## On algebraic models for homotopy 3-types

Venue: | J. Homotopy Relat. Struct |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Arvasi_onalgebraic,

author = {Z. Arvasi and E. Ulualan},

title = {On algebraic models for homotopy 3-types},

journal = {J. Homotopy Relat. Struct},

year = {},

pages = {2006}

}

### OpenURL

### Abstract

We explore the relations among quadratic modules, 2-crossed modules, crossed squares and simplicial groups with Moore complex of length 2.

### Citations

292 |
Simplicial Objects in Algebraic Topology
- May
- 1967
(Show Context)
Citation Context ...ing the constructions given below. Acknowledgements. The authors wishes to thank the referee for helpful comments and improvements to the paper. 1 Preliminaries We refer the reader to May’s book (cf. =-=[17]-=-) and Artin-Mazur’s, [1], article for the basic properties of simplicial groups, bisimplicial groups, etc. A simplicial group G consists of a family of groups Gn together with face and degeneracy maps... |

68 | On the 3-type of a complex - Lane, Whitehead |

65 |
Spaces with finitely many nontrivial homotopy groups
- Loday
- 1982
(Show Context)
Citation Context ...n obvious way. We thus define the category of 2-crossed modules denoting it by X2Mod. The following theorem, in some sense, is known. We do not give the proof since it exists in the literature, [10], =-=[15]-=-, [18], [21]. Theorem 2.1 The category X2Mod of 2-crossed modules is equivalent to the category SimpGrp �2 of simplicial groups with Moore complex of length 2. ✷ 3 Cat 2 -Groups and Crossed Squares Al... |

41 | Simplicial methods and the interpretation of triple cohomology - Duskin - 1970 |

39 |
Algebraic models for 3-types and automorphism structures for crossed modules
- Brown, Gilbert
- 1989
(Show Context)
Citation Context ...s. A quadratic module is thus a ‘nilpotent’ algebraic model of connected 3-types. Another algebraic model of connected 3-types is ‘braided regular crossed module’ introduced by Brown and Gilbert (cf. =-=[5]-=-). These notions are then related to simplicial groups. Conduché has shown that the category of simplicial groups with Moore complex of length 2 is equivalent to that of 2-crossed modules. Baues gives... |

39 |
Group-theoretic algebraic models for homotopy types
- Carrasco, Cegarra
- 1991
(Show Context)
Citation Context ...loop spaces. Some light on the 2-crossed module structure was also shed by Mutlu and Porter, [20], who suggested ways of generalising Conduché’s construction to higher n-types. Also Carrasco-Cegarra, =-=[9]-=-, gives a generalisation of the Dold-Kan theorem to an equivalence between simplicial groups and a non-Abelian chain complex with a lot of extra structure, generalising 2-crossed modules. The present ... |

36 |
Higher dimensional crossed modules and the homotopy groups of n+ 1-ads
- Ellis, Steiner
- 1987
(Show Context)
Citation Context ...p with big group (L ⋊ N) ⋊ (M ⋊ P) and induced endomorphisms s1, t1, s2, t2. ✷ A generalisation of a crossed square to higher dimensions called a “crossed n-cube”, was given by Ellis and Steiner (cf. =-=[14]-=-), but we use only the case n = 2. The following result for groups was given by Mutlu and Porter (cf. [18]). Let G be a simplicial group. Then the following diagram NG2/∂3NG3 ∂2 �� NG1 ∂ ′ 2 �� NG1 µ ... |

34 |
Modules croises generalises de longueur 2
- Conduche
- 1984
(Show Context)
Citation Context ...odels of connected (weak homotopy) 2-types. Crossed squares as introduced by Loday and Guin-Walery, [22], model connected 3-types. Crossed n-cubes model connected (n + 1)-types, (cf. [21]). Conduché, =-=[10]-=-, gave an alternative model for connected 3-types in terms of crossed modules of groups of length 2 which he calls ‘2-crossed module’. Conduché also constructed (in a letter to Brown in 1984) a 2-cros... |

24 |
n-types of simplicial groups and crossed n-cubes
- Porter
- 1993
(Show Context)
Citation Context ..., are algebraic models of connected (weak homotopy) 2-types. Crossed squares as introduced by Loday and Guin-Walery, [22], model connected 3-types. Crossed n-cubes model connected (n + 1)-types, (cf. =-=[21]-=-). Conduché, [10], gave an alternative model for connected 3-types in terms of crossed modules of groups of length 2 which he calls ‘2-crossed module’. Conduché also constructed (in a letter to Brown ... |

23 | Kampen theorems for diagram of spaces - Van - 1987 |

23 | crossed modules and the fundamental groupoid of a topological group - Brown, Spencer, et al. - 1976 |

21 | Double loop spaces, braided monoidal categories and algebraic 3-type of space - Berger - 1996 |

19 |
Obstructions á l’excision en K-theories algébrique
- Guin-Walery, Loday
- 1981
(Show Context)
Citation Context ... Moore complex of length 2. Introduction Crossed modules defined by Whitehead, [23], are algebraic models of connected (weak homotopy) 2-types. Crossed squares as introduced by Loday and Guin-Walery, =-=[22]-=-, model connected 3-types. Crossed n-cubes model connected (n + 1)-types, (cf. [21]). Conduché, [10], gave an alternative model for connected 3-types in terms of crossed modules of groups of length 2 ... |

15 |
Algebraic homotopy, Cambridge
- Baues
- 1989
(Show Context)
Citation Context ...he homology groups of the complex L ∂2 −→ M ⋊ N ∂1 −→ P −→ 1 where ∂1 and ∂2 are defined above. 5 Quadratic Modules from 2-Crossed Modules Quadratic modules of groups were initially defined by Baues, =-=[2, 3]-=-, as models for connected 3types. In this section we will define a functor from the category X2ModX2Mod of 2- crossed modules to that of quadratic modules QM. Before giving the definition of quadratic... |

12 |
On the van Kampen theorem
- Artin, Mazur
- 1966
(Show Context)
Citation Context ...en below. Acknowledgements. The authors wishes to thank the referee for helpful comments and improvements to the paper. 1 Preliminaries We refer the reader to May’s book (cf. [17]) and Artin-Mazur’s, =-=[1]-=-, article for the basic properties of simplicial groups, bisimplicial groups, etc. A simplicial group G consists of a family of groups Gn together with face and degeneracy maps d n i : Gn → Gn−1, 0 � ... |

10 | Applications of Peiffer pairing in the Moore complex of a simplicial group
- Mutlu, Porter
- 1998
(Show Context)
Citation Context ... n⋂ ) = Hn(NG, ∂) = . i=0 kerd n i /d n+1 n+1 i=0 kerd n+1 i The Moore complex carries a lot of fine structure and this has been studied, e.g. by Carrasco and Cegarra (cf. [9]), Mutlu and Porter (cf. =-=[18, 19, 20]-=-). Consider the product category ∆ × ∆ whose objects are pairs ([p], [q]) and whose maps are pairs of weakly increasing maps. A (contravariant) functor G., . : (∆ × ∆) op → Grp is called a bisimplicia... |

10 |
Combinatorial homotopy II
- Whitehead
- 1949
(Show Context)
Citation Context ... Abstract We explore the relations among quadratic modules, 2-crossed modules, crossed squares and simplicial groups with Moore complex of length 2. Introduction Crossed modules defined by Whitehead, =-=[23]-=-, are algebraic models of connected (weak homotopy) 2-types. Crossed squares as introduced by Loday and Guin-Walery, [22], model connected 3-types. Crossed n-cubes model connected (n + 1)-types, (cf. ... |

8 |
Crossed squares and combinatorial homotopy
- Ellis
- 1993
(Show Context)
Citation Context ... kernel isomorphic to 1 −→ kerd0 = −→ kerd0 which is thus contractible. Note: The construction given above from a crossed square to a 2-crossed module preserves the homotopy type. In fact, Ellis (cf. =-=[13]-=-) defined the homotopy groups of the crossed square is the homology groups of the complex L ∂2 −→ M ⋊ N ∂1 −→ P −→ 1 where ∂1 and ∂2 are defined above. 5 Quadratic Modules from 2-Crossed Modules Quadr... |

5 |
On the equivariant 2-type of a G-space
- Bullejos, Cabello, et al.
(Show Context)
Citation Context ..., (1, (m, p))). 8For the verification of the simplicial identities, see appendix. Remark: The construction given above may be shortened in terms of the W construction or ‘bar’ construction (cf. [1], =-=[8]-=-), but we have not attempted this method. Loday, [15], defined the mapping cone of a complex as analogous to the construction of the Moore complex of a simplicial group. (for further work see also [11... |

5 |
Simplicial crossed modules and mapping cones
- Conduché
(Show Context)
Citation Context ...[8]), but we have not attempted this method. Loday, [15], defined the mapping cone of a complex as analogous to the construction of the Moore complex of a simplicial group. (for further work see also =-=[11]-=-). We next describe the mapping cone of a crossed square of groups as follows: Proposition 4.1 The Moore complex of the simplicial group G (2) is the mapping cone, in the sense of Loday, of the crosse... |

3 |
Combinatorial homotopy and 4-dimenional complexes, Walter de Gruyter
- Baues
- 1991
(Show Context)
Citation Context ... 3-types in terms of crossed modules of groups of length 2 which he calls ‘2-crossed module’. Conduché also constructed (in a letter to Brown in 1984) a 2-crossed module from a crossed square. Baues, =-=[3]-=-, gave the notion of quadratic module which is a 2-crossed module with additional ‘nilpotency’ conditions. A quadratic module is thus a ‘nilpotent’ algebraic model of connected 3-types. Another algebr... |

3 | Crossed squares and 2-crossed modules - Mutlu, Porter - 2002 |