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Results tagged with gerbes
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"Gerbe" is a construct in homological algebra and topology. They can be seen as a generalization of principal bundles to the setting of 2-categories. "Gerbe" is a French (and archaic English) word that literally means wheat sheaf. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2.
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Crossed modules in context of gerbes
Question : How does Crossed modules comes into the set up of gerbes.
I am reading notes on 1- and 2-gerbes by Lawrence Breen. Once he defines torsors, he introduces notion of crossed modules. … As gerbes are seen as elements of third cohomology class here, I am expecting that understanding about crossed modules in same setup as that of gerbes might give better idea of gerbes. …
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Is a gerbe over a manifold is a special case of a gerbe over a stack?
There is a notion of Gerbe over a Manifold and a notion of Gerbe over a stack. Given a manifold $M$, there is a way to associate a stack $\underline{M}$ with it and this gives an embedding of cat …
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Trivializations of gerbes as generalisation of trivializations of line bundles
I understood gerbes as generalization of line bundle here.
In this, I am trying to understand how to generalize notion of trivialization of line bundle to the notion of trivialization of gerbes. … In case of gerbes it has to be on $1$ fold intersection extra i.e.,
$f_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow S^1$. …
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Cohomological description of gerbes over stacks
basically in gerbe territory (for smooth manifolds) if any one of the following is being considered
a cohomology class in $H^3(X,\mathbb{Z})$
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In similar manner, When reading about gerbes … Can some one give me some outline of how and what cohomology comes in when studying about gerbes over stacks? …
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Confusion in definition of Gerbes in Hitchin's notes
I am reading Nigel Hitchin's notes to understand about gerbes. … To understand gerbes, we need to consider the other creatures in a hierarchy to which gerbes belong, and here the lowest form of life consists of circle valued functions $f:X\rightarrow S^1$. …
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Cocycle description of gerbes
I am trying to understand cocycle description of gerbes as in https://arxiv.org/pdf/math/0611317.pdf. …
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What is there in the book Cohomologie non abélienne by Jean Giraud
These days I am trying to understand about stacks and gerbes.
Most of the articles that has something to do with gerbes cite this work Cohomologie non abélienne by Jean Giraud. …
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Connection on a Principal bundle and transition functions, as in Hitchin's notes
This is along the lines of this question
Gerbes are not just topological objects: we can do differential geometry with them too. We shall next describe what a connection on a gerbe is. …
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Understanding definition of gerbe over a stack
I am reading Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu.
They define gerbe over a stack as follows.
Let $\mathfrak{X}$ be a differentiable stack. …
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Central extension gives a gerbe over stack
Consider a central extension of Lie groups $1\rightarrow S^1\rightarrow \hat{G}\xrightarrow{\pi} G\rightarrow 1$.
I understand that this mean $\pi:\hat{G}\rightarrow G$ is a surjective homomorphism o …
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Lie groupoid $G$ extensions and principal $\text{Out}(G)$ bundles over Lie groupoids
I am reading the paper Non abelian differentiable gerbes by C. Laurent-Gengoux et.al. …
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Examples of of gerbe over stacks in terms of manifolds
I am looking for some examples of gerbes over stacks (as defined in Understanding definition of gerbe over a stack) that comes from manifolds. …
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How should one think about the band of a gerbe?
I have had a look at the notes on 1-gerbes and 2-gerbes by Lawrence Breen.
Question :
How should one think about the band of a gerbe? …
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Understanding the definition of $G$-gerbe
In Differentiable Stacks and Gerbes Kai Behrend and Ping Xu defines an $S^1$-gerbe as the following. …
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Roadmap to understand gerbe in the sense of Lurie’s Higher Topos Theory
Definition $7.2.2.20$ : Let $\mathfrak{X} $ be an $\infty$-topos. An $n$-gerbe on $\mathfrak{X}$ is an object in $\mathfrak{X}$ which is $n$-connective and $n$-truncated.
Above is the definition …
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