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

Abstract

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 $\A \to \A'$. We also prove that if the domain 2-crossed module $\A$ 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 $\A \to \A'$ and their homotopies and 2-fold homotopies.

24 pages, to appear in Communications in Algebra

References in corpus (3)

• Pointed homotopy and pointed lax homotopy of 2-crossed module maps
• Pointed Homotopy of Maps Between 2-Crossed Modules of Commutative Algebras
• Mapping Spaces of Gray-Categories