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12
votes
2answers
331 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 ...
3
votes
1answer
282 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
60 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 ...
3
votes
1answer
155 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, ...