# 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.

67 questions
Filter by
Sorted by
Tagged with
758 views

• 1,084
200 views

• 5,973
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 ...
• 1,185
417 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 ...
• 1,459
450 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 ...
• 5,973
108 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 ...
• 1,406
424 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 ...
• 5,973
213 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 ...
• 5,973
970 views

• 5,973
389 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 ...
• 5,973
321 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 ...
• 5,973
236 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 ...
• 5,973
353 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 ...
• 5,973
562 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, ...
• 5,973
602 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, ...
• 5,973
1 vote
430 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 ...
• 5,973
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 ...
• 5,973
1k 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 ...
448 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 ...
• 1,406
336 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 ...
• 7,629
235 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 ...
• 34.3k
353 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 ...
1 vote
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 $G$-...
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 ...
• 1,700
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 ...
• 3,545
248 views

• 15.3k