TY - JOUR

T1 - The algebra of oriented simplexes

AU - Street, Ross

PY - 1987

Y1 - 1987

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0001656380&partnerID=8YFLogxK

U2 - 10.1016/0022-4049(87)90137-X

DO - 10.1016/0022-4049(87)90137-X

M3 - Article

AN - SCOPUS:0001656380

VL - 49

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - C

ER -