Components of the fundamental category. II. (English) Zbl 1131.55011
The fundamental category of a pospace, i.e. a topological space equipped with a partial ordering, is the small category with objects the points of the space and with morphisms the dihomotopy classes of nondecreasing continuous maps (called dimaps) from the segment \([0,1]\) endowed with the usual ordering to the pospace. The composition law is induced by the concatenation of paths. A dihomotopy between two dimaps is in this paper a dimap in the space of dimaps preserving the two extremities [M. Grandis, Cah. Topol. Géom. Différ. Catég. 44, No. 4, 281–316 (2003; Zbl 1059.55009)]. A van Kampen theorem is known for fundamental categories. This result is useful for inductively computing the fundamental categories of pospaces representing concurrent systems. Unfortunately, the fundamental category is often not small enough, with uncountably many objects and morphisms. A notion of component category of a fundamental category, actually of any small category satisfying an additional hypothesis (the set of isomorphisms must be pure), is introduced in this paper. It is an improvement of the preceding construction given in [L. Fajstrup, M. Raussen, E. Goubault and E. Haucourt, Appl. Categ. Struct. 12, No. 1, 81–108 (2004; Zbl 1078.55020)] which enables the authors to compress the geometric information contained in the fundamental category. It is defined as the categorical localization of the fundamental category by the greatest Yoneda system. A Yoneda system is a set of morphisms (called Yoneda morphisms) satisfying various conditions. In particular, every Yoneda morphism preserves the past and future cones. The existence of the greatest Yoneda system comes from the fact that the set of Yoneda systems of the fundamental category of a pospace is a locale.
The main result of the paper is a van Kampen theorem for component categories allowing inductive computations of the component category of pospaces associated with concurrent systems. Other detailed results about component categories are available in [E. Haucourt, Theory Appl. Categ. 16, 736–770, electronic only (2006; Zbl 1117.18006)]. An application to static analysis of concurrent programs was described in [E. Goubault and E. Haucourt, CONCUR 2005 – concurrency theory. 16th international conference, CONCUR 2005, San Francisco, CA, USA, August 23–26, 2005. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 3653, 503–517 (2005; Zbl 1134.68440)].
The main result of the paper is a van Kampen theorem for component categories allowing inductive computations of the component category of pospaces associated with concurrent systems. Other detailed results about component categories are available in [E. Haucourt, Theory Appl. Categ. 16, 736–770, electronic only (2006; Zbl 1117.18006)]. An application to static analysis of concurrent programs was described in [E. Goubault and E. Haucourt, CONCUR 2005 – concurrency theory. 16th international conference, CONCUR 2005, San Francisco, CA, USA, August 23–26, 2005. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 3653, 503–517 (2005; Zbl 1134.68440)].
Reviewer: Philippe Gaucher (Paris)
MSC:
| 55U40 | Topological categories, foundations of homotopy theory |
Keywords:
partially ordered space; pospace; dihomotopy; fundamental category; category of fractions; component; pure systemReferences:
| [1] | Bednarczyk, M., Borzyszkowski, A., Pawlowski, W.: Generalized congruences-Epimorphisms in CAT. Theory Appl. Categ. 5(11), (1999) · Zbl 0936.18002 |
| [2] | Borceux, F.: Handbook of categorical algebra 1 : basic category theory. In: Encyclopedia of Mathematics and Its Applications, vol. 50. Cambridge University Press, Cambridge, MA (1994a) · Zbl 0803.18001 |
| [3] | Borceux, F.: Handbook of categorical algebra 3 : categories of sheaves. In: Encyclopedia of Mathematics and Its Applications, vol. 52. Cambridge University Press, Cambridge, MA (1994b) · Zbl 0911.18001 |
| [4] | Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin Heidelberg New York (1999) · Zbl 0988.53001 |
| [5] | Cohen, R.L., Jones, D.S., Segal, G.B.: Morse theory and classifying spaces. Technical report, Warwick University Preprint (1995) · Zbl 0843.58019 |
| [6] | Dijkstra, E.: Cooperating Sequential Processes. Academic, New York (1968) |
| [7] | Dodson, C., Poston, T.: Tensor Geometry, 2nd edn. Springer, Berlin Heidelberg New York (1997) · Zbl 0732.53002 |
| [8] | Eilenberg, S.: Ordered topological spaces. Amer. J. Math. (63), 39–45 (1941) · Zbl 0024.19203 |
| [9] | Fajstrup, L., Goubault, E., Haucourt, E., Raussen, M.: Component categories and the fundamental category. APCS 12(1), 81–108 (2004) · Zbl 1078.55020 |
| [10] | Fajstrup, L., Goubault, E., Raussen, M.: Algebraic topology and concurrency. Theor. Comp. Sci. 357, 241–278 (2006) (and also technical report, Aalborg University 1999) · Zbl 1099.55003 · doi:10.1016/j.tcs.2006.03.022 |
| [11] | Freyd, P., Scedrov, A.: Categories, allegories. In: Mathematical Library, vol. 39. North-Holland, Amsterdam (1990) · Zbl 0698.18002 |
| [12] | Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theory. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, no. 35. Springer, Berlin Heidelberg New York (1967) · Zbl 0186.56802 |
| [13] | Gaucher, P.: A model category for the homotopy theory of concurrency. Homology, Homotopy Appl. 5(1), 549–599 (2003) · Zbl 1069.55008 |
| [14] | Goubault, E.: Some geometric perspectives in concurrency theory. Homology, Homotopy Appl. 5(2), 95–136 (2003) · Zbl 1034.68059 |
| [15] | Goubault, E., Haucourt, E.: A practical application of geometric semantics to static analysis of concurrent programs. In: Abadi, M., de Alfaro, L. (eds.), CONCUR 2005 – Concurrency Theory: 16th International Conference, San Francisco, USA, August 23–26. Lecture Notes in Computer Science, vol. 3653, pp. 503–517. Springer, Berlin Heidelberg New York (2005) · Zbl 1134.68440 |
| [16] | Goubault, E., Raussen, M.: Dihomotopy as a tool in state space analysis. In: Rajsbaum, S. (ed.), LATIN 2002: Theoretical Informatics. Lecture Notes in Computer Science, Cancun, Mexico, vol. 2286, pp. 16–37. Springer, Berlin Heidelberg New York (2002) · Zbl 1059.68072 |
| [17] | Grandis, M.: Directed homotopy theory, I. The fundamental category. Cahiers Topologie Géom. Différentielle Catég 44, 281–316 (2003) (Preliminary version: Dip. Mat. Univ. Genova, Preprint 443 (Oct 2001); Revised: 5 Nov 2001, 26 pp (2003)) |
| [18] | Grandis, M.: The shape of a category up to directed homotopy. Theory Appl. Categ. 15(4), 95–146 (2005) · Zbl 1091.55006 |
| [19] | Haefliger, A.: Extension of complexes of groups. Ann. Inst. Fourier 42(1–2), 275–311 (1992) · Zbl 0762.20018 |
| [20] | Haucourt, E.: Topologie algébrique dirigée et Concurrence. Ph.D. thesis, Université Paris 7, Denis Diderot (2005) |
| [21] | Haucourt, E.: Categories of components and loop-free categories. Theory Appl. Categ. 16(27), 736–770 (2006) · Zbl 1117.18006 |
| [22] | Hawkins, S., Ellis, G.: The large scale structure of spacetime. Cambridge monographs on mathematical physics. Cambridge University Press, Cambridge (1973) |
| [23] | Higgins, P.J.: Categories and Groupoids. Van Nostrand Reinhold, London (1971) (available as TAC reprint at http://www.tac.mta.ca/tac/reprints/ ) · Zbl 0226.20054 |
| [24] | Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge, MA (1982) · Zbl 0499.54001 |
| [25] | Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin Heidelberg New York (1971) · Zbl 0232.18001 |
| [26] | Martin, K., Panangaden, P.: A domain of spacetime intervals in general relativity. In: Kopperman, R., Smyth, M.B., Spreen, D., Webster, J. (eds.), Spatial Representation: Discrete vs. Continuous Computational Models. Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany (2005) · Zbl 1188.83071 |
| [27] | Nachbin, L.: Topology and Order. Van Nostrand, Princeton (1965) · Zbl 0131.37903 |
| [28] | Smale, S.: On gradient dynamical systems. Ann. of Math. 74, 199–206 · Zbl 0136.43702 |
| [29] | Takesaki, M.: Theory of Operator Algebra I, 2nd edn. In: Encyclopaedia of Mathematical sciences, vol. 124. Springer, Berlin Heidelberg New York (2002) (first edition 1979) · Zbl 0436.46043 |
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