Let be a commutative ring with unit. To every pair of -algebras and one can associate a simplicial set so that equals the set of polynomial homotopy classes of morphisms from to . We prove that is the set of homotopy classes of morphisms from to , where is the ind-algebra of polynomials on the -dimensional cube with coefficients in vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortiñas-Thom, which addresses the cases . As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic -theory space in terms of polynomial homotopy classes of morphisms.
Cite this article
Emanuel Darío Rodríguez Cirone, The Homotopy Groups of the Simplicial Mapping Space between Algebras. Doc. Math. 24 (2019), pp. 251–270DOI 10.4171/DM/680