Abstract Anm-simplex x in ann-category A consists of the assignment of anr-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 anm-categoryOm such that anm-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 freen-categories are carefully defined. Each Om is free.
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