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Pair component categories for directed spaces

  • Martin RaussenEmail author
Article
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Abstract

The notion of a homotopy flow on a directed space was introduced in Raussen (Appl Categ Struct 15(4):355–386, 2007) as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all directed maps along such a 1-parameter deformation preserve the homotopy types of path spaces, such a flow and the parameter maps are called inessential. For a directed space, one may consider various categories whose objects are pairs of reachable points to which a functor associates the space of directed paths between them. The monoid of all inessential maps acts on such a category by endofunctors leaving the associated path spaces invariant up to homotopy. We construct a pair component category as quotient category: it has as objects pair components along which the homotopy type is invariant—for a coherent and transparent reason. This paper follows up Fajstrup et al. (J Homotopy Relat Struct 12(1):81–108, 2004); Goubault and Haucourt (Appl Categ Struct 15(4):387–414, 2007); Raussen (Appl Categ Struct 15(4):355–386, 2007) and removes some of the restrictions for their applicability. At least in several examples, it gives reasonable results for spaces with non-trivial directed loops. If one uses homology equivalence instead of homotopy equivalence as the basic relation, it yields an alternative to computable versions of “natural homology” introduced in Dubut et al (in: Automata, languages, and programming. 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015. Proceedings. Part II, 171–183, Springer, Berlin 2015) and elaborated in Dubut (Directed homotopy and homology theories for geometric models of true concurrency. École normale supérieure Paris-Saclay 2017). It refines, for good and for evil, the stable components introduced and investigated in Ziemiański (Appl Categ Struct 27:217–244, 2019).

Keywords

d-Space Homotopy flow Pair categories Localization Component category Cubical complex 

Mathematics Subject Classification

18B35 55P60 55U40 68Q85 

Notes

Compliance with ethical standards

Conflict of interest

The author states that there is no conflict of interest

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborg ØstDenmark

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