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

Abstract

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 Crans' 1-transfors between strict Gray functors, however being pointed, thus this corresponds to Baues' 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 $A \to A'$ yields, in this case, an equivalence relation. Furthermore, if a chosen basis $B$ is specified for $F$, then we can define a 2-groupoid $HOM_B(A,A')$ of 2-crossed module maps $A \to A'$, homotopies connecting them, and 2-fold homotopies between homotopies, where the latter correspond to (pointed) Crans' 2-transfors between 1-transfors. We define a partial resolution $Q^1(A)$, for a 2-crossed module $A$, whose underlying group is free, with a canonical chosen basis, together with a projection map ${\rm proj}\colon Q^1(A) \to A$, 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 ${HOM}_{\rm LAX}(A,A')$ of (strict) 2-crossed module maps $A \to A'$, and their lax homotopies and lax 2-fold homotopies. The associated notion of a (strict) 2-crossed module map $f\colon A \to A'$ to be a lax homotopy equivalence has the two-of-three property, and it is closed under retracts.