Mapping Spaces of Gray-Categories

We define a mapping space for Gray-enriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential ingredients are a path space construction for Gray-categories and a kind of comonadic resolution of the 1-dimensional structure of a given Gray-category obtained by lifting the resolution of ordinary categories along the canonical fibration of GrayCat over Cat.Comment: 87 page

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