In this paper, we will give a description of the functor from the category of crossed $n$-cubes to that of simplicial commutative algebras and study a commutative algebra version of Loday's theorem on $n$-types of simplicial groups.
ARVASI, Z.; KOÇAK, M.; and ALP, M. (1998) "A Combinatorial Definition of n-Types of Simplicial Commutative Algebras," Turkish Journal of Mathematics: Vol. 22: No. 3, Article 1. Available at: https://journals.tubitak.gov.tr/math/vol22/iss3/1