30
votes
Phenomena of gerbes
At least when the group $G$ is discrete, and when the base is a topological space (as opposed to e.g. a scheme), I would like to advertise the fact that:
a $G$-gerbe is the same thing as a ...
- 43.2k
18
votes
Phenomena of gerbes
You can get a lot of examples by dimension shifting. Namely, consider any exact sequence of groups $$1\to K\to G \to H\to 1 \; .$$ Fix a $H$-torsor $T$. The stack $\mathcal G_T$ of liftings of the ...
- 3,998
14
votes
Accepted
Gerbes over finite fields
Suppose that $x$ is an object of $\mathcal{G}$ over a finite extension $k'/k$. Denote $\sigma : k' \to k'$ the $q$-power frobenius. Let $x^\sigma$ be the pullback of of $x$ by $\sigma$. As $\mathcal{G}...
- 1,144
10
votes
Phenomena of gerbes
The following is a fancy way of saying that every elliptic curve contains a copy of $\mathbb Z/2$ in its automorphism group:
The moduli stack of elliptic curves is a $\mathbb Z/2$-gerbe over some ...
- 43.2k
9
votes
References on Gerbes
My personal impression is that at least on the level of foundational theory, Higher Topos Theory of Lurie is a good source. I guess this also explain the hard time you feel finding references: Gerbes ...
- 4,189
8
votes
How should one think about the band of a gerbe?
if $X$ is a connected topological space without a chosen base point, then what is $\pi_1(X)$? The good answer is of course that it's a groupoid. [Actually, let's also assume that $\pi_n(X)=0$ for $n\...
- 43.2k
7
votes
Accepted
References on Gerbes
The book of Giraud is a fundamental reference on the subject, but you have to be used to the language of Grothendieck. A reference more accessible, for example for a differential geometer is the ...
- 3,652
6
votes
Phenomena of gerbes
My favorite one, in the sense that I am trying to really understand it for many years, is the determinantal gerbe of a locally linearly compact vector space, as described by Kapranov in "Semiinfinite ...
6
votes
References on Gerbes
Urs Schreiber has written a lot
on gerbes and their applications to physics:
https://ncatlab.org/nlab/show/Urs+Schreiber
See, for instance, the expository works
“Differential cohomology in a cohesive ...
- 37.8k
5
votes
Second nonabelian group cohomology: cocycles vs. gerbes
Nonabelian $H^2$ in Galois cohomology can be defined in terms of: (1) cocycles, (2) extensions, (3) gerbes. The relations between these three definitions are described in Section 2.2 of Le principe de ...
- 14.1k
5
votes
Second nonabelian group cohomology: cocycles vs. gerbes
I am not sure if this is the most general definition, but my proposal for 2nd nonabelian cohomology is combinatorial descent data in a cosimplicial crossed gropoid, modulo gauge equivalence. This is a ...
- 600
5
votes
Crossed modules in context of gerbes
This answer is meant to elaborate on my remark above concerning crossed modules as models for 2-groupoids. A crossed module consists of a group $G$ acting on a group $H$, together with a $G$-...
- 9,481
5
votes
How should one think about the band of a gerbe?
Like for sheaf of groups, you can try to describe the patching data required to glue sheaves of groupoids ("stacks"). A grebe is a sheaf of groupoids, locally equivalent to a sheaf of the form $BG$ ...
- 4,189
4
votes
Accepted
Confusion in definition of Gerbes in Hitchin's notes
Isn't the condition you mention just saying that the $g_{\alpha\beta\gamma}$ depends only on the corresponding intersection and not on the order in which the three sets are listed in writing down ...
- 9,597
4
votes
Accepted
Cocycle description of gerbes
It might help to divide this process into finer steps. We choose our open cover so that the groupoids $\mathcal{P}(U_i)$ and $\mathcal{P}(U_i \cap U_j)$ are connected and nonempty. Choose objects $...
- 45.7k
4
votes
References on Gerbes
The following reference might be helpful for you:
Hitchin, Lectures on Special Lagrangian submanifolds, $\S1$.
- 6,825
4
votes
Are deformations of a scheme some kind of a "derived gerbe" under the cotangent complex?
A related question is answered in https://arxiv.org/abs/1101.4069.
edit: My paper https://arxiv.org/abs/1712.01384 takes a very similar approach. It deforms modules instead of algebras, so it's ...
- 1,084
4
votes
Phenomena of gerbes
A typical example from deformation theory : fix $i:X_0\to X$ of first order thickening defined by a square zero ideal $\mathcal I$, and let $\mathcal E_0$ be a locally free sheaf of finite rank on $...
- 3,998
4
votes
Geometric models for 2-gerbes
One fairly concrete way to view these things is via the Cech model (= transition functions). But maybe this isn't what you were looking for...
Say the base space $X$ is a manifold or finite $CW$-...
- 1,150
4
votes
Geometric models for 2-gerbes
Two possible ideas. One is that you can realize $K(\mathbb{Z},3)$ as the quotient $U(HS)/PU(\infty)$ where $U(HS)$ is the unitary group on the Hilbert space of Hilbert-Schmidt operators. However, I ...
- 4,217
4
votes
How should one think about the band of a gerbe?
I will start by explaining the easiest possible case of bundle gerbes,
when the band A (alias structure group) is an abelian Lie group.
A bundle n-gerbe with band A over a smooth manifold M
is a ...
- 37.8k
4
votes
Accepted
Connections on bundle gerbes from cocycle data
A gerbe on a manifold $M$ is a morphism of simplicial presheaves
$$\def\tB{{\sf B}}\def\U{{\rm U}}\def\cC{{\sf\check C}}\cC(U)→\tB^2\U(1),$$
where $\{U_i\}_{i∈I}$ is an open cover of $M$, $\cC(U)$ is ...
- 37.8k
3
votes
Gerbes on the multiplicative group
I believe the answer is "yes, such a gerbe is necessarily trivial".
We first note that for any field $K$ and open subscheme $U$ of $\mathbb{A}^{1}_{K}$, the inclusion $\operatorname{Br}(U) \subseteq \...
- 2,017
3
votes
Phenomena of gerbes
If $\mathscr{M}$ is the moduli stack of mathematical objects $X$ of some specified kind such that $\mathrm{Aut}(X)$ always (naturally) contains a group (object) $G\subseteq\mathrm{Aut}(X)$, then $\...
- 23.3k
3
votes
Phenomena of gerbes
The root gerbes $${^r}\sqrt{\mathscr{L}/X}$$ associated to a line bundle on a scheme (or stack) $X$
- 23.3k
3
votes
Accepted
Examples of of gerbe over stacks in terms of manifolds
There are no other such gerbes. If $M$ and $N$ are manifolds, and $p\colon \underline{M}\to \underline{N}$ is a gerbe, then the corresponding map of manifolds is a diffeomorphism. The same holds if ...
- 35.4k
3
votes
Accepted
Is a gerbe over a manifold is a special case of a gerbe over a stack?
Yes. The category of manifolds embeds fully faithfully into the 2-category of stacks on Mfld, essentially by Yoneda, and the site structures are likewise compatible, so when restricted to the special ...
- 35.4k
3
votes
Accepted
Connection on a Principal bundle and transition functions, as in Hitchin's notes
A principal $G$-bundle $P\to M$ can be described by an open cover $(U_\alpha)$ of $M$ and a cocycle $g_{\beta\alpha}: U_\alpha\cap U_\beta\to G$. The total space is the quotient of the disjoint ...
- 34.7k
3
votes
holonomy of connection on gerbes
Take a closed surface, $\Sigma$, and map it into your manifold. If $\Sigma \hookrightarrow U_{i}$, then you can simply integrate $F_{i}$ over $\Sigma$ and obtain the holonomy in $S^1$, evaluated on $\...
- 1,466
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