# Questions tagged [crossed-modules]

The crossed-modules tag has no usage guidance.

19
questions

3
votes

1
answer

231
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### Is Deligne's braiding functorial?

$\newcommand{\ssc}{{\rm sc}}
\newcommand{\ad}{{\rm ad}}
\newcommand{\Fbar}{{\overline F}}
$
Let $F$ be a field and $\Fbar$ be a fixed algebraic closure of $F$.
Let $G$ be a (connected) reductive group ...

1
vote

1
answer

113
views

### Braided R-matrices for finite action groupoids

1. Algebra from action groupoids
Let $G$ be a finite group acting on a finite set $X$ from the
right (denoted in element as $x^{g}$). We have an algebra (of the
action groupoid) over $\mathbb{C}$: the ...

6
votes

1
answer

378
views

### Long exact sequence of cohomology from 2-groups

I am trying to understand the following from Principal Infinity Bundles - General by Nikolaus, Schreiber and Stevenson.
So following the reference there to Nikolaus-Waldorf tells us that given any (...

5
votes

2
answers

263
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### First time appearance of Lie crossed module (crossed module of Lie groups) in literature

Can someone point me to a reference where the notion of "Lie crossed module" appeared for the first time?
I see many papers "recall" the definition of the Lie crossed module but, I ...

7
votes

0
answers

162
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### Stabilisation of crossed modules?

D. Conduché has introduced a notion of a stable crossed module ("Modules croisés généralisés de longueur 2", JPAA 34 (1984) pp155–178, doi:10.1016/0022-4049(84)90034-3, Def. 3.1).
Is there a ...

1
vote

1
answer

125
views

### Classification of finite crossed modules

A finite crossed module is a 4-tuple
$$(G_1,G_2,\delta: G_2 \to G_1, \alpha: G_1 \to Aut(G_2))$$
satisfying certain compatible conditions, where the $G_i$ are finite groups and the maps are group ...

2
votes

0
answers

118
views

### On 2 crossed modules

Let $(G,H,\alpha,\tau)$ and $(H,J,\alpha',\tau')$ be two crossed modules of groups. It is given that $Kernel(\tau) \cap Image(\tau')$ =$e$. For every $h\in H$ does there always exist a $j_h \in J$ ...

3
votes

0
answers

100
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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 ...

9
votes

1
answer

384
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### Automorphisms of $GL_n(\mathbb{Z})$

I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand ...

3
votes

1
answer

216
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 ...

8
votes

1
answer

180
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### Can I recover a crossed module by its homomorphisms?

This is a follow up to this question.
Imagine there is a finitely presented crossed module $\mathcal{G} = (G,H, -\triangleright-\colon G \to \operatorname{Aut}(H), \delta\colon H \to G)$ which I don'...

13
votes

2
answers

627
views

### Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?

It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which ...

2
votes

1
answer

102
views

### Why does vertical multiplication in 2-groups not follow the same order as horizontal one when constructed from crossed modules?

Consider a 2-group (seen as a 2-category with only one object $\star$) constructed from a crossed module $(G,H,t,\rhd)$ ($\circ$ will denote 1-morphisms composition, $\circ_{h}$ 2-morphisms horizontal ...

2
votes

0
answers

291
views

### Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$?
Is it enough to consider the injection via Hochschild:
$Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$?
Edit: The ...

8
votes

3
answers

766
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### strict 2-groups VS crossed modules

nLab defines a strict 2-groups in many different but equivalent ways, among them:
an internal group object in Cat,
an internal group object in Grpd
Also, it is known that strict 2-groups may be ...

13
votes

2
answers

877
views

### Group cohomology without G-modules (a.k.a. what does this bar construction compute?)

Without any prior exposure to the cohomology of groups, one might naively proceed by replacing a group by a sort of resolution.
For instance, let's take $G = \mathbb{Z}^2$, and "resolve":
$$ 0 \to \...

7
votes

2
answers

1k
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### Maps between classifying spaces

Let $G$ be a discrete group and let $BG \simeq K(G,1)$ be its classifying space. Let $H$ be a topological group with classifying space $BH$.
In case $H$ is also discrete, it was pointed out in the (...

2
votes

0
answers

84
views

### Crossed modules quasi-isomorphic to a quasi-abelian crossed module

This is a follow-up of this question,
where the definition of a quasi-abelian crossed module was given.
Namely, a crossed module $\partial\colon F\to G$ is quasi-abelian
if the embedding $\partial_Z\...

3
votes

1
answer

196
views

### Non-quasi-abelian braided crossed modules

A right crossed module is a homomorphism of groups
$\partial\colon F\to G$
together with a right action of $G$ on $F$, written
$(g,f)\mapsto f^g$,
satisfying certain conditions.
The question is, ...