Notes on 1- and 2-Gerbes

@article{Breen2010NotesO1,
  title={Notes on 1- and 2-Gerbes},
  author={Lawrence S. Breen},
  journal={arXiv: Category Theory},
  year={2010},
  pages={193-235}
}
  • L. Breen
  • Published 10 November 2006
  • Mathematics
  • arXiv: Category Theory
The aim of these notes is to discuss in an informal manner the construction and some properties of 1- and 2-gerbes. They are for the most part based on the author’s texts [1–4]. Our main goal is to describe the construction which associates to a gerbe or a 2-gerbe the corresponding non-abelian cohomology class. 
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References

SHOWING 1-10 OF 26 REFERENCES
Differential geometry of gerbes
NOTES ON 2-GROUPOIDS, 2-GROUPS AND CROSSED MODULES
This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of…
On the correspondence between gerbes and bouquets
Duskin [ 1, 2 ] showed that gerbes over a Grothendieck topos E can be interpreted by certain internal groupoids of E which he called bouquets. He proved that for any bouquet B of E the fibred…
On weak maps between 2-groups
We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define…
Lectures on Special Lagrangian Submanifolds
These notes consist of a study of special Lagrangian submanifolds of Calabi-Yau manifolds and their moduli spaces. The particular case of three dimensions, important in string theory, allows us to…
On cocycle bitorsors and gerbes over a Grothendieck topos
The aim of this paper is to give a new description of Giraud's nonabelian cohomology set H 2 ( L ), [7], defined for a band L of a Grothendieck topos E as the set of L -equivalence classes of L…
Bundle gerbes
Just as C principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral…
Spaces with finitely many non-trivial homotopy groups
An outline of non-abelian cohomology in a topos : (I) The theory of bouquets and gerbes
© Andrée C. Ehresmann et les auteurs, 1982, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les…
Homotopy Transition Cocycles
For locally homotopy trivial fibrations, one can define transition functions gfifl : Ufi \ Ufl ! H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle…
...
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