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The algebra of directed complexes

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The theory of directed complexes is a higher-dimensional generalisation of the theory of directed graphs. In a directed graph, the simple directed paths form a subset of the free category which they generate; if the graph has no directed cycles, then the simple directed paths constitute the entire category. Generalising this, in a directed complex there is a class of split subsets which is contained in and generates a free ω-category; when a simple loop-freeness condition is satisfied, the split sets constitute the entire ω-category. The class of directed complexes is closed under the natural product and join constructions. The free ω-categories generated by directed complexes include the important examples associated to cubes and simplexes.

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Steiner, R. The algebra of directed complexes. Appl Categor Struct 1, 247–284 (1993). https://doi.org/10.1007/BF00873990

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Mathematics Subject Classification (1991)

  • 18D05

Key words

  • Directed complex
  • parity complex
  • ω-category
  • ∞-category
  • globelike
  • atom
  • molecule
  • split set
  • frame dimension
  • loop-free
  • totally loop-free