14
votes
Accepted
Non-Abelian Hodge theory
I personally like the notes by Eugene Xia: Abelian and Non-Abelian Cohomology to build intuition.
But for a definitive source, I would read Simpson: Moduli of representations of the fundamental group ...
- 8,529
12
votes
primary decomposition for nonabelian cohomology of finite groups
Yes, this is true.
Let $F$ be the free product of all the Sylow subgroups of $G$:
$$
F = \mathop\ast_{P\text{ Sylow}} P
$$
The inclusions $P \to G$ together give a group homomorphism $F \to G$. On ...
- 52.6k
9
votes
Accepted
Long exact sequence of cohomology from 2-groups
$c$ is your crossed module, or 2-group, in a sense. Anything more concrete will depend on a choice of a cocycle description of the pointed set $[BH,B^3A]$.
For example, $[BH,B^3A]$ classifies central ...
- 4,519
9
votes
Accepted
Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site?
Take $U=\coprod_{i∈I}Y(U_i)$, where $Y\colon C\to\mathop{\rm Presh}(C,{\rm Set})$ is the Yoneda embedding.
We have a canonical morphism $U→Y(X)$.
The Čech groupoid of $J_c$ can now be defined as
the ...
- 37.8k
8
votes
Accepted
primary decomposition for nonabelian cohomology of finite groups
It turns out the answer is no. Here I'll sketch a counter-example in which $H^1(G;M)$ is non-trivial (in fact infinite), while $H^1(H;M)$ is trivial for all proper subgroups $H<G$.
Let $G=A_5$, ...
- 35.9k
6
votes
Accepted
Non-abelian Ext functor and non-abelian $H^2$
EDITED, taking into account the comments of Donu Arapura.
As JLA wrote, a homomorphism $f\colon G\to N$ gives an extension
\begin{equation}\label{e:E}
1\to K\to E\to G\to 1.\tag{E}
\end{equation}
This ...
- 14.2k
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 ...
- 14.2k
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 ...
- 610
5
votes
Non-Abelian Hodge theory
I recently attented a nice online talk by Pengfei Huang and he indicated two sources:
the first chapter of his own phd Non-abelian Hodge theory and some specializations - TEL - Thèses en ligne
...
- 4,008
4
votes
Accepted
What extra structure does the group of automorphisms of a torsor carry?
I am not sure what kind of criterion you are after. You can give an answer in terms of cocycles or an answer in terms of killing an obstruction. Briefly, if $\mathcal{G}$ is a bundle of groups on $Y$ ...
- 6,239
3
votes
Minimal parabolic subgroups are $G(k)$-conjugate: a cohomological interpretation?
For simplicity we write $G$ for $G^{\rm der}$ and $M_0$ for $M_0^{\rm der}$, then $G$ is a connected reductive group, and $M_0$ is a Levi subgroup of a minimal parabolic $P_0$ of $G$.
Let $$\xi\in{\rm ...
- 14.2k
1
vote
Accepted
Connecting homomorphism in non-abelian cohomology
$\newcommand{\diag}{{\rm diag}}
\newcommand{\sH}{{\mathcal H}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\HH}{\sf H}
\newcommand{\V}{{\sf V}}
\newcommand{\B}{{\sf B}}
\newcommand{\C}{{\Bbb C}}
$No, ...
- 14.2k
1
vote
Accepted
Connection between the classifications of group extensions and group-graded algebras in terms of non-abelian cohomology
In the meantime I found the right nlab pages to read... The answer seems to be 2-groups! Specifically automorphism 2-groups. I will write up what I have come to understand so far (though I have not ...
- 9,650
1
vote
Non-abelian Ext functor and non-abelian $H^2$
If you have a morphism $f:G\to N\,,$ then you get a $K$-extension of $G$ by pulling back the $K$-extension of $N\,.$ The morphism $f$ lifts to a morphism into $M$ if and only if this extension is ...
- 1,198
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