# Pair component categories for directed spaces

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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

## References

- Borceux, T.: Handbook of Categorial Algebra I: Basic Category Theory, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
- Bednarczyk, M., Borzyskowski, A., Pawlowski, W.: Generalized congruences—epimorphisms in \(\cal{C}\)
*at*. Theory Appl. Categ.**5**(11), 266–280 (1999)MathSciNetzbMATHGoogle Scholar - Calk, C., Goubault, É., Malbos, Ph.: Time-reversal homotopical properties of concurrent systems. Homol. Homotopy Appl. (2020)
**(to appear)**arXiv:1812.05062 - Dubut, J.: Directed homotopy and homology theories for geometric models of true concurrency. Ph.D.-thesis, École normale supérieure Paris-Saclay (2017)Google Scholar
- Dubut, J., Goubault, É., Goubault-Larrecq, J.: Natural homology, in: Automata, languages, and programming. 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015. Proceedings. Part II, 171–183, Springer, Berlin (2015)CrossRefGoogle Scholar
- Fahrenberg, U.: Directed homology. Electron. Notes Theor. Comput. Sci.
**100**, 111–125 (2004)MathSciNetCrossRefGoogle Scholar - Fahrenberg, U., Raussen, M.: Reparametrizations of continuous paths. J. Homotopy Relat. Struct.
**2**(2), 93–117 (2007)MathSciNetzbMATHGoogle Scholar - Fajstrup, L., Raussen, M., Goubault, É., Haucourt, E.: Components of the Fundamental Category. Appl. Categ. Struct.
**12**(1), 81–108 (2004)MathSciNetCrossRefGoogle Scholar - Fajstrup, L., Goubault, É., Raussen, M.: Algebraic Topology and Concurrency. Theor. Comput. Sci.
**357**, 241–278 (2006). Revised version of Aalborg University preprint, 1999MathSciNetCrossRefGoogle Scholar - Fajstrup, L., Goubault, É., Haucourt, E., Mimram, S., Raussen, M.: Directed Algebraic Topology and Concurrency. Springer, Cham (2016)CrossRefGoogle Scholar
- Farber, M.: Invitation to Topological Robotics, Zurich Lectures in Advanced Mathematics. EMS Publishing House, Zurich (2008)CrossRefGoogle Scholar
- van Glabbeek, R.: Bisimulation semantics for higher dimensional automata. Technical report, Stanford University, (1991)Google Scholar
- van Glabbeek, R.: On the Expressiveness of Higher Dimensional Automata. Theor. Comput. Sci.
**368**(1–2), 168–194 (2006)MathSciNetCrossRefGoogle Scholar - Goubault, É., Haucourt, E.: Components of the Fundamental Category II. Appl. Categ. Struct.
**15**(4), 387–414 (2007)MathSciNetCrossRefGoogle Scholar - Goubault, É., Farber, M., Sagnier, A.: Directed topological complexity . J. Appl. Comput. Topol. (2019). https://doi.org/10.1007/s41468-019-00034-x
- Grandis, M.: Directed homotopy theory I. The fundamental category. Cah. Topol. Géom. Différ. Catég.
**44**, 281–316 (2003)MathSciNetzbMATHGoogle Scholar - Grandis, M.: Directed Algebraic Topology. Models of Non-Reversible Worlds. New Mathematical Monographs, vol. 13. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
- Mitchell, B.: Rings with several objects. Adv. Math.
**8**(1), 1–161 (1972)MathSciNetCrossRefGoogle Scholar - Pratt, V.: Modelling concurrency with geometry. In: Proceedings of the 18th ACM Symposium on Principles of Programming Languages, pp. 311–322 (1991)Google Scholar
- Raussen, M.: Invariants of directed spaces. Appl. Categ. Struct.
**15**(4), 355–386 (2007)MathSciNetCrossRefGoogle Scholar - Raussen, M.: Trace spaces in a pre-cubical complex. Topol. Appl.
**156**(9), 1718–1728 (2009)MathSciNetCrossRefGoogle Scholar - Raussen, M.: Simplicial models for trace spaces II: General higher-dimensional automata. Algebr. Geom. Topol.
**12**(3), 1745–1765 (2012)MathSciNetCrossRefGoogle Scholar - Raussen, M.: Inessential directed maps and directed homotopy equivalences, arXiv:1906.09031 (2019)
- Ziemiański, K.: Stable components of directed spaces. Appl. Categ. Struct.
**27**, 217–244 (2019)MathSciNetCrossRefGoogle Scholar