Stable Components of Directed Spaces

  • Krzysztof ZiemiańskiEmail author
Open Access


In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces.


D-space Stable component system Enriched category Trace category 

Mathematics Subject Classification

18B35 18D20 55U40 


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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