The algebra of oriented simplexes

Ross Street*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    156 Citations (Scopus)


    An m-simplex x in an n-category A consists of the assignment of an r-cell x(u) to each (r + 1)-element subset u of {0, 1, ..., m} such that the source and target (r-1)-cells of x(u) are appropriate composites of x(v) for v a proper subset of u. As m increases, the appropriate composites quickly become hard to write down. This paper constructs an m-category Om such that an m-functor x: Om → A is precisely an m-simplex in A. This leads to a simplicial set ΔA, called the nerve of A, and provides the basis for cohomology with coefficients in A. Higher order equivalences in A as well as free n-categories are carefully defined. Each Om is free.

    Original languageEnglish
    JournalJournal of Pure and Applied Algebra
    Issue numberC
    Publication statusPublished - 1987


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