## (2002)

### BibTeX

@MISC{Mutlu02,

author = {A. Mutlu and T. Porter},

title = {},

year = {2002}

}

### OpenURL

### Abstract

Crossed squares and 2-crossed modules

### Citations

169 |
Complexe cotangent et déformations. I
- Illusie
- 1971
(Show Context)
Citation Context ...om the category of simplicial groups to that of crossed n-cubes of groups. We will summarise its construction. The functor is constructed using the décalage functor studied by Duskin [13] and Illusie =-=[22]-=- and is a π0-image of a functor taking values in a category of simplicial normal (n+1)-ads. The décalage functor will be denoted by Dec. Given any simplicial group G, DecG is the augmented simplicial ... |

41 |
Combinatorial homotopy and 4-dimensional CW-complexes. De Gruyter Expositions
- Baues
- 1991
(Show Context)
Citation Context ...2-types. Crossed squares model 3-types. Crossed n-cubes model (n + 1)−types, cf. [35] and the references therein. Conduché [10] has an alternative model for 3-types namely 2-crossed modules and Baues =-=[3]-=- uses a variant of this, the quadratic module, in some of his work. Another model for 3-types was introduced by Brown and Gilbert, [5], (involving a particular subdivided triangular diagram in its der... |

40 |
A combinatorial definition of homotopy groups
- Kan
- 1958
(Show Context)
Citation Context ...ssification: 18G30, 18G55. Key words and phrases : Simplicial Group, Crossed n-cubes, Crossed Squares, 2-Crossed Modules. 1 Introduction Simplicial groups were first studied by D. M. Kan in the 1950s =-=[24]-=-. Early work by Kan himself, Moore, Milnor, and Dold showed that (a) these objects have a well structured homotopy theory, (b) they modelled all homotopy types of connected spaces, (c) abelian simplic... |

38 |
Simplicial methods and the interpretation of “triple” cohomology
- Duskin
- 1975
(Show Context)
Citation Context ...ibed a functor from the category of simplicial groups to that of crossed n-cubes of groups. We will summarise its construction. The functor is constructed using the décalage functor studied by Duskin =-=[13]-=- and Illusie [22] and is a π0-image of a functor taking values in a category of simplicial normal (n+1)-ads. The décalage functor will be denoted by Dec. Given any simplicial group G, DecG is the augm... |

37 |
Algebraic models of 3-types and automorphism structures for crossed modules
- Brown, Gilbert
- 1989
(Show Context)
Citation Context ... alternative model for 3-types namely 2-crossed modules and Baues [3] uses a variant of this, the quadratic module, in some of his work. Another model for 3-types was introduced by Brown and Gilbert, =-=[5]-=-, (involving a particular subdivided triangular diagram in its derivation). This model, braided crossed modules and their lax counterpart, the braided categorical groups, have been further studied by ... |

34 |
Group-theoretic Algebraic Models for Homotopy Types
- Carrasco, Cegarra
- 1991
(Show Context)
Citation Context ...in of n-equivalences linking them. A simplicial group G is an n-type if πi(G) = 1 for i > n. The Moore complex carries a lot of fine structure and this has been studied, e.g. by Carrasco and Cegarra, =-=[8]-=-, Wu [38], and the present authors in earlier papers in this series, [30–34]. The specific structure of the k-truncation of the Moore complex for k = 1 and 2 is now well known. For k = 1 this gives a ... |

32 |
Higher Dimensional Crossed Modules and the Homotopy
- Ellis, Steiner
- 1987
(Show Context)
Citation Context ...ares were difficult to generalise to higher order structures, although cat 2 -groups could clearly and easily be generalised to cat n -groups. The following generalisation is due to Ellis and Steiner =-=[19]-=- and includes a reformulation of crossed squares as a special case.. Let 〈n〉 denote the set {1, ..., n}. A crossed n-cube of group is a family {MA : A ⊆ 〈n〉} of groups, together with homomorphisms µi ... |

30 |
Modules Croisés Généralisés de Longueur 2
- Conduché
- 1984
(Show Context)
Citation Context ...s we will often use that shortened form here. Crossed modules model homotopy 2-types. Crossed squares model 3-types. Crossed n-cubes model (n + 1)−types, cf. [35] and the references therein. Conduché =-=[10]-=- has an alternative model for 3-types namely 2-crossed modules and Baues [3] uses a variant of this, the quadratic module, in some of his work. Another model for 3-types was introduced by Brown and Gi... |

22 | Homotopy Coherent Category Theory
- Cordier, Porter
(Show Context)
Citation Context ...× ∆ → ∆ be the ordinal sum functor, then for a bisimplicial group X, it is well known that ∇X has a description as a coend (∇X)n = ∫ [p],[q] ∆([n], [p]or[q]) × Xp,q, (cf. for example, Cordier-Porter, =-=[11]-=-). A corresponding codiagonal for a m-fold simplicial group X∗, ∗ an m-fold index, is (∇ (m) ∫ p X)n = ∆([n], orp) × Xp, where by abuse of notation, we indicate by or : ∆ ×m → ∆, the m-fold ordinal su... |

22 |
n-Types of Simplicial Groups and Crossed n-cubes
- Porter
- 1991
(Show Context)
Citation Context ...plicially enriched groupoids. None the less we will often use that shortened form here. Crossed modules model homotopy 2-types. Crossed squares model 3-types. Crossed n-cubes model (n + 1)−types, cf. =-=[35]-=- and the references therein. Conduché [10] has an alternative model for 3-types namely 2-crossed modules and Baues [3] uses a variant of this, the quadratic module, in some of his work. Another model ... |

20 | Double loop spaces, braided monoidal categories and algebraic 3-type of space. Higher homotopy structures in topology and mathematical physics - Berger - 1996 |

20 |
Braided) tensor structures on homotopy groupoids and nerves of (braided) categorical groups
- Carrasco, Cegarra
(Show Context)
Citation Context ...erivation). This model, braided crossed modules and their lax counterpart, the braided categorical groups, have been further studied by members of the Granada Algebra group (cf. Carrasco and Cegarra, =-=[9]-=-, and Garzon and Miranda, [20], for example). Similar object, Gray groupoids, have been studied by Joyal and Tierney (unpublished) and have recently started to receive some attention in the TQFT liter... |

18 |
Obstructions a l'excision en K-theorie algebrique
- Guin-Walery, Loday
- 1981
(Show Context)
Citation Context ... equivalent to the category SimpGrp≤2 of simplicial groups with Moore complex of length 2. □ 4 Cat 2 -groups and crossed squares The following definition is due to D. Guin-Walery and J.-L. Loday, see =-=[21]-=- and also [27]. Definition: A crossed square of groups is a commutative square of groups λ ′ L λ �� N µ ′ together with actions of P on L, M and N. There are thus actions of N on L and M via µ ′ and M... |

15 |
Simplicial homotopy theory, Adv
- Curtis
- 1971
(Show Context)
Citation Context ... a family of groups {Gn} together with face and degeneracy maps di = d n i : Gn −→ Gn−1, 0 ≤ i < n (n ̸= 0) and si = s n i : Gn −→ Gn+1, 0 ≤ i ≤ n, satisfying the usual simplicial identities given in =-=[12]-=-, [24] and [25]. It can be completely described as a functor G : ∆ op −→ Grp where ∆ is the category of finite ordinals, [n] = {0 < 1 < 2 < · · · < n}, and increasing maps. We will denote the category... |

11 |
On the van Kampen theorem
- Artin, Mazur
- 1966
(Show Context)
Citation Context ...ithin M. 6.2 From bisimplicial groups to simplicial groups. There are two useful ways of passing from bisimplicial groups to simplicial groups. One is the diagonal, the other, due to Artin and Mazur, =-=[2]-=-, is the ‘codiagonal’ and, for us, is more useful. In the bisimplicial group X, the corresponding Moore bicomplex has relatively few non-zero terms. In fact, N(X)p,q will be zero if p or q is bigger t... |

10 | Applications of Peiffer pairing in the Moore complex of a simplicial group
- Mutlu, Porter
- 1998
(Show Context)
Citation Context ... of this, elementwise: The simplicial group Ner(M) has Ner(M)0 = P Ner(M)n = M ⋊ (. . .(M ⋊ P) . . .), with n semidirect factors of M (see Conduché, [10], Carrasco and Cegarra, [8], or our own papers =-=[30,34]-=- for more on semidirect decompositions and simplicial groups). If (m, p) ∈ Ner(M)1, then and If (m2, m1, p) ∈ Ner(M)2, then d 1 0 (m, p) = µ(m)p d 1 1 (m, p) = p s 0 0 (p) = (1, p). d 2 0 (m2, m1, p) ... |

9 |
On cat”-groups and homotopy types
- Bullejos, Cegarra, et al.
- 1993
(Show Context)
Citation Context ...y its 2-coskeleton). However we have not attempted to give this here. 8 Higher dimensions Loday’s mapping complex was defined for cat n -groups (see [27]) and results of Bullejos, Cegarra and Duskin, =-=[7]-=- suggest that a similar multiple codiagonal would give his mapping complex. This raises the question of what would be the result on taking a crossed n-cube M and forming its n-fold nerve Ner (n) M, wh... |

8 | une notion de 3-categorie adaptee a l’homotopie, U. Montpellier II preprint - Leroy, Sur |

7 |
A relation between CW-complex and free c.s.s. groups
- Kan
- 1959
(Show Context)
Citation Context ...oups {Gn} together with face and degeneracy maps di = d n i : Gn −→ Gn−1, 0 ≤ i < n (n ̸= 0) and si = s n i : Gn −→ Gn+1, 0 ≤ i ≤ n, satisfying the usual simplicial identities given in [12], [24] and =-=[25]-=-. It can be completely described as a functor G : ∆ op −→ Grp where ∆ is the category of finite ordinals, [n] = {0 < 1 < 2 < · · · < n}, and increasing maps. We will denote the category of simplicial ... |

6 |
Homotopy theory for (braided) Cat-groups, Cahiers de Top. et Géom. Diff. cat
- Garzon, Miranda
(Show Context)
Citation Context ...ed crossed modules and their lax counterpart, the braided categorical groups, have been further studied by members of the Granada Algebra group (cf. Carrasco and Cegarra, [9], and Garzon and Miranda, =-=[20]-=-, for example). Similar object, Gray groupoids, have been studied by Joyal and Tierney (unpublished) and have recently started to receive some attention in the TQFT literature. The proof of the equiva... |

5 |
Varieties of Simplicial Groupoids, I: Crossed Complexes
- Ehlers, Porter
- 1997
(Show Context)
Citation Context ...n of the Moore complexes of the vertex simplicial groups of 2G together with the groupoid G0 providing elements that allow conjugation between (some of) these vertex complexes (cf. Ehlers and Porter =-=[15]-=-). Consider the product ∆ × ∆ whose objects are pairs ([p], [q]) and whose maps are pairs of weakly increasing maps. A functor G : (∆ × ∆) op −→ Grp is called a bisimplicial group. To give G is equiva... |

5 |
Crossed Squares and
- Ellis
- 1993
(Show Context)
Citation Context ...uations. Some success has been achieved in this area by Alp and Wensley, [1], and by Ellis, [18]. 6.4 Squared complexes and 2-crossed complexes In 1993, Ellis defined the notion of a squared complex, =-=[17]-=-. A crossed complex combines a crossed module at its ‘base’ with a continuation by a chain complex of modules further up. They thus have good descriptive power, being able to model 2-types via their c... |

5 | Pairings in the Moore Complex of a Simplicial Group - Mutlu, Peiffer - 1997 |

5 | T.: Freeness conditions for 2-crossed modules and complexes
- Mutlu, Porter
- 1998
(Show Context)
Citation Context ...ralise crossed complexes to include the so-called quadratic information available in a 3-type. Other versions include double crossed complexes (cf. Tonks, [36]), 2-crossed complexes (cf. the authors, =-=[33]-=-) and quadratic complexes, (cf. Baues, [3]). We will need to examine 2-crossed complexes in detail later. A squared complex consists of a diagram of group homomorphisms . . . �� C4 ∂4 �� C3 ∂3 �� L N ... |

4 |
Homotopy theory and simplicial
- Dwyer, Kan
- 1984
(Show Context)
Citation Context ...oids have a more recent birth, but have the same sort of attributes plus being able to model non-connected homotopy types. The shortened form of their name ‘simplicial groupoid’ used by Dwyer and Kan =-=[14]-=- is more usually used for these objects although not strictly correct as simplicial objects in the category of groupoids form a much larger setting than do simplicially enriched groupoids. None the le... |

3 |
Spaces having finitely many non-trivial homotopy groups
- Loday
- 1982
(Show Context)
Citation Context ...etter to Brown and Loday, dated in the mid 1980s, Conduché pointed out that given a crossed square, ⎛ ⎞ L �� M ⎜ ⎟ M = ⎝ ⎠ , the mapping cone complex of M, N � P L → M ⋊ N → P constructed by Loday in =-=[27]-=-, has a 2-crossed module structure. He ended his letter by pointing out that this seemed to give a canonical and direct way to link the categories of crossed squares and 2-crossed modules, and his res... |

3 | Free crossed resolutions from simplicial resolutions with given CW-basis, Cahiers Top
- Mutlu, Porter
- 1999
(Show Context)
Citation Context ...poids. We thus can consider the category of 2-crossed modules denoting it as X2Mod. Conduché [10] proved that 2-crossed modules give algebraic models of connected homotopy 3-types. Theorem 3.1 ([10], =-=[32]-=-) The category, X2Mod, of 2-crossed modules is equivalent to the category SimpGrp≤2 of simplicial groups with Moore complex of length 2. □ 4 Cat 2 -groups and crossed squares The following definition ... |

3 |
On combinatorial descriptions of π∗(ΣK(π, 1)), MSRI preprint 069
- Wu
- 1995
(Show Context)
Citation Context ...equivalences linking them. A simplicial group G is an n-type if πi(G) = 1 for i > n. The Moore complex carries a lot of fine structure and this has been studied, e.g. by Carrasco and Cegarra, [8], Wu =-=[38]-=-, and the present authors in earlier papers in this series, [30–34]. The specific structure of the k-truncation of the Moore complex for k = 1 and 2 is now well known. For k = 1 this gives a crossed m... |

1 |
share package for GAP, available from http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/preprint.html, preprint no. 97.14 or from http://www-gap.dcs.st-and.ac.uk/ gap/Info/share.html (see the list there
- Alp, Wensley, et al.
(Show Context)
Citation Context ...f both combinatorial and computational group theory, adapting the methods of the classical case to these higher dimensional situations. Some success has been achieved in this area by Alp and Wensley, =-=[1]-=-, and by Ellis, [18]. 6.4 Squared complexes and 2-crossed complexes In 1993, Ellis defined the notion of a squared complex, [17]. A crossed complex combines a crossed module at its ‘base’ with a conti... |

1 |
Various software packages, available at http://hamilton.nuigalway.ie
- Ellis
(Show Context)
Citation Context ...l and computational group theory, adapting the methods of the classical case to these higher dimensional situations. Some success has been achieved in this area by Alp and Wensley, [1], and by Ellis, =-=[18]-=-. 6.4 Squared complexes and 2-crossed complexes In 1993, Ellis defined the notion of a squared complex, [17]. A crossed complex combines a crossed module at its ‘base’ with a continuation by a chain c... |

1 | Approche en dimension supérieure des 3-catégories augmentées d’Olivier LeRoy - Marty - 1999 |

1 | Freeness Conditions for Crossed Squares and Squared Complexes, K-Theory
- Mutlu, Porter
(Show Context)
Citation Context ... of this, elementwise: The simplicial group Ner(M) has Ner(M)0 = P Ner(M)n = M ⋊ (. . .(M ⋊ P) . . .), with n semidirect factors of M (see Conduché, [10], Carrasco and Cegarra, [8], or our own papers =-=[30,34]-=- for more on semidirect decompositions and simplicial groups). If (m, p) ∈ Ner(M)1, then and If (m2, m1, p) ∈ Ner(M)2, then d 1 0 (m, p) = µ(m)p d 1 1 (m, p) = p s 0 0 (p) = (1, p). d 2 0 (m2, m1, p) ... |

1 |
Theory and applications of crossed complexes: the Eilenberg-Zilber theorem and homotopy colimits
- Tonks
- 1994
(Show Context)
Citation Context ...xes are one of the possible notions that generalise crossed complexes to include the so-called quadratic information available in a 3-type. Other versions include double crossed complexes (cf. Tonks, =-=[36]-=-), 2-crossed complexes (cf. the authors, [33]) and quadratic complexes, (cf. Baues, [3]). We will need to examine 2-crossed complexes in detail later. A squared complex consists of a diagram of group ... |