Group crossed modules \(\mathcal{G}=(\partial : E\to G,\,\circ)\) are given by a group homomorphism \(\partial : E\to G\) together with an action \(\circ\) of \(G\) on \(E\). The \(2\)-crossed modules of groups are given by a complex \(\mathcal{A}=(L\stackrel{\delta}{\to}E\stackrel{\partial}{\to}G,\,\circ,\,\{,\})\) together with actions \(\circ\) of \(G\) on \(L\) and \(E\), and a map \(\{-,-\} : E\times E\to L\).
Crossed modules and \(2\)-crossed modules of algebras (over a ring) are defined see e.g., [T. Porter, J. Algebra 99, 458–465 (1986; Zbl 0594.13011)] in the same way as in the group case. The authors study the homotopy theory of \(2\)-crossed modules of commutative algebras. In particular, the concept of a two-fold homotopy between a pair of one-fold homotopies connecting \(2\)-crossed module maps \(\mathcal{A}\to\mathcal{B}\) is defined. The main result states: if the domain \(2\)-crossed module \(\mathcal{A}\) is free up to order one (i.e., if the bottom algebra is a polynomial algebra) then we have a \(2\)-groupoid \(\text{HOM}(\mathcal{A},\mathcal{B})_2\) of \(2\)-crossed module maps \(\mathcal{A}\to\mathcal{B}\), homotopies between maps, and two-fold homotopies between homotopies.