# 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 $\...

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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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:
...

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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)...

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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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### 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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### 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\...

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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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### 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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### 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 ...

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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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### 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 ...

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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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### 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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### "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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### 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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### 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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### 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 ...

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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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### 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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### 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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### 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, ...

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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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### 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 ...

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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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### 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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### 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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### 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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### 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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### 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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### 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 $G$-...

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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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### 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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### 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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### 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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### 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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### 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 ...

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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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### 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 ...

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### 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?

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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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### 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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### 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 ...