Conduch´e’s property and T ree-based categories Stefano Kasangian

Anna Labella

Dip. di Matematica Universit` a di Milano [email protected]

Dip. di Informatica Universit`a di Roma “La Sapienza” [email protected]

November 29, 2007 Abstract According to Lawvere [7], a functor strictly reflecting morphism factorization entails a notion of state on its domain, when it is considered as a control functor. This intuition works both in the case of physical processes and computing processes [1, 5]. In the same vein in this note we aim to investigate a more general property in the family of models we proposed elsewhere for communicating processes in order to assess them and bisimulation relations thereof [6, 3]. The framework is enriched category theory, evolved from the seminal work by Eilenberg and Kelly [4], and we adapted the notion of “Conduch´e condition”[2] to that context. Actually this notion, weaker than the original “Moebius condition” used by Lawvere, seems to be more suitable when one is interested into the parametrisation of models w.r.t. a base category obtained via the change of base machinery. The base category is a monoidal 2-category and a category of generalised trees, T ree, is obtained from it. We consider Conduch´e T reebased categories, where enrichement reflects factorization of objects in the base category. We prove that the analogous of Conduch´e’s theorem holds for those categories. We also show how Conduch´e condition plays a crucial role in modeling concurrent processes and bisimulations between them, in strict relation with the notions of “state preservation” and “determinacy” [8].

References [1] M. Bunge and M.P. Fiore. Unique factorization lifting functors and categories of linearly-controlled processes. Mathem. Structures in Comp. Sci., 10(2):137–163, 2000. [2] F. Conduch´e. Au sujet de l’existence d’adjoints `a droˆıte aux foncteurs “image reciproque” dans la cat´egorie des cat´egories. C.R. Acad. Sci. Paris, 275 (1972), A891-894. [3] R. De Nicola, D. Gorla and A. Labella. T ree-Functors, Determinacy and Bisimulations. Submitted, 2007. [4] S. Eilenberg and G. Kelly. Closed Categories. In Proc. of of the Conference on Categorical Algebra, La Jolla 1965, pages 421–562, Springer, 1966. [5] M.P. Fiore. Fibered Models of Processes: Discrete, Continuous and Hybrid Systems. In Proc. of IFIP TCS 2000, LNCS 1872, pages 457–473, 2000. [6] S. Kasangian, A. Labella. Observational trees as models for concurrency. Math. Struct. in Comp. Sci., 9: 687–718, 1999. [7] F.W. Lawvere. State categories and response functors. Unpublished manuscript, 1986. [8] R. Milner. Communication and concurrency. Prentice Hall International, 1989.

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