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

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### 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.
Original language Unknown 986-1049 Advances In Mathematics 248 NA https://doi.org/10.1016/j.aim.2013.08.020 Published - 1 Jan 2013

### Cite this

@article{57c2fa4d8d7a4423b82dcb5b871d138d,
title = "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.",
keywords = "Crossed module, Gray category, Quadratic module, Homotopy 3-type, Peiffer lifting, 2-crossed module, Simplicial group, Tricategory",
author = "Martins, {Jo{\~a}o Nuno Gon{\cc}alves Faria}",
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month = "1",
day = "1",
doi = "10.1016/j.aim.2013.08.020",
language = "Unknown",
volume = "248",
pages = "986--1049",
issn = "0001-8708",
publisher = "Elsevier Science B.V., Amsterdam.",
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In: Advances In Mathematics, Vol. 248, No. NA, 01.01.2013, p. 986-1049.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Pointed homotopy and pointed lax homotopy of 2-crossed module maps

AU - Martins, João Nuno Gonçalves Faria

N1 - Sem PDF conforme despacho.

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

KW - Crossed module

KW - Gray category

KW - Homotopy 3-type

KW - Peiffer lifting

KW - 2-crossed module

KW - Simplicial group

KW - Tricategory

U2 - 10.1016/j.aim.2013.08.020

DO - 10.1016/j.aim.2013.08.020

M3 - Article

VL - 248

SP - 986

EP - 1049