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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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 $\...
Daniel Loughran's user avatar
2 votes
0 answers
75 views

Sections of a vector bundle taking values in a gerbe

We can make sense of sections of a vector bundle taking values in a line bundle; an oversimplified answer might be sections of the vector bundle tensored with the line bundle. However, I can't see the ...
Partha's user avatar
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1 answer
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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 $c_{ijk}$ ...
Hollis Williams's user avatar
4 votes
0 answers
214 views

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 $[\...
Leo Herr's user avatar
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5 votes
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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, \...
Exit path's user avatar
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3 votes
1 answer
241 views

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 $\mathbb{F}_{p}$-...
The Thin Whistler's user avatar
9 votes
1 answer
740 views

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 ...
Harry Gindi's user avatar
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8 votes
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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 ...
Harry Gindi's user avatar
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1 vote
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Confusion in understanding the notion of $G$ Principal bundle where $G$ is a geometric group over a site

The first paragraph of the section Overview in the paper Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson https://arxiv.org/abs/1207.0248 precisely reads the following: ...
Adittya Chaudhuri's user avatar
2 votes
2 answers
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What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?

Let $X$ be a topological space and $G$ be a topological group. Let $\tilde{G}$ be the sheaf of groups defined by the sheaf of sections of the product $G$ bundle $ \pi_1:X\times G \rightarrow X $. (1)...
Adittya Chaudhuri's user avatar
4 votes
0 answers
359 views

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 ...
curious math guy's user avatar
6 votes
3 answers
903 views

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 $\mathcal{F}$...
Praphulla Koushik's user avatar
2 votes
0 answers
56 views

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. The definition $3.4$ of the paper goes as follows: Definition : Let $X_1\xrightarrow{\phi} Y_1\...
Praphulla Koushik's user avatar
7 votes
0 answers
244 views

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 ...
Adittya Chaudhuri's user avatar
13 votes
3 answers
412 views

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 ...
Anton Kapustin's user avatar
3 votes
0 answers
439 views

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 ...
Praphulla Koushik's user avatar
3 votes
0 answers
105 views

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 ...
cheyne's user avatar
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5 votes
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413 views

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 ...
Praphulla Koushik's user avatar
2 votes
1 answer
211 views

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 ...
Praphulla Koushik's user avatar
7 votes
1 answer
941 views

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}\...
Praphulla Koushik's user avatar
3 votes
1 answer
249 views

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 ...
Praphulla Koushik's user avatar
8 votes
0 answers
191 views

"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 ...
wonderich's user avatar
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4 votes
1 answer
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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 ...
Praphulla Koushik's user avatar
3 votes
2 answers
540 views

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 $\...
Praphulla Koushik's user avatar
2 votes
2 answers
381 views

Cohomological description of gerbes over stacks

When understanding about gerbe over a manifold $X$ from Hitchin - Lectures on special Lagrangian submanifolds it is said that We are basically in gerbe territory (for smooth manifolds) if any one ...
Praphulla Koushik's user avatar
6 votes
2 answers
319 views

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 ...
Praphulla Koushik's user avatar
3 votes
1 answer
229 views

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 ...
Praphulla Koushik's user avatar
3 votes
2 answers
342 views

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 ...
Praphulla Koushik's user avatar
2 votes
1 answer
544 views

Confusion in definition of Gerbes in Hitchin's notes

I am reading Nigel Hitchin's notes to understand about gerbes. It starts the article by saying the following : Before giving a definition, it’s worthwhile to recognize when we, as mathematicians, ...
Praphulla Koushik's user avatar
5 votes
1 answer
589 views

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, ...
Praphulla Koushik's user avatar
1 vote
0 answers
428 views

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. I am ...
Praphulla Koushik's user avatar
16 votes
4 answers
2k views

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 ...
Praphulla Koushik's user avatar
11 votes
0 answers
991 views

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 ...
15 votes
0 answers
442 views

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 ...
cheyne's user avatar
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6 votes
0 answers
333 views

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 ...
Saal Hardali's user avatar
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4 votes
0 answers
234 views

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 ...
David Roberts's user avatar
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5 votes
1 answer
350 views

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 ...
FrankJ112's user avatar
1 vote
0 answers
160 views

On $G$-gerbes over the punctured disk

Let $G$ be a finite (not necessarily abelian) group and let $\mathcal{X}\to D^*$ be a $G$-gerbe over the punctured disk $D^*$. Is there a finite etale cover $D^*\to \mathcal{X}$? I think of $G$-...
Porto Vino's user avatar
40 votes
10 answers
4k views

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 ...
tttbase's user avatar
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8 votes
1 answer
381 views

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 ...
Justin Campbell's user avatar
4 votes
1 answer
246 views

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 $\...
Christian's user avatar
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7 votes
1 answer
366 views

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 ...
Marci Senf's user avatar
3 votes
1 answer
266 views

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 ...
Marci Senf's user avatar
3 votes
0 answers
195 views

Writing down gerbes explicitly over the projective line

Let $X = [\mathbb P^1/(\mathbb Z/2\mathbb Z)]$, where we take the trivial action of $\mathbb Z/2\mathbb Z$ on $\mathbb P^1$. Is this DM stack over $\mathbb C$ a gerbe over $\mathbb P^1$? Is it the ...
user2's user avatar
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9 votes
3 answers
2k views

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 $U_{ij}$, ...
Matthias Ludewig's user avatar
3 votes
0 answers
60 views

Connection on 3-bundle given as triplet of forms

A connection on a bundle is given locally by a Lie algebra-valued 1-form. Gauge transformations act in the usual way on the forms, and form a groupoid. A connection on a 2-bundle is given locally by ...
geodude's user avatar
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0 votes
0 answers
164 views

Where are there defined objects between gerbes and bundle gerbes?

Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorphism. Does this exist in the literature?
Jim Stasheff's user avatar
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14 votes
1 answer
974 views

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 ...
David Roberts's user avatar
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16 votes
1 answer
818 views

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 ...
Piotr Achinger's user avatar
5 votes
0 answers
396 views

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 ...
cheyne's user avatar
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