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Results tagged with gerbes
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user 118688
"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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What is the geometric description of the set of isomorphism class of $G$-torsors over a site...
The notion of principal bundle over an differentiable stack; that is a special kind of fibered category $\mathcal{C}\rightarrow \text{Man}$ can be found in section $4$ of Differentiable stacks and gerbes …
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Phenomena of gerbes
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of topological spaces on $X$.
Then, the map $U\mapsto \pi_1(\mathcal{F}(U))$ for $U\subseteq X$ open is a gerbe over $X$.
I learned this e …
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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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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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Prerequisites for understanding algebraic geometry of “algebraic gerbes”
I am trying to learn about algebraic geometry of gerbes.
I am familiar with set up of gerbes in the case of differential geometry. … Though there is some similarity between differentiable gerbes and gerbes as mentioned above, they are not quite same. …
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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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Understanding the definition of $G$-gerbe
In Brauer Groups and Quotient stacks, they define $G$-gerbe as follows.
Set up : Fix a Noeth. scheme $X$. Let $G$ be a group scheme (flat, separated and of finite type) over $X$.
A $G$-gerbe over …
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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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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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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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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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Understanding definition of gerbe over a stack
I am not very comfortable to use the definition of epimorphism as in Differentiable Stacks and Gerbes.
I use the definition of epimorphism as in Principal actions of stacky Lie groupoids. …
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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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holonomy of connection on gerbes
I am reading this notes of Hitchin to understand about gerbes. … Any reference for concept of holonomy on gerbes would be useful.
EDIT : I thank user Tsemo for proving the equality that I said I was not able to prove. …
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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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