Abstract
In a previous paper (Kasangian and Labella, J Pure Appl Algebra, 2009) we proved a form of Conduché’s theorem for LSymcat-categories, where L was a meet-semilattice monoid. The original theorem was proved in Conduché (CR Acad Sci Paris 275:A891–A894, 1972) for ordinary categories. We showed also that the “lifting factorisation condition” used to prove the theorem is strictly related to the notion of state for processes whose semantics is modeled by LSymcat-categories. In this note we resume the content of Kasangian and Labella (J Pure Appl Algebra, 2009) in order to generalise the theorem to other situations, mainly arising from computer science. We will consider PSymcat-categories, where P is slightly more general than a meet-semilattice monoid, in which the lifting factorisation condition for a PSymcat-functor still implies the existence of a right adjoint to its corresponding inverse image functor.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bunge, M., Fiore, M.P.: Unique factorization lifting functors and categories of linearly-controlled processes. Math. Structures Comput. Sci. 10(2), 137–163 (2000)
Conduché, F.: Au sujet de l’existence d’adjoints à droîte aux foncteurs “image reciproque” dans la catégorie des catégories. C. R. Acad. Sci. Paris 275, A891–A894 (1972)
Eilenberg, S., Kelly, G.: Closed categories. In: Proc. of the Conference on Categorical Algebra, La Jolla 1965, pp. 421–562. Springer, New York (1966)
Fiore, M.P.: Fibered models of processes: discrete, continuous and hybrid systems. In: Proc. of IFIP TCS 2000, LNCS, vol. 1872, pp. 457–473 (2000)
Kasangian, S., Labella, A.: On continuous time agents. In: Mathematical Foundations of Programming Semantics, LNCS, vol. 598, pp. 403–425. Springer, Berlin (1992)
Kasangian, S., Labella, A.: Observational trees as models for concurrency. Math. Structures Comput. Sci. 9, 687–718 (1999)
Kasangian, S., Labella, A.: Conduché’s property and Tree-based categories. J. Pure Appl. Algebra (2009, in press)
Kasangian, S., Labella, A., Murphy, D.: Process synchronisation as fusion. Appl. Categ. Structures 4(4), 403–421 (1996)
Kelly, M., Labella, A., Schmitt, V., Street, R.: Categories enriched on two sides. J. Pure Appl. Algebra 168(1&2), 53–98 (2002)
Kelly, G.: Basic Concepts of Enriched Category Theory. Cambridge University Press, Cambridge (1982)
Labella, A.: Categories with sums and right distributive tensor product. J. Pure Appl. Algebra 178(3), 273–296 (2003)
Lawvere, F.W.: Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano XLIII, 135–166 (1973). Available in Reprints in Theory and Applications of Categories
Lawvere, F.W.: Thermodynamics of deformations of continuous bodies, non homogeneous, with memory, far from equilibrium. Handwritten notes from a seminar held in Trieste, July (1982)
Milner, R.: Communication and Concurrency. Prentice Hall, Englewood Cliffs (1989)
Park, D.: Concurrency and automata on infinite sequences. In: Proc. of Theoretical Computer Science, LNCS, vol. 104, pp. 167–183. Springer, Berlin (1981)
Walters, R.: Sheaves and Cauchy-complete categories. Cahiers Topologie Géom. Différentielle 22, 283–286 (1981)
Author information
Authors and Affiliations
Corresponding author
Additional information
As a further output of the “Australian-Italian axis”, this paper is dedicated to Max Kelly whose invitation to the first author commenced this international collaboration,which continued also with the second author.
Rights and permissions
About this article
Cite this article
Kasangian, S., Labella, A. & Montoli, A. Generalising Conduché’s Theorem. Appl Categor Struct 19, 277–292 (2011). https://doi.org/10.1007/s10485-009-9200-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-009-9200-9