## (2004)

### BibTeX

@MISC{Breen04,

author = {Notes Prepared L. Breen},

title = {},

year = {2004}

}

### OpenURL

### Abstract

A word of warning in lieu of introduction: The aim of the following notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. The material pertaining to the construction of the associated cocycles is mainly based on the author’s texts [1] and [2]. A notable improvement here is that some diagrams in [2] have been refomulated here as (hyper)cubical diagrams, which mirror in the present context certain diagrams introduced by W. Messing and the author in [3]. Since the concepts discussed are very general, it has at times not been made explicit to precisely which mathematical objects they apply. For example, when we refer to “a space ” this might mean a topological space, but also “a scheme ” when one prefers to work in an algebro-geometric context. Similarly, in computing 1 2 cocycles, we will always refer to spaces X endowed with a covering U: = (Ui)i∈I, but the entire discussion remains valid when � i Ui is replaced by a covering morphism Y − → X in an appropriate Grothendieck topology. Finally, there has been no attempt at a serious bibliography, or at making careful attributions of the results mentioned. 1. Torsors and bitorsors Let G be a bundle of groups on a space X. Definition 1.1. A right principal G-bundle (or right G-torsor) on X is a space P π − → X above X, together with a right group action P ×X G − → P of G on P such that the induced morphism

### Citations

88 | Bundle gerbes
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- 1996
(Show Context)
Citation Context ...h a (p ∗ 2G, p ∗ 1G)bitorsor P above Y ×X Y satisfying the coherence condition analogous to (6) above Y ×X Y ×X Y . A bitorsor P satisfying the coherence condition is also 17s18 called a bundle gerbe =-=[10]-=- , and this therefore corresponds to the giving of a gerbe P on X, together with a trivialization of its pullback to Y .s5. Cocyclic description of a gerbe Let us now choose arrows xj φij 19 �� xi (7)... |

58 |
On the classification of 2-gerbes and 2-stacks, Astérisque 225
- Breen
- 1994
(Show Context)
Citation Context ...uss in an informal manner the construction and some properties of 1- and 2-gerbes. The material pertaining to the construction of the associated cocycles is mainly based on the author’s texts [1] and =-=[2]-=-. A notable improvement here is that some diagrams in [2] have been refomulated here as (hyper)cubical diagrams, which mirror in the present context certain diagrams introduced by W. Messing and the a... |

50 |
Bitorseurs et cohomologie non abélienne. The Grothendieck Festschrift, Vol
- Breen
- 1990
(Show Context)
Citation Context ... to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. The material pertaining to the construction of the associated cocycles is mainly based on the author’s texts =-=[1]-=- and [2]. A notable improvement here is that some diagrams in [2] have been refomulated here as (hyper)cubical diagrams, which mirror in the present context certain diagrams introduced by W. Messing a... |

47 |
Cohomologie non abelienne, Grundlehren der
- Giraud
- 1971
(Show Context)
Citation Context ...ing proposition is known as the Morita theorem by analogy with the corresponding characterization in terms of bimodules of equivalences between certain categories of modules. Proposition 1.5. ( Giraud=-=[8]-=- ) i) An (H, G)-bitorsor Q on X determines an equivalence Tors(H) −→ Tors(G) P ↦→ P ∧ H Q between the corresponding categories of right torsors on X. ii) Any such equivalence Φ between two categories ... |

37 |
Algebraic models of 3-types and automorphism structures for crossed modules
- Brown, Gilbert
- 1989
(Show Context)
Citation Context ...5. i) When G is the gr-stack associated to a crossed module δ : G −→ Π, this coefficient crossed module of gr-stacks is a stackified version of the crossed square associated by K.J.Norrie ( see [11], =-=[4]-=-) to the crossed module G −→ π: G δ �� π �� Der ∗ (π, G) �� �� Aut(G → π) 41 (17)s42 It is however less restrictive than Norrie’s version, since the latter would correspond to the diagram of gr-stacks... |

4 |
An outline of non-abelian cohomology in a topos (1): the theory of bouquets and gerbes, Cahiers de topologie et géométrie différentielle XXIII
- Duskin
- 1982
(Show Context)
Citation Context ...orphisms x |Uα � xα compatible with the morphisms φαβ.s3. 1-gerbes We begin with the global description of the 2-category of gerbes, due to Giraud [8]. For other early discussions of gerbes, see [6], =-=[7]-=-. Definition 3.1. i) A (1)-gerbe on a space X is a stack in groupoids G on X which is locally nonempty and locally connected. ii) A morphism of gerbes (resp. a natural transformation between a pair of... |

4 |
On cocycle bitorsors and gerbes over a Grothendieck topos
- Ulbrich
- 1991
(Show Context)
Citation Context ... ∧ Pkl �� �� Pil (6) i) The isomorphism (5), satisfying the coherence condition (6), may be viewed as a 1-cocycle on X with values in the monoidal stack of G-bitorsors on X. We say, following Ulbrich =-=[12]-=- , that the family of bitorsors Pij form a bitorsor cocycle on X. ii) In the case of abelian G-gerbes ([2] definition 2.9), with G an abelian group, the monoidal stack of bitorsors on Uij may be repla... |

3 |
2-Cohomologie Galoisienne des Groupes semi-simples The`se Univ. de Lille
- Douai
- 1976
(Show Context)
Citation Context ... isomorphisms x |Uα � xα compatible with the morphisms φαβ.s3. 1-gerbes We begin with the global description of the 2-category of gerbes, due to Giraud [8]. For other early discussions of gerbes, see =-=[6]-=-, [7]. Definition 3.1. i) A (1)-gerbe on a space X is a stack in groupoids G on X which is locally nonempty and locally connected. ii) A morphism of gerbes (resp. a natural transformation between a pa... |

2 |
de France 118
- Math
- 1989
(Show Context)
Citation Context ...ark 6.5. i) When G is the gr-stack associated to a crossed module δ : G −→ Π, this coefficient crossed module of gr-stacks is a stackified version of the crossed square associated by K.J.Norrie ( see =-=[11]-=-, [4]) to the crossed module G −→ π: G δ �� π �� Der ∗ (π, G) �� �� Aut(G → π) 41 (17)s42 It is however less restrictive than Norrie’s version, since the latter would correspond to the diagram of gr-s... |

1 |
A 3-dimensional Non-Abelian Cohomology of Groups with Applications to Homotopy Classification
- Dedecker
- 1969
(Show Context)
Citation Context ...m) [ �mijk, gklm] g ijk νiklm This is an equation satisfied by elements with values in the fibre of Ar (G) above Uijklm. Similar non-abelian 3-cocycle relations first appeared in the work of Dedecker =-=[5]-=- , in the context of cohomology of groups, rather than as here for Čechcohomology. The definition of νijkl as the front face of the cube (15) induces by conjugation the following equation among elemen... |