@MISC{Breen04, author = {Notes Prepared L. Breen}, title = {}, year = {2004} }

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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