Conduché property and Tree-based categories

Abstract This paper focuses on a property of enriched functors reflecting the factorisation of morphisms, used in concurrency semantics. According to Lawvere [F.W. Lawvere, State categories and response functors, 1986, Unpublished manuscript], a functor strictly reflecting morphism factorisation induces a notion of state on its domain, when it is considered as a control functor. This intuition works both in case of physical and computing processes [M. Bunge, M.P. Fiore, Unique factorisation lifting functors and categories of linearly-controlled processes, Math. Structures Comput. Sci. 10 (2) 2000 137–163; M.P. Fiore, Fibered models of processes: Discrete, continuous and hybrid systems, in: Proc. of IFIP TCS 2000, in: LNCS, vol. 1872, 2000, pp. 457–473]. In this note we investigate a more general property in the family of models we proposed elsewhere for communicating processes, and we assess their bisimulation relations [S. Kasangian, A. Labella, Observational trees as models for concurrency, Math. Structures. Comput. Sci. 9 (1999) 687–718; R. De Nicola, D. Gorla, A. Labella, Tree-Functors, determinacy and bisimulations, Technical Report, 02/2006, Dip. di Informatica, Univ. di Roma “La Sapienza” (Italy), 2008 (submitted for publication), http://www.dsi.uniroma1.it/%7Egorla/papers/DGL-TR0206.pdf ]. Hence, we adapt the notion of “Conduche condition” [F. Conduche, Au sujet de l’existence d’adjoints a droite aux foncteurs image reciproque dans la categorie des categories, C. R. Acad. Sci. Paris 275 (1972) A891–894] to the context of enriched category theory. This notion, weaker than the original “Moebius condition” used by Lawvere, seems to be more suitable for the description of the concurrency models parametrised w.r.t. a base category via the mechanism of change of base, actually. The base category is a monoidal 2-category; a category of generalised trees, T r e e , is obtained from it. We consider Conduche  T r e e -based categories, where enrichment reflects factorisation of objects in the base category. We prove that a form of Conduche’s theorem holds for Conduche  T r e e -functors. We also show how the Conduche condition plays a crucial role in modelling concurrent processes and bisimulations between them. The notions of “state preservation” and “determinacy” [R. Milner, Communication and Concurrency, Prentice Hall International, 1989] are formally characterised.

[1]  Anna Labella,et al.  On Continuous Time Agents , 1991, MFPS.

[2]  Charles Ehresmann,et al.  Sheaves and Cauchy-complete categories , 1981 .

[3]  Rob J. van Glabbeek,et al.  Branching time and abstraction in bisimulation semantics , 1996, JACM.

[4]  Anna Labella,et al.  Categories with sums and right distributive tensor product , 2003 .

[5]  Glynn Winskel,et al.  Bisimulation and open maps , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[6]  Marcelo P. Fiore,et al.  Unique factorisation lifting functors and categories of linearly-controlled processes , 2000, Math. Struct. Comput. Sci..

[7]  Glynn Winskel,et al.  Weak bisimulation and open maps , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[8]  Rocco De Nicola,et al.  Tree-functors, determinacy and bisimulations , 2010, Mathematical Structures in Computer Science.

[9]  Anna Labella,et al.  Observational trees as models for concurrency , 1999, Mathematical Structures in Computer Science.

[10]  David Park,et al.  Concurrency and Automata on Infinite Sequences , 1981, Theoretical Computer Science.

[11]  Robin Milner,et al.  Communication and concurrency , 1989, PHI Series in computer science.

[12]  Anna Labella,et al.  Enriched categorial semantics for distributed calculi , 1992 .

[13]  Ross Street,et al.  Variation through enrichment , 1983 .

[14]  Marcelo P. Fiore Fibred Models of Processes: Discrete, Continuous, and Hybrid Systems , 2000, IFIP TCS.

[15]  Anca Muscholl,et al.  Trace Theory , 2011, Encyclopedia of Parallel Computing.

[16]  Stefano Kasangian,et al.  A quasi-universal realization of automata , 1982 .

[17]  G. M. Kelly,et al.  BASIC CONCEPTS OF ENRICHED CATEGORY THEORY , 2022, Elements of ∞-Category Theory.