The algebra of oriented simplexes

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 O_m such that an m-functor x: O_m → 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 O_m is free.