Trace spaces in a pre-cubical complex

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MR2521708 (2010f:54049) 54F65 (54D30 54E35 55Q99 68Q85) Raussen, Martin (DK-ALBG) Trace spaces in a pre-cubical complex. (English summary) Topology Appl. 156 (2009), no. 9, 1718–1728. Summary: “In directed algebraic topology, directed irreversible (d)-paths and spaces consisting of d-paths are studied from a topological and from a categorical point of view. Motivated by models for concurrent computation, we study in this paper spaces of d-paths in a pre-cubical complex. Such paths are equipped with a natural arc length which moreover is shown to be invariant under directed homotopies. D-paths up to reparametrization (called traces) can thus be represented by arc length parametrized d-paths. Under weak additional conditions, it is shown that trace spaces in a pre-cubical complex are separable metric spaces which are locally contractible and locally compact. Moreover, they have the homotopy type of a CW-complex.” References 1. D.R. Andersson, F.X. Conolly, H.J. Munkholm, A comparison of continuously controlled and controlled K-theory, Topology Appl. 71 (1) (1996) 9–46. MR1391956 (97h:57054) 2. M.R. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss., vol. 319, Springer-Verlag, 1999. MR1744486 (2000k:53038) 3. R. Brown, P.J. Higgins, Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11–41. MR0621285 (82m:55015b) 4. R. Brown, P.J. Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233–260. MR0617135 (82m:55015a) 5. J. Dugundji, Topology, Allyn & Bacon, 1966. MR0193606 (33 #1824) 6. U. Fahrenberg, M. Raussen, Reparametrizations of continuous paths, J. Homotopy Relat. Struct. 2 (2) (2007) 93–117. MR2369163 (2008j:55006) 7. L. Fajstrup, Dipaths and dihomotopies in a cubical complex, Adv. in Appl. Math. 35 (2) (2005) 188–206. MR2152887 (2006b:52014) ´ Goubault, E. Haucourt, M. Raussen, Components of the fundamental category, 8. L. Fajstrup, E. Appl. Categ. Structures 12 (1) (2004) 81–108. MR2057412 (2005a:55010) ´ Goubault, M. Raussen, Algebraic topology and concurrency, Theoret. Comput. 9. L. Fajstrup, E. Sci. 357 (2006) 241–278. MR2242768 (2007d:68133) ´ Goubault, E. Haucourt, Components of the fundamental category II, Appl. Categ. Structures 10. E. 15 (2007) 387–414. MR2350213 (2008h:18003) 11. M. Grandis, Directed homotopy theory II. Homotopy constructs, Theory Appl. Categ. 10 (14) (2002) 369–391. MR1921751 (2004b:55030) 12. M. Grandis, Directed homotopy theory I. The fundamental category, Cahiers Topologie G´eom. Diff´erentielle Cat´egoriques 44 (2003) 281–316. MR2030049 (2004k:55019) 13. M. Grandis, L. Mauri, Cubical sets and their site, Theory Appl. Categ. 11 (8) (2003) 185–211.

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