Applied Categorical Structures

, Volume 1, Issue 3, pp 247–284 | Cite as

The algebra of directed complexes

  • Richard Steiner


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.

Mathematics Subject Classification (1991)


Key words

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


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Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Richard Steiner
    • 1
  1. 1.Department of MathematicsUniversity of Glasgow, University GardensGlasgowScotland

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