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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11 votes
1 answer
744 views

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 $\...
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 ...
3 votes
1 answer
142 views

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 $c_{ijk}$ ...
15 votes
1 answer
806 views

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 ...
4 votes
0 answers
215 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 $[\...
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 ...
5 votes
0 answers
198 views

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, \...
3 votes
1 answer
242 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 $\mathbb{F}_{p}$-...
9 votes
1 answer
744 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 ...
8 votes
0 answers
188 views

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 ...
2 votes
2 answers
473 views

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)...
1 vote
0 answers
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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: ...
19 votes
2 answers
3k views

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....
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 ...
4 votes
0 answers
360 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 ...
6 votes
3 answers
911 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 $\mathcal{F}$...
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\...
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 ...
16 votes
1 answer
819 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 ...
3 votes
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440 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 ...
7 votes
1 answer
947 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}\...
3 votes
0 answers
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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 ...
2 votes
2 answers
385 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 ...
5 votes
0 answers
414 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 ...
2 votes
1 answer
212 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 ...
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 ...
3 votes
2 answers
542 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 $\...
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 $G$-...
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 ...
19 votes
3 answers
2k views

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 ...
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 ...
3 votes
2 answers
344 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 ...
4 votes
1 answer
356 views

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 ...
3 votes
1 answer
233 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 ...
6 votes
2 answers
320 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 ...
5 votes
1 answer
593 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, ...
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 ...
2 votes
1 answer
549 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, ...
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 ...
11 votes
0 answers
997 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 ...
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 ...
15 votes
0 answers
443 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 ...
5 votes
1 answer
352 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 ...
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 ...
4 votes
1 answer
247 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 $\...
7 votes
1 answer
368 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 ...
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 ...
3 votes
0 answers
196 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 ...
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 $U_{ij}$, ...
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 ...