Questions tagged [gerbes]
"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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Phenomena of gerbes
What is your favourite example of Gerbes?
I would like to know Where do we find Gerbes in "nature"?
The examples could vary from String theory to Galois theory. For example my favourite examples of ...
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Why do gerbes live in H^2?
Line bundles on a scheme $X$ live in $H^1(X,O_X^*)$, where $O_{X}^{*}$ is the sheaf of invertible functions. If $X$ is noetherian separated, then we can think of this $H^1$ to be Čech cohmology w.r.t....
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Second nonabelian group cohomology: cocycles vs. gerbes
In 1965 Jean Giraud published two Comptes Rendus notes titled "Cohomologie non abélienne", and in 1971 he published a book with the same title.
In 1966 Tonny A. Springer's paper "Nonabelian $H^2$ in ...
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Algebraic versus Analytic Brauer Group
Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$. Then I think that someone (Serre?) showed that the Cohomological Etale Brauer Group agrees with the torsion part of the Analytic ...
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What do gerbes and complex powers of line bundles have to do with each other?
We all know how to take integer tensor powers of line bundles. I claim that one should be able to also take fractional or even complex powers of line bundles. These might not be line bundles, but ...
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References on Gerbes
I am looking for some references related to gerbes and their differential geometry. Almost every article I have seen that is related to gerbes there is a common reference that is Giraud's book ...
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Are deformations of a scheme some kind of a "derived gerbe" under the cotangent complex?
(This is probably a very naive question. My understanding of the cotangent complex is quite vague.)
Let me first recall the picture for deformations of a smooth morphism:
If $f:X_0\to S_0$ is a ...
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Is the first differential Pontryagin class a morphism of stacks?
In Cech Cocycles for Characteristic Classes, Jean-Luc Brylinski and Dennis McLaughlin provide explicit formulas for Cech cocycles for characteristic classes of real and complex vector bundles, and ...
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Are bundle gerbes bundles of algebras?
The category of line bundles (possibly with connection)
on a smooth manifold M can be defined in two different ways:
The first definition uses transition functions that satisfy a cocycle condition
(...
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What is an example of a non-abelian gerbe with connection?
Abelian gerbes can arise from obstructions to lifting a principal $C$-bundle to a principal $B$-bundle given some central extension $0\to A \to B \to C \to 0$ or as a representative of a cohomology ...
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Gabber's proof of Br' = Br for quasiprojective schemes
In a note by deJong showing the cohomological and ordinary Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the ...
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Geometric models for 2-gerbes
One can think of a complex line bundle as a geometric model for an integral cohomology class of degree 2. Similarly, a locally-trivial bundle of $C^*$-algebras with fiber B(H) (the $C^*$-algebra of ...
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Recovering classical Tannaka duality from Lurie's version for geometric stacks
In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback ...
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Gerbes over finite fields
Let $k$ be a field with algebraic closure $\bar{k}$.
Recall that a gerbe over $k$ is an algebraic stack $\mathcal{G}$ over $k$ such that the groupoid $\mathcal{G}(\bar{k})$ is connected. We say that $\...
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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.
I do not read the ...
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Gerbes and Stacks
The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, ...
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Bundle Gerbes as Characteristic Classes
Perhaps this is a bit naïve, but I was wondering if it possible to (at least formally) represent Bundle Gerbes as Characteristic Classes. Disclaimer: My understanding of Bundle Gerbes is limited to ...
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Giraud's proper base change for Gerbes - Elimination of Noetherian hypotheses
I was looking through Giraud's book Cohomologie Non-abelienne, and there is a very nice theorem that Giraud proves in the Noetherian case (Cohomologie Non-Abelienne VII.2.2):
Let $f:X\to Y$ be a ...
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Geometrizing the Third Cohomology of a Complex Lie Group
If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural $\...
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Gerbes for a cyclic group. (or maybe G_m too)
Let μn be the group scheme of n-th roots of unity. If X is a scheme and L is a line bundle on X, then I can construct a μn-gerbe Y over X by letting the S-points of Y be a S-point of X, a line ...
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Gerbes on the multiplicative group
Let $k$ be an arbitrary field with absolute Galois group $\Gamma$. The group $\text{Hom}(\Gamma,\mathbb{Q}/\mathbb{Z})$ injects into $H^2(\mathbb{A}^1 \setminus \{ 0 \},\mathbb{G}_m)$, as one can see ...
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Closed immersion → Pro-open immersion factorization for residual gerbes
Let $X$ be a quasi-separated algebraic stack. Then it is a theorem of Rydh that every point $x$ in $X$ admits a residual gerbe. More or less, the construction proceeds by first taking the closure ...
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"Gerbes" in the cobordism theory
In a lecture I attended today, I heard the use of gerbes in the cobordism theory.
Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group ...
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What's a good reference about gerbes and bands?
I've seen several papers that I would like to read that use the language of gerbes and bands. The wiki page on gerbes is useful, but doesn't even contain the word 'band', so I'm left confused even ...
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Fiberwise criterion for a stack to be a gerbe
Let $f:X\to Y$ be a morphism of algebraic stacks.
If the geometric fibres of $f$ are algebraic spaces, then $f$ is representable by algebraic spaces.
I'm wondering about analogues of this fiberwise ...
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Understanding the definition of $G$-gerbe
In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following.
Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\...
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Weak 2-groups and non-abelian gerbe over a manifold
In literature, a strict 2-group is defined as a group object in the category of categories or groupoids. It has many well known equivalent descriptions viz:
1. A strict monoidal category in which all ...
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How should one think about the band of a gerbe?
Let $X$ be a topological space. Let $\mathcal{F}$ be a fibered category over $X$; seen as an assignment of a category $\mathcal{F}(U)$ for each open $U\subseteq X$.
A fibered catgeory $\mathcal{F}$...
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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.
Let $\mathcal{P}$ be a gerbe on a topological space $X$ i.e., $\mathcal{P}$ is a stack over ...
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Categorical formalism for higher non-abelian group cohomology / obstruction theory for gerbes?
I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question:
What's the neat abstract framework for obstruction theory for non-abelian ...
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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.
To begin with, ...
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Residual gerbe and field of moduli
I am studying residual gerbes from Laumon Moret-Bailly and I would like to know if the residue field of the residue gerbe has the following property. I am a beginner in this subject so I find ...
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Geometric interpretation of $\mathbb{C}^{\times}$-gerbes
Let $X$ be a (nice enough, e.g. smooth etc.) variety over the complex numbers, and let $\mathcal{G}$ be a gerbe on $X$. Then $\mathcal{G}$ is classified by a cohomology class in $\alpha \in H^2(X, \...
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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 of ...
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Differential Geometry of (Non-Abelian) Gerbes in the language of Brylinski
Context
In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there ...
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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 ...
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Smooth algebraic stacks with precisely two $\mathbb C$-objects
In my quest of "understanding" stacks, I recently tried to figure out the structure of a smooth algebraic stack of finite type $\mathcal X$ over $\mathbb C$ with affine diagonal and precisely one $\...
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K theoretic pushforward along gerbes
I have a nontrivial gerbe $\pi : \mathscr{G} \to X$ banded by a cyclic group $G = \mathbb{Z}/r$. I'm working over $\mathbb{C}$. I want to describe $\pi_\ast$ and relate the fundamental class $[\...
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Kottwitz global gerbes
I've been trying to understand global gerbes constructed by Kottwitz in $B(G)$ for all local and global fields (arXiv:1401.5728). Scholze explained in $p$-adic geometry (arXiv:1712.03708) why I would ...
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Sections of gerbes that can "vanish"
The notion of bundle gerbe is a categorification of line bundles/principal $U(1)$-bundles, and comes in two presentations: a linear version (with $Line_\mathbb{C}$-enriched underlying groupoid) and a ...
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holonomy of connection on gerbes
I am reading this notes of Hitchin to understand about gerbes. He defines gerbe by giving a collection of $2$ cocycles $g_{\alpha\beta\gamma}:U_\alpha\cap U_\beta\cap U_\gamma\rightarrow S^1$ with ...
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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. It was ...
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Connections on bundle gerbes from cocycle data
I am reading a 2007 article of Bressler et al. on deformation quantization of gerbes. In the article, the authors state that a gerbe on a manifold is defined using certain two-cocycles $c_{ijk}$ ...
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Are there any non-trivial $G$-gerbes over the analytic space $\mathbb C$
Does there exist a finite (abstract) group $G$ and a non-trivial $G$-gerbe $\mathcal X\to \mathbb C$, where we work in the category of analytic stacks.
My guess is that $G$-gerbes for $G$ an abelian ...
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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.
Let $M$ be a manifold then $\underline{M}$ is a stack ...
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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. An $\mathfrak{S}$-stack $\...
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Twisting an object P by an H-Torsor I
I am reading Brylinski's Loop Spaces, Characteristic Classes, and Geometric Quantization.
The Statement
Let $C$ be a gerbe on a space $X$ with "abelian" band $H$, $f: Y \to X$ a local ...
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Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
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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 ...
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Concerning the definition of a 2-crossed module
Question:
Is there some generalization of the definition of crossed-module which appropriately fits into the holonomy-considerations I am interested in and has, as an example, the generalization of ...