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NOTES ON 1 AND 2GERBES
, 2006
"... The aim of these notes is to discuss in an informal manner the construction and some properties of 1 and 2gerbes. 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 2gerbe the corresponding nonabelian coh ..."
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The aim of these notes is to discuss in an informal manner the construction and some properties of 1 and 2gerbes. 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 2gerbe the corresponding nonabelian cohomology class. We begin by reviewing the wellknown theory for principal bundles and show how to extend this to biprincipal bundles (a.k.a bitorsors). After reviewing the definition of stacks and gerbes, we construct the cohomology class associated to a gerbe. While the construction presented is equivalent to that in [4], it is clarified here by making use of diagram (5.1.9), a definite improvement over the corresponding diagram [4] (2.4.7), and of (5.2.7). After a short discussion regarding the role of gerbes in algebraic topology, we pass from 1 − to 2−gerbes. The construction of the associated cohomology classes follows the same lines as for 1gerbes, but with the additional degree of complication entailed by passing from 1 to 2categories, so that it now involves diagrams reminiscent of those in [5]. Our emphasis will be on explaining how the fairly elaborate equations which define cocycles and coboundaries may be reduced to terms which can be described in the tradititional formalism of nonabelian cohomology. Since the concepts discussed here are very general, we have at times not made explicit the mathematical
Galois Groupoids and Covering Morphisms in Topos Theory
"... The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given ..."
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The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S which is general enough to include, in addition to the covering projections determined by the locally constant objects, also the unramified morphisms of topos theory given by those local homeomorphisms which are at the same time complete spreads in the sense of BungeFunk (1996, 1998).
The simplicial interpretation of bigroupoid 2torsors
, 902
"... Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk’s classification of regular Lie groupoids in foliation theory [34] to Waldmann’s work on deformation quantization [38]. For any such action we intr ..."
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Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk’s classification of regular Lie groupoids in foliation theory [34] to Waldmann’s work on deformation quantization [38]. For any such action we introduce an action bicategory, together with a canonical projection (strict) 2functor to the bicategory which acts. When the bicategory is a bigroupoid, we can impose the additional condition that action is principal in bicategorical sense, giving rise to a bigroupoid 2torsor. In that case, the Duskin nerve of the canonical projection is precisely the DuskinGlenn simplicial 2torsor, introduced in [25]. 1
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, 2004
"... 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 2gerbes. The material pertaining to the construction of the associated cocycles is mainly based on the author’s texts [1] and [2]. A notable i ..."
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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 2gerbes. 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 algebrogeometric 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 Gbundle (or right Gtorsor) 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