Pointed Homotopy of Maps Between 2-Crossed Modules of Commutative Algebras

TL;DR

This paper develops a homotopy theory for 2-crossed modules of commutative algebras, establishing a homotopy relation that forms a groupoid under certain conditions, thus advancing the understanding of their algebraic and homotopical structure.

Contribution

It introduces a homotopy relation for 2-crossed modules of commutative algebras and proves it forms a groupoid under specific freeness conditions, extending previous algebraic homotopy concepts.

Findings

Homotopy relation is an equivalence relation under freeness conditions.

The homotopy relation forms a groupoid with fixed domain and codomain.

Applicable to 2-crossed modules with free up to order one domain.

Abstract

We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crossed modules (with free up to order one domain), the morphisms being the homotopies between 2-crossed module maps.