Conduché property and Tree-based categories

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 According to Lawvere [F.W. Lawvere, State categories and response functors, 1986, Unpublished manuscript], a functor strictly reflecting morphism factorisation induces a notion of state on its domain, when it is considered as a control functor.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 “Conduché condition” [F. Conduché, Au sujet de l'existence d'adjoints à droîte aux foncteurs image reciproque dans la catégorie des catégories, 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, Tree, is obtained from it. We consider Conduché Tree-based categories, where enrichment reflects factorisation of objects in the base category. We prove that a form of Conduché's theorem holds for Conduché Tree-functors. We also show how the Conduché 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.