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

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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.
Original languageUnknown
Pages (from-to)986-1049
JournalAdvances In Mathematics
Issue numberNA
Publication statusPublished - 1 Jan 2013

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