9,11] 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 a large issue and the ideas of how to extend weak bisimulation to presheaf models ....

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

....more usual to consider nitely branching trees, or trees for which the branching is limited, for instance to being countable. Second, in the case of the combined presence of higher order and names as in the Higher Order calculus, it is not clear whether the presheaf construction is su cient (see [9,7] for a more detailed discussion) If we can give an axiomatic account of some of the relevant constructions, we are in a better position to address such issues. An axiomatic account also clari es the reasoning behind the various decisions. So in this paper, we do not take presheaf categories for ....

....generally, Proposition 5 For any regular cardinal , the category of synchronisation trees with branching less than is the free completion T L of L under colimits of size less than . This line of argument applies not only to strings but to a range of notions of path objects (see [34,9,7]) giving one reason to consider, for any small category C of path objects, the categories T C , T 1 C , and more generally T C of free colimit completions of C under nite, countable, and less than size colimits respectively. We shall soon have other reasons to consider T C for ....

[Article contains additional citation context not shown here]

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

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