Pointed homotopy and pointed lax homotopy of 2-crossed module maps

We address the (pointed) homotopy theory of 2-crossed modules (of groups), which are known to faithfully represent Gray 3-groupoids, with a single object, and also connected homotopy 3-types. The homotopy relation between 2-crossed module maps will be defined in a similar way to Cransʼ 1-transfors between strict Gray functors, however being pointed, thus this corresponds to Bauesʼ homotopy relation between quadratic module maps. Despite the fact that this homotopy relation between 2-crossed module morphisms is not, in general, an equivalence relation, we prove that if A and A′ are 2-crossed modules, with the underlying group F of A being free (in short A is free up to order one), then homotopy between 2-crossed module maps A→A′ yields, in this case, an equivalence relation. Furthermore, if a chosen basis B is specified for F, then we can define a 2-groupoid HOMB(A,A′) of 2-crossed module maps A→A′, homotopies connecting them, and 2-fold homotopies between homotopies, where the latter correspond to (pointed) Cransʼ 2-transfors between 1-transfors. We define a partial resolution Q1(A), for a 2-crossed module A, whose underlying group is free, with a canonical chosen basis, together with a projection map proj:Q1(A)→A, defining isomorphisms at the level of 2-crossed module homotopy groups. This resolution (which is part of a comonad) leads to a weaker notion of homotopy (lax homotopy) between 2-crossed module maps, which we fully develop and describe. In particular, given 2-crossed modules A and A′, there exists a 2-groupoid HOMLAX(A,A′) of (strict) 2-crossed module maps A→A′, and their lax homotopies and lax 2-fold homotopies, leading to the question of whether the category of 2-crossed modules and strict maps can be enriched over the monoidal category Gray. The associated notion of a (strict) 2-crossed module map f:A→A′ to be a lax homotopy equivalence has the two-of-three property, and it is closed under retracts. This discussion leads to the issue of whether there exists a model category structure in the category of 2-crossed modules (and strict maps) where weak equivalences correspond to lax homotopy equivalences, and any free up to order one 2-crossed module is cofibrant.