Two-Fold Homotopy of 2-Crossed Module Maps of Commutative Algebras

We address the homotopy theory of 2-crossed modules of commutative algebras. In particular, we define the concept of a 2-fold homotopy between a pair of 1-fold homotopies connecting 2-crossed module maps $\text{A}\to {\text{A}}^{\mathrm{\prime }}$. We also prove that if the domain 2-crossed module $\text{A}$ is free up to order one (i.e. if the bottom algebra is a polynomial algebra) then we have a 2-groupoid of 2-crossed module maps $\text{A}\to {\text{A}}^{\mathrm{\prime }}$ and their homotopies and 2-fold homotopies.