Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [crossed-modules]

The tag has no usage guidance.

10
votes
1answer
252 views

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
1answer
132 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
1answer
161 views

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'...
12
votes
2answers
360 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
1answer
78 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
0answers
185 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 ...
5
votes
3answers
468 views

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
2answers
581 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
2answers
778 views

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 (...
1
vote
0answers
74 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
1answer
166 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, ...