## Determination Of A Double Lie Groupoid By Its Core Diagram (1992)

Venue: | J. Pure Appl. Algebra |

Citations: | 27 - 14 self |

### BibTeX

@ARTICLE{Brown92determinationof,

author = {Ronald Brown and Kirill C. H. Mackenzie},

title = {Determination Of A Double Lie Groupoid By Its Core Diagram},

journal = {J. Pure Appl. Algebra},

year = {1992},

volume = {80},

pages = {237--272}

}

### Years of Citing Articles

### OpenURL

### Abstract

In a double groupoid S, we show that there is a canonical groupoid structure on the set of those squares of S for which the two source edges are identities; we call this the core groupoid of S. The target maps from the core groupoid to the groupoids of horizontal and vertical edges of S are now base-preserving morphisms whose kernels commute, and we call the diagram consisting of the core groupoid and these two morphisms the core diagram of S. If S is a double Lie groupoid, and each groupoid structure on S satisfies a natural double form of local triviality, we show that the core diagram determines S and, conversely, that a locally trivial double Lie groupoid may be constructed from an abstractly given core diagram satisfying some natural additional conditions. In the algebraic case, the corresponding result includes the known equivalences between crossed modules, special double groupoids with special connection (Brown and Spencer), and cat 1 --groups (Loday). These cases correspon...

### Citations

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Categories for the working mathematician
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Citation Context ...). In the case H = V; OE = id, we write \Theta(H ) and call it the comma double groupoid of H . The horizontal structure of \Theta(H; OE; V ) is precisely the comma category, in the sense of Mac Lane =-=[18]-=-, arising from the diagram H \Gamma! V /\Gamma H . In the differentiable setting, this construction arose from the study of extensions of principal bundles [15, x2]. This construction is less special ... |

206 |
Lie groupoids and Lie algebroids in differential geometry
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Citation Context ... between locally trivial double Lie groupoids and locally trivial core diagrams. We should first explain that we are using the term "Lie groupoid" in a sense different to that in which it wa=-=s used in [14]-=- and elsewhere in the work of the second-named author. In [14] a Lie groupoid was taken to be a differentiable groupoid satisfying a local triviality condition, and Lie groupoids were accordingly esse... |

128 |
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Citation Context ...passing by Pradines ([19], [20] and elsewhere). Very recently double Lie groups have been studied by several authors in connection with Poisson Lie groups and related structures (see Lu and Weinstein =-=[13]-=- and references given there); these may be regarded as double Lie groupoids in which the double base is a singleton, so that the side groupoids are in fact groups, and which satisfy a further strong c... |

124 | The algebra of cubes
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Citation Context ...motopy. This result is easily extended to give an equivalence between arbitrary special double groupoids with special connection and crossed modules over groupoids; this is included in the results of =-=[3]-=-. We recall these results in more detail in x3 below. Special double groupoids with special connection and a singleton double base are also equivalent to the cat 1 -groups of Loday. Cat n -groups, for... |

83 |
Algebraic Constructions in the Category of Lie Algebroids
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(Show Context)
Citation Context ...hen proceed to x2, referring back to the examples as necessary. We begin by recalling the main classes of morphisms of (ordinary) Lie groupoids; the terminology which follows is an amalgam of that of =-=[10]-=- with that of Pradines [23]. p.244 We will often use the notation G \Gamma! \Gamma! B to indicate briefly that G is a groupoid on base B. Here the two arrows should be thought of as the source ff: G !... |

76 | Colimits theorems for relative homotopy groups - Brown, Higgins - 1981 |

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Spaces with finitely many nontrivial homotopy groups
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Citation Context ...x3 below. Special double groupoids with special connection and a singleton double base are also equivalent to the cat 1 -groups of Loday. Cat n -groups, for any positive integer n, were introduced in =-=[12]-=- as algebraic models of homotopy (n + 1)-types; a cat 1 -group is a group G together with endomorphisms s; t: G ! G such that st = t; ts = s and such that ker(s) and ker(t) commute elementwise. It was... |

62 |
Coisotropic calculus and Poisson groupoids
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Citation Context ...e regarded as double Lie groupoids in which the double base is a singleton, so that the side groupoids are in fact groups, and which satisfy a further strong condition. Also quite recently, Weinstein =-=[24]-=- has introduced a notion of symplectic double groupoid. The double groupoids which arise in homotopy theory are of a particularly special type, and admit several equivalent descriptions. The special d... |

51 | On the connection between the second relative homotopy groups of some related space
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Citation Context ...]), but primarily as instances of double categories, and as a part of a general exploration of categories with structure. Since that time their main use has been in homotopy theory. Brown and Higgins =-=[2] gave the -=-earliest example of a "higher homotopy groupoid", by associating to a pointed pair of spaces (X; A) a special double groupoid with special connection,sae(X; A). Such double groupoids have id... |

34 |
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Citation Context ...) for all composable h; h 0 2 H (where e 1 H h 0 ; e 1 V h 0 denote identities for the horizontal and vertical structures on S; see x1). Such maps \Gamma had already been studied by Brown and Spencer =-=[6] under the-=- name of "special connections"; condition (2) is called the transport law. The special connection encodes the basic properties characteristic of the homotopy double groupoidsae(X; A) [2]; in... |

25 | Fibrations of groupoids
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Citation Context ...e speak of an s-fibration or an s-action morphism. This concept of fibration, introduced by Pradines [23] under the name "exacteur", is a smooth form of the algebraic notion of fibration of =-=groupoids [1]. See also-=- [10], [11]. In the algebraic setting, action morphisms were formerly often called "coverings"(for example [1]). When f is a surjective submersion one can also form the pullback (which is in... |

15 |
Classification of principal bundles and Lie groupoids with prescribed gauge group bundle
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Citation Context ...ication of such core diagrams. This is the analogue for double groupoids of the problem of describing all locally trivial Lie groupoids with prescribed base and prescribed gauge group bundle (compare =-=[17]-=-). In principal bundle terms, it is the problem solved by the concept of transition function. We treat a very special case of the first part of this problem in the next section. 3 SPLIT DOUBLE GROUPOI... |

14 |
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Citation Context ...r the p.239 fundamental group �� 1 (A). In differential geometry, double Lie groupoids, but usually with one of the structures totally intransitive, have been considered in passing by Pradines ([1=-=9], [20]-=- and elsewhere). Very recently double Lie groups have been studied by several authors in connection with Poisson Lie groups and related structures (see Lu and Weinstein [13] and references given there... |

11 |
Représentation des jets non holonomes par des morphismes vectoriels doubles soudés
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Citation Context ...lt. The general case will be taken up elsewhere. The concept of core groupoid may also be regarded as a generalization of Pradines' concept of the core (French: coeur) of a double vector bundle [20], =-=[21]-=-, [22]; this is, of course, the origin of our terminology. The core of a double vector bundle is the intersection of the kernels of the two bundle projections; it inherits a unique vector bundle struc... |

10 |
Géométrie différentielle au-dessus d’un groupoïde
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Citation Context ...le over the p.239 fundamental group �� 1 (A). In differential geometry, double Lie groupoids, but usually with one of the structures totally intransitive, have been considered in passing by Pradin=-=es ([19]-=-, [20] and elsewhere). Very recently double Lie groups have been studied by several authors in connection with Poisson Lie groups and related structures (see Lu and Weinstein [13] and references given... |

7 |
Fibrations and quotients of differentiable groupoids
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Citation Context ...ation or an s-action morphism. This concept of fibration, introduced by Pradines [23] under the name "exacteur", is a smooth form of the algebraic notion of fibration of groupoids [1]. See a=-=lso [10], [11]. In the a-=-lgebraic setting, action morphisms were formerly often called "coverings"(for example [1]). When f is a surjective submersion one can also form the pullback (which is in fact the pullback gr... |

6 |
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(Show Context)
Citation Context ...les (over groups) and to group objects in the category of groupoids. That group objects in the category of groupoids are equivalent to crossed modules had been shown much earlier by Brown and Spencer =-=[5]-=-, in a result there attributed to Verdier. There is thus a commuting square of equivalences between the concepts of special double groupoid with special connection and a singleton base, group object i... |

4 | Ehresmann, Multiple functors IV. Monoidal closed structures on Catn, Cahiers de Topologie et Géométrie Différentielle Catégoriques 20 - Ehresmann, Charles - 1979 |

4 |
Quotients de groupoïdes différentiables
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(Show Context)
Citation Context ...g back to the examples as necessary. We begin by recalling the main classes of morphisms of (ordinary) Lie groupoids; the terminology which follows is an amalgam of that of [10] with that of Pradines =-=[23]-=-. p.244 We will often use the notation G \Gamma! \Gamma! B to indicate briefly that G is a groupoid on base B. Here the two arrows should be thought of as the source ff: G ! B and target fi: G ! B map... |

3 | A note on Lie algebroids which arise from groupoid actions. Cahiers Topologie Géom - Mackenzie - 1987 |

3 |
Suites exactes vectorielles doubles et connexions
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Citation Context ...e general case will be taken up elsewhere. The concept of core groupoid may also be regarded as a generalization of Pradines' concept of the core (French: coeur) of a double vector bundle [20], [21], =-=[22]-=-; this is, of course, the origin of our terminology. The core of a double vector bundle is the intersection of the kernels of the two bundle projections; it inherits a unique vector bundle structure a... |

2 |
Cat'egories structur'ees. Ann. Sci. ' Ecole Norm
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Citation Context ...tegories, where the basic laws (such as associativity, and the existence of identities) hold only up to equivalence. Double groupoids were introduced by Ehresmann in the early 1960's (see for example =-=[9]-=-, [7]), but primarily as instances of double categories, and as a part of a general exploration of categories with structure. Since that time their main use has been in homotopy theory. Brown and Higg... |

2 |
On extensions of principal bundles
- Mackenzie
- 1988
(Show Context)
Citation Context ...ion theory. In the same way it seems reasonable to expect that the connection theory of a locally trivial double Lie groupoid can be studied in terms of connections (in the slightly extended sense of =-=[16]-=-) in the exact sequences of its core diagram. The correspondence given in x3 between special connections in the sense of [6] and splittings in the core diagram may be regarded as the flat case of this... |