Questions tagged [crossed-modules]
The crossed-modules tag has no usage guidance.
23 questions
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Postnikov invariant of crossed square
Is there a reference where Postnikov invariants of the classifying space of a crossed square have been computed ? I am especially interested in the computation of the third Postnikov invariant $B\...
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150
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Local-to-global philosophy for crossed modules
In the survey Groupoids and crossed objects in algebraic topology Ronald Brown made after Corollary 5.17 (p 30) an very interesting remark I not fully understand. He stated that this
Corollary 5.17 ...
- 5,998
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Geometric realization of crossed square
Given a crossed square of groups, you can "totalize" it and get a 2 crossed module in the sense of Conduché "Modules croisés généralisés de longueur 2", then you can apply his ...
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Explicit examples of 4-cocycles over finite 2-groups
By a (finite) 2-group $X$, I mean a finite group $G$, a finite abelian group $A$, an action of $G$ on $\operatorname{Aut}(A)$, as well as a 3-cocycle $\alpha\in H^3(BG, A)$. They are also equivalent ...
- 3,001
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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 ...
- 14.1k
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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 ...
- 5,230
6
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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 (...
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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 ...
- 6,023
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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 ...
- 1,407
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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 ...
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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 $j_h \in J$ ...
- 1,215
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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 ...
- 1,466
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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 ...
- 1,466
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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 ...
- 6,023
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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'...
- 5,565
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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 ...
- 5,565
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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 ...
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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 ...
- 1,466
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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 ...
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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 \...
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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 (...
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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\...
- 14.1k
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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, ...
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