Abstract
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
L. Breen, Bitorseurs et cohomologie non abélienne in The Grothendieck Festschrift I, Progress in Mathematics 86:401–476 (1990), Birkhaüser.
L. Breen, Tannakian Categories, in Motives, Proc. Symp. Pure Math. 55(1):337–376 (1994).
L. Breen, Théorie de Schreier supérieure, Ann. Scient. École Norm. Sup. 25(4):465–514 (1992).
L. Breen, Classification of 2-gerbes and 2-stacks, Astérisque 225 (1994), Société Mathématique de France.
L. Breen and W. Messing, Differential Geometry of Gerbes, Advances in Math. 198:732–846 (2005).
R. Brown and N.D. Gilbert, Algebraic Models of 3-types and Automorphism structures for Crossed modules, Proc. London Math. Soc. 59:51–73 (1989).
P. Dedecker, Three-dimensional Non-Abelian Cohomology for Groups, in Cate-gory theory, homology theory and their applications II, Lecture Notes in Mathematics 92:32–64 (1969).
E. Dror and A. Zabrodsky, Unipotency and Nilpotency in Homotopy Equiva-lences, Topology 18:187–197 (1979).
J. Duskin, An outline of non-abelian cohomology in a topos (1): the theory of bouquets and gerbes, Cahiers de topologie et géométrie différentielle XXIII (1982).
J. Giraud, Cohomologie non abélienne, Grundlehren 179, Springer-Verlag (1971).
N. Hitchin, Lectures on special Lagrangians on submanifolds, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge MA, 1999), AMS/IP Stud. Adv. Math 23:151–182, American Math. Soc. (2001).
J.F. Jardine, Cocycle categories, arXiv:math.AT/0605198.
J.F. Jardine, Homotopy classification of gerbes arXiv:math.AT/0605200.
J.-L. Loday, Spaces with finitely many homotopy groups, J. Pure Appl. Alg. 24:179–202 (1982).
M.K. Murray, Bundle gerbes, J. of the London Math. Society 54:403–416 (1996).
B. Noohi, Notes on 2-groupoids, 2-groups and crossed modules, Homology, Homotopy and Applications 9(1):75–106 (2007).
B. Noohi, On weak maps between 2-groups, arXiv:math/0506313.
K. Norrie, Actions and automorphisms, Bull. Soc. Math. de France 118:109–119 (1989).
O. Schreier, Uber die Erweiterung von Gruppen, I Monatsh. Math. Phys. 34:165–180 (1926); II Abh. Math. Sem. Hamburg 4:321–346 (1926).
J.-P. Serre, Cohomologie galoisienne, Lecture Notes in Math 5 (1964), fifth revised and completed edition (1994), Springer-Verlag.
K.-H. Ulbrich, On the correspondence between gerbes and bouquets, Math. Proc. Cambridge Phil. Soc. 108:1–5 (1990).
K.-H. Ulbrich, On cocycle bitorsors and gerbes over a Grothendieck topos, Math. Proc. Cambridge Phil. Soc. 110:49–55 (1991).
J. Wirth and J. Stasheff, Homotopy Transition Cocycles, J. Homotopy and Related Structure 1:273–283 (2006).
A. Vistoli, Grothendieck topologies, fibered categories and descent theory, in Fundamental Algebraic Geometry: Grothendieck’s FGA explained, Mathematical Monographs 123:1–104, American Math. Soc. (2005).
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Breen, L. (2010). Notes on 1- and 2-Gerbes. In: Baez, J., May, J. (eds) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications, vol 152. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1524-5_5
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