The Homotopy Groups of the Simplicial Mapping Space between Algebras

Let \ell be a commutative ring with unit. To every pair of \ell -algebras A and B one can associate a simplicial set \text{Hom}(A,B^\Delta) so that \pi_0\text{Hom}(A,B^\Delta) equals the set of polynomial homotopy classes of morphisms from A to B . We prove that \pi_n\text{Hom}(A,B^\Delta) is the set of homotopy classes of morphisms from A to B^{\mathfrak{S}_n}_\bullet , where B^{\mathfrak{S}_n}_\bullet is the ind-algebra of polynomials on the n -dimensional cube with coefficients in B vanishing at the boundary of the cube. This is a generalization to arbitrary dimensions of a theorem of Cortiñas-Thom, which addresses the cases n\leq 1 . As an application we give a simplified proof of a theorem of Garkusha that computes the homotopy groups of his matrix-unstable algebraic KK -theory space in terms of polynomial homotopy classes of morphisms.