## (2008)

### BibTeX

@MISC{Brown08,

author = {Ronald Brown and George Janelidze and Uwb Maths Preprint},

title = {},

year = {2008}

}

### OpenURL

### Abstract

Galois theory and a new homotopy double groupoid of a map of spaces

### Citations

327 | Homotopical algebra - Quillen - 1967 |

124 | On the algebra of cubes
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...... ,tn) = (t1,t2,... ,ti−1,A(ti,ti+1),ti+2,... ,tn) where A(s,t) = max(s,t),min(s,t) as α = −,+ respectively. The enrichment with connections Γ + i for the traditional cubical sets was introduced in =-=[5]-=-. The full properties of these structures are set out in for example [1]. Here we will assume only the obvious geometric properties in the range n = 0,... ,3. Let q : M → B be a map of spaces. We reca... |

94 | Calculus of Fractions and Homotopy Theory - Gabriel, Zisman - 1967 |

52 | On the connection between the second relative homotopy groups of some related spaces
- BROWN, HIGGINS
- 1978
(Show Context)
Citation Context ...of pointed spaces was made with P.J. Higgins in 1974, and exploited to obtain a 2-dimensional Van Kampen type theorem for this double groupoid, and hence for Whitehead’s crossed module of a pair (see =-=[4]-=-). The double groupoid constructed in [4] is edge symmetric and has a connection, and so is not the same as that constructed here. A classification of certain double groupoids is given in [11], but th... |

44 |
Kampen theorems for diagrams of spaces. Topology
- Brown, Loday
- 1987
(Show Context)
Citation Context ...in [17]; his cat n -groups were shown there to model connected homotopy (n + 1)-types. These higher groupoid methods yield new calculations in homotopy theory through higher order Van Kampen theorems =-=[6, 10]-=-, as well as suggesting new algebraic constructions. 1 Galois groupoids Later we will be considering the category C = Sets ∆op of simplicial sets and the fundamental groupoid functor I = π1 : C → X fr... |

27 | Determination of a double Lie groupoid by its core diagram
- Brown, Mackenzie
- 1992
(Show Context)
Citation Context ...air (see [4]). The double groupoid constructed in [4] is edge symmetric and has a connection, and so is not the same as that constructed here. A classification of certain double groupoids is given in =-=[11]-=-, but this does not yield much information for the double groupoid considered here. Thus there is still a way to go in the understanding and in the use of double groupoids. Higher homotopy groupoids w... |

23 |
Pure Galois theory in categories
- Janelidze
- 1990
(Show Context)
Citation Context ...te cyclic group, since there are compositions where the ai are alternately (−1α) and β. a1 ◦2 a2 ◦2 · · · ◦2 an The idea for this double groupoid arose from the Generalised Galois Theory of Janelidze =-=[14, 15]-=-, which under certain conditions gives a Galois groupoid from a pair of adjoint functors. The standard fundamental group arises from the adjoint pair between topological spaces and sets given by discr... |

18 |
Galois Theories. Cambridge
- Borceux, Janelidze
- 2001
(Show Context)
Citation Context ...er references in [1]), to define the Galois groupoid of (E,p) as GalI(E,p) = I(Eq(p)). (10) In particular this applies to the following situation studied by the authors before (see Proposition 3.5 in =-=[2]-=-): Proposition 1.3 Let I : C → X be the fundamental groupoid functor from the category C = Sets ∆op of simplicial sets to the category X = Grpd of (small) groupoids, and p : E → B a surjective fibrati... |

14 |
Höherdimensionale Homotopiegruppen
- Čech
- 1932
(Show Context)
Citation Context ...oups, using maps of spheres. However these groups were quickly proved to be abelian in dimensions > 1, and on this ground Čech was asked to withdraw his paper, so that only a small paragraph appeared =-=[12]-=-. Thus the dream of these topologists seemed to fail, and was widely felt to be a mirage, although the abelian higher homotopy groups became and still are very important. J.H.C. Whitehead in the 1940s... |

14 |
An abstract setting for homotopy pushouts and pullbacks. Cahier de topologie et géométrie différentielle catégorique
- Spencer
- 1977
(Show Context)
Citation Context ...π v 0(R2(q) + ∂ × − 1 ∂1 defined by the projections, is a bijection. R2(q)) → ρ2(q) + ∂ × − 1 ∂ ρ2(q) (21) 1 For the proof we use properties of the connections, and we use the following notation from =-=[21]-=-. We write: j j j+1 for Γ + j ; j+1 for εj; j j j+1 for Γ − j j+1 for εj+1. Thus the thick lines denote degenerate faces. We shall use inversions applied to connections, for example 2 −2 , , 1 10�� �... |

13 |
Multiple categories: the equivalence between a globular and cubical approach
- Al-Agl, Brown, et al.
- 2002
(Show Context)
Citation Context ...goes back to A. Grothendieck’s observation “the fundamental groupoids are to be defined as quotients of equivalence relations”, is used in categorical Galois theory and its various special cases (see =-=[1]-=-, [4], and other references in [1]), to define the Galois groupoid of (E,p) as GalI(E,p) = I(Eq(p)). (10) In particular this applies to the following situation studied by the authors before (see Propo... |

13 |
The equivalence of ∞-groupoids and crossed
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...died by J.H.C. Whitehead. However we do not have a reconstruction method for ρ(q) from ¯ρ(q), whereas the 2-groupoid can be reconstructed from the crossed module of groupoids it contains, as shown in =-=[7]-=-. ✷ Example 6.3 Foliations Let F be a foliation on a space M. Thus the leaves of the foliation define an equivalence relation R = R(F). Let q : M → B be a map of spaces. The foliation defines a finer ... |

13 |
Spaces with finitely many homotopy groups
- Loday
- 1982
(Show Context)
Citation Context ...1) 1� �� �� �� �� �� This double groupoid contains the 2-groupoid associated to a map defined by Kamps and Porter in [16], and hence also includes the cat 1 -group of a fibration defined by Loday in =-=[17]-=-, the 2-groupoid of a pair defined by Moerdijk and Svensson in [19], and the classical fundamental crossed module of a pair of pointed spaces defined by J.H.C. Whitehead. Advantages of our constructio... |

12 |
Galois theories
- Borceux, Janelidze
- 2001
(Show Context)
Citation Context ...te cyclic group, since there are compositions where the ai are alternately (−1α) and β. a1 ◦2 a2 ◦2 · · · ◦2 an The idea for this double groupoid arose from the Generalised Galois Theory of Janelidze =-=[14, 15]-=-, which under certain conditions gives a Galois groupoid from a pair of adjoint functors. The standard fundamental group arises from the adjoint pair between topological spaces and sets given by discr... |

11 | Kampen theorems for categories of covering morphisms in lextensive categories
- Brown, Janelidze
- 1997
(Show Context)
Citation Context ...ions gives a Galois groupoid from a pair of adjoint functors. The standard fundamental group arises from the adjoint pair between topological spaces and sets given by discrete and π0, see for example =-=[8]-=-. The adjoint pair between simplicial sets and crossed complexes given by nerve and π1 was studied in [9] and shown to lead to a Galois double groupoid of a fibration of simplicial sets. We are now gi... |

6 | Galois theory of second order covering maps of simplicial sets
- Brown, Janelidze
- 1999
(Show Context)
Citation Context ...the adjoint pair between topological spaces and sets given by discrete and π0, see for example [8]. The adjoint pair between simplicial sets and crossed complexes given by nerve and π1 was studied in =-=[9]-=- and shown to lead to a Galois double groupoid of a fibration of simplicial sets. We are now giving a topological version of this construction. We show that if p : E → B is a Serre fibration then the ... |

6 | The Holonomy of gerbes with connections
- Mackaay, Picken
(Show Context)
Citation Context ...ion of this construction could be used in association with the ‘thin fundamental groupoids’ and their smooth structures in differential geometrical situations, as exemplified by Mackaay and Picken in =-=[18]-=-. Here is some background to the search for higher groupoid models of homotopical structures (for more detailed references, see [3]). Geometers in the early part of the 20th century were aware that in... |

4 |
Algebraic classification of equivariant 2-types
- Moerdijk, Svensson
- 1993
(Show Context)
Citation Context ...associated to a map defined by Kamps and Porter in [16], and hence also includes the cat 1 -group of a fibration defined by Loday in [17], the 2-groupoid of a pair defined by Moerdijk and Svensson in =-=[19]-=-, and the classical fundamental crossed module of a pair of pointed spaces defined by J.H.C. Whitehead. Advantages of our construction are: (i) it contains information on the map q, and (ii) we get di... |

3 |
Groupoids and crossed objects in algebric topology’, Homotopy, homology and applications 1
- Brown
- 1999
(Show Context)
Citation Context ...ial geometrical situations, as exemplified by Mackaay and Picken in [18]. Here is some background to the search for higher groupoid models of homotopical structures (for more detailed references, see =-=[3]-=-). Geometers in the early part of the 20th century were aware that in the connected case the first homology group was the fundamental group made abelian, and that homology 2�� �� � �� � groups existe... |

3 |
A homotopy 2-groupoid from a fibration’, Homotopy, homology and applications
- Kamps, Porter
- 1999
(Show Context)
Citation Context ...i, Georgia. 1 2000 Maths Subject Classification: 18D05, 20L05, 55 Q05, 55Q35 � B (1) 1� �� �� �� �� �� This double groupoid contains the 2-groupoid associated to a map defined by Kamps and Porter in =-=[16]-=-, and hence also includes the cat 1 -group of a fibration defined by Loday in [17], the 2-groupoid of a pair defined by Moerdijk and Svensson in [19], and the classical fundamental crossed module of a... |