## THREE CROSSED MODULES (812)

### BibTeX

@MISC{I812threecrossed,

author = {Z. Arvas I and T. S. Kuzpinari and E. Ö. Uslu},

title = {THREE CROSSED MODULES},

year = {812}

}

### OpenURL

### Abstract

We introduce the notion of 3-crossed module, which extends the notions of 1-crossed module (Whitehead) and 2-crossed module (Conduché).

### Citations

266 |
Simplicial Objects in Algebraic Topology
- May
- 1967
(Show Context)
Citation Context ...es points out that a “nilpotent”algebraic model for 4-types is not known. 3-crossed modules go some way towards that aim. 2 Simplicial groups, Moore Complexes, Peiffer pairings We refer the reader to =-=[21]-=- and [14] for the basic properties of simplicial structures. 2.1 Simplicial Groups A simplicial group G consists of a family of groups {Gn} together with face and degeneracy maps dn i : Gn → Gn−1, 0 ≤... |

63 |
Spaces with finitely many non-trivial homotopy groups, Topology
- Loday
- 1993
(Show Context)
Citation Context ...) were first defined by Whitehead in [25]. They model connected homotopy 2-types. Conduché [12] in 1984 described the notion of 2-crossed module as a model of connected 3-types. More generally Loday, =-=[20]-=-, gave the foundation of a theory of another algebraic model, which is called cat n -groups, for connected (n + 1)-types. Ellis-Steiner [17] shown that cat n -groups are equivalent to crossed n-cubes.... |

60 |
Simplicial homotopy theory
- Curtis
- 1971
(Show Context)
Citation Context ...potent”algebraic model for 4-types is not known. 3-crossed modules goes some way towards that aim. 2 Simplicial groups, Moore Complexes, Peiffer pairings Surveys of these materials are given in [21], =-=[14]-=- for most of the basic properties of simplicial structures. 2.1 Simplicial Groups A simplicial group G consists of a family of groups {Gn} together with face and degeneracy maps dn i : Gn → Gn−1, 0 ≤ ... |

41 |
Combinatorial homotopy and 4-dimensional CW-complexes. De Gruyter Expositions
- Baues
- 1991
(Show Context)
Citation Context ... gives a relation between crossed 2-cubes (i.e. crossed squares) and 2-crossed modules. 2-crossed modules were known to be equivalent to that of simplicial groups whose Moore complex has length 2. In =-=[4, 5]-=- Baues introduced a related notion of quadratic module. The first author and Ulualan [2] also explored some relations among these algebraic models for (connected) homotopy 3-types. The most general in... |

38 |
Simplicial methods and the interpretation of triple cohomology
- Duskin
- 1975
(Show Context)
Citation Context ...cates hypercrossed complexes at level n, throwing away terms of higher dimension, the resulting n-hypercrossed complexes form a category equivalent to the n-hyper groupoids of groups given by Duskin, =-=[15]-=-, Glenn [18] and give algebraic models for n-types. For n = 1, a 1-hypercrossed complex gives a crossed module, whilst a subcategory of the category of hypercrossed 2-complexes is equivalent to Conduc... |

37 | Algebraic models of 3-types and automorphism structures for crossed modules - Brown, Gilbert - 1989 |

35 | Group theoretic algebraic models for homotopy types - Carrasco, Cegarra - 1991 |

32 |
Higher Dimensional Crossed Modules and the Homotopy
- Ellis, Steiner
- 1987
(Show Context)
Citation Context ...dule as a model of connected 3-types. More generally Loday, [20], gave the foundation of a theory of another algebraic model, which is called cat n -groups, for connected (n + 1)-types. Ellis-Steiner =-=[17]-=- shown that cat n -groups are equivalent to crossed n-cubes. A link between simplicial groups and crossed n-cubes were given by Porter [23]. Conduché [13] gives a relation between crossed 2-cubes (i.e... |

31 |
Modules croisés généralisés de longueur 2
- Conduché
- 1984
(Show Context)
Citation Context ...up, Moore complex. A. M. S. C.: 18D35 18G30 18G50 18G55. 1 Introduction Crossed modules (or 1-crossed modules) were first defined by Whitehead in [25]. They model connected homotopy 2-types. Conduché =-=[12]-=- in 1984 described the notion of 2-crossed module as a model of connected 3-types. More generally Loday, [20], gave the foundation of a theory of another algebraic model, which is called cat n -groups... |

23 |
n-Type of simplicial groups and crossed n-cubes
- Porter
- 1993
(Show Context)
Citation Context ... lifting is given by {(x, y), (x ′ , y ′ )} = h(x, yy ′ y −1 ). Crossed squares were generalised by G. Ellis in [16, 17] called “Crossed ncubes” which was related to simplicial groups by T. Porter in =-=[23]-=-. Here we only consider this construction for n = 3 and look at the relation between crossed 3-cubes (see Appendix B) and 3-crossed modules. Let K ∂3 −→ L ∂2 −→ M ∂1 −→ N be a 3-crossed module and let... |

23 |
The inner automorphism 3-group of a strict 2-group
- Roberts, Schreiber
(Show Context)
Citation Context ...homotopy 4-types; (ii) It is easy to handle with respect to other models such as the 3-hypercrossed complex; (iii) Give a possible way to generalising n-crossed modules (or equivalently n-groups (see =-=[24]-=- )) which is analogues to a n-hypercrossed complex. (iv) In [5], Baues points out that a “nilpotent”algebraic model for 4-types is not known. 3-crossed modules go some way towards that aim. 2 Simplici... |

22 |
Kampen theorems for diagrams of spaces
- BROWN, LoDAY, et al.
- 1987
(Show Context)
Citation Context ... modules can be defined in an obvious way. We thus define the category of 2-crossed modules denoting it by X2Mod. A crossed square as defined by D. Guin-Waléry and J.-L. Loday in [19] (see also [20], =-=[8]-=-), can be seen as a mapping cone in [13]. Furthermore 2-crossed modules are related to simplicial groups. This relation can be found in [12], [22]. Theorem 6 The category X2Mod of 2-crossed modules is... |

19 |
Obstructions á l’excision en K-theories algébrique
- Guin-Walery, Loday
- 1981
(Show Context)
Citation Context ...morphism of 2-crossed modules can be defined in an obvious way. We thus define the category of 2-crossed modules denoting it by X2Mod. A crossed square as defined by D. Guin-Waléry and J.-L. Loday in =-=[19]-=- (see also [20], [8]), can be seen as a mapping cone in [13]. Furthermore 2-crossed modules are related to simplicial groups. This relation can be found in [12], [22]. Theorem 6 The category X2Mod of ... |

15 |
Simplicial homotopy theory, Adv
- Curtis
- 1971
(Show Context)
Citation Context ... out that a “nilpotent”algebraic model for 4-types is not known. 3-crossed modules go some way towards that aim. 2 Simplicial groups, Moore Complexes, Peiffer pairings We refer the reader to [21] and =-=[14]-=- for the basic properties of simplicial structures. 2.1 Simplicial Groups A simplicial group G consists of a family of groups {Gn} together with face and degeneracy maps dn i : Gn → Gn−1, 0 ≤ i ≤ n, (... |

14 |
Realization of cohomology classes in arbitrary exact categories
- Glenn
- 1982
(Show Context)
Citation Context ...rossed complexes at level n, throwing away terms of higher dimension, the resulting n-hypercrossed complexes form a category equivalent to the n-hyper groupoids of groups given by Duskin, [15], Glenn =-=[18]-=- and give algebraic models for n-types. For n = 1, a 1-hypercrossed complex gives a crossed module, whilst a subcategory of the category of hypercrossed 2-complexes is equivalent to Conduche’s categor... |

10 |
Homotopy types, Handbook of Algebraic Topology
- Baues
- 1995
(Show Context)
Citation Context ... gives a relation between crossed 2-cubes (i.e. crossed squares) and 2-crossed modules. 2-crossed modules were known to be equivalent to that of simplicial groups whose Moore complex has length 2. In =-=[4, 5]-=- Baues introduced a related notion of quadratic module. The first author and Ulualan [2] also explored some relations among these algebraic models for (connected) homotopy 3-types. The most general in... |

10 |
Crossed Modules and their Higher Dimensional Analogues
- Ellis
- 1984
(Show Context)
Citation Context ...) = (f(z) −1 , u(z)) for z ∈ L, ∂1(x, y) = g(x)g(y) for x ∈ M and y ∈ N, and the Peiffer lifting is given by {(x, y), (x ′ , y ′ )} = h(x, yy ′ y −1 ). Crossed squares were generalised by G. Ellis in =-=[16, 17]-=- called “Crossed ncubes” which was related to simplicial groups by T. Porter in [23]. Here we only consider this construction for n = 3 and look at the relation between crossed 3-cubes (see Appendix B... |

10 | Applications of Peiffer pairing in the Moore complex of a simplicial group
- Mutlu, Porter
- 1998
(Show Context)
Citation Context ...For n = 1, a 1-hypercrossed complex gives a crossed module, whilst a subcategory of the category of hypercrossed 2-complexes is equivalent to Conduche’s category of 2-crossed modules. 1Mutlu-Porter, =-=[22]-=-, introduced a Peiffer pairing structure within the Moore complexes of a simplicial group. They applied this structure to the study of algebraic models for homotopy types. In this article we will defi... |

9 |
Complejos Hipercruzados, Cohomologia y Extensiones
- Carrasco
- 1987
(Show Context)
Citation Context ...ions among these algebraic models for (connected) homotopy 3-types. The most general investigation into the extra structure of the Moore complex of a simplicial group was given by Carrasco-Cegarra in =-=[9]-=- to construct the Non-Abelian version of the classical Dold-Kan theorem. A much more general context of their work was given by Bourn in [6]. Carrasco and Cegarra arrived at a notion of a hypercrossed... |

5 |
Simplicial crossed modules and mapping cones
- Conduché
(Show Context)
Citation Context ...ay. We thus define the category of 2-crossed modules denoting it by X2Mod. A crossed square as defined by D. Guin-Waléry and J.-L. Loday in [19] (see also [20], [8]), can be seen as a mapping cone in =-=[13]-=-. Furthermore 2-crossed modules are related to simplicial groups. This relation can be found in [12], [22]. Theorem 6 The category X2Mod of 2-crossed modules is equivalent to the category of SimpGrp ≤... |

4 | Higher dimensional Peiffer elements in simplicial commutative algebras
- Arvasi, Porter
- 1997
(Show Context)
Citation Context ... normal subgroup ∂n(NGn ∩ Dn) by Fα,β elements which were defined first by Carrasco in [9]. Castiglioni and Ladra [11] gave a general proof for the inclusions partially proved by Arvasi and Porter in =-=[1]-=-, Arvasi and Akça in [3] and Mutlu and Porter in [22]. Their approach to the problem is different from that of cited works. They have succeeded with a proof, for the case of algebras, over an operad b... |

3 | On algebraic models for homotopy 3-types
- Arvasi, Ulualan
- 2006
(Show Context)
Citation Context ...rossed modules were known to be equivalent to that of simplicial groups whose Moore complex has length 2. In [4, 5] Baues introduced a related notion of quadratic module. The first author and Ulualan =-=[2]-=- also explored some relations among these algebraic models for (connected) homotopy 3-types. The most general investigation into the extra structure of the Moore complex of a simplicial group was give... |

3 |
Simplicial and Crossed Lie Algebras
- Akça, Arvasi
- 2002
(Show Context)
Citation Context ...∩ Dn) by Fα,β elements which were defined first by Carrasco in [9]. Castiglioni and Ladra [11] gave a general proof for the inclusions partially proved by Arvasi and Porter in [1], Arvasi and Akça in =-=[3]-=- and Mutlu and Porter in [22]. Their approach to the problem is different from that of cited works. They have succeeded with a proof, for the case of algebras, over an operad by introducing a differen... |

2 |
normalization and Dold-Kan theorem for semiabelian categories
- Moore
- 2007
(Show Context)
Citation Context ...plex of a simplicial group was given by Carrasco-Cegarra in [9] to construct the Non-Abelian version of the classical Dold-Kan theorem. A much more general context of their work was given by Bourn in =-=[6]-=-. Carrasco and Cegarra arrived at a notion of a hypercrossed complexes and proved that the category of such hypercrossed complexes is equivalent to that of simplicial groups. If one truncates hypercro... |

2 |
Applications of Peiffer Pairing in The Moore Complex of A
- Mutlu, Porter
- 1998
(Show Context)
Citation Context .... For n = 1, a 1-hypercrossed complex gives a crossed module, whilst a subcategory of the category of hypercrossed 2-complexes is equivalent to Conduche’s category of 2-crossed modules. Mutlu-Porter, =-=[22]-=-, introduced a Peiffer pairing structure within the Moore complexes of a simplicial group. They applied this structure to the study of algebraic models for homotopy types. 1In this article we will de... |

1 | Peiffer Elements in Simplicial Groups and Algebras
- Castiglioni, Ladra
- 2008
(Show Context)
Citation Context ...− {α} J = [n − 1] − {β} where (α, β) ∈ P(n). Remark 4 Shortly in [22] they defined the normal subgroup ∂n(NGn ∩ Dn) by Fα,β elements which were defined first by Carrasco in [9]. Castiglioni and Ladra =-=[11]-=- gave a general proof for the inclusions partially proved by Arvasi and Porter in [1], Arvasi and Akça in [3] and Mutlu and Porter in [22]. Their approach to the problem is different from that of cite... |

1 |
27 Arvasi zarvasi@ogu.edu.tr Osmangazi University Department of Mathematics and Computer Sciences Art and Science Faculty Eski¸sehir/Turkey Tufan Sait Kuzpinari stufan@dpu.edu.tr Dumlupinar University Department of Mathematics Art and Science Faculty Küta
- I, Soc
- 1949
(Show Context)
Citation Context ...ords: Crossed module, 2-crossed module, Simplicial group, Moore complex. A. M. S. C.: 18D35 18G30 18G50 18G55. 1 Introduction Crossed modules (or 1-crossed modules) were first defined by Whitehead in =-=[25]-=-. They model connected homotopy 2-types. Conduché [12] in 1984 described the notion of 2-crossed module as a model of connected 3-types. More generally Loday, [20], gave the foundation of a theory of ... |

1 |
Combinatorial Homotopy types, and 4-Dimensional Complexes. Handbook of Algebraic Topology, Edited by
- Baues
- 1995
(Show Context)
Citation Context ...13] gives a relation between crossed 2-cubes (i.e. crossed squares) and 2-crossed modules. 2-crossed modules were known to equivalent to that of simplicial groups whose Moore complex has length 2. In =-=[4, 5]-=- Baues introduced a related notion of quadratic module. The first author and Ulualan [2] also explored the some relations among these algebraic models for (connected) homotopy 3-types. The most genera... |