The Homotopy Groups of the Simplicial Mapping Space between Algebras

Abstract

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 $B_{\bullet }^{{\mathfrak{S}}_n}$, where $B_{\bullet }^{{\mathfrak{S}}_n}$ 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 $n\le 1$. 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.