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Results tagged with classifying-spaces
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user 4177
The classifying space BG of a group G classifies principal G-bundles, in that homotopy classes of maps [X, BG] are naturally identified with isomorphism classes of principal G-bundles P ⭢ X.
1
vote
Low dimensional integral cohomology of $BPSO(4n)$
There is something special about $H^4$ that doesn't work in general. This way I can rely on a result proved by Henriques in
The classification of chiral WZW models by $H^4_+(BG,\mathbb{Z})$, arXiv:16 …
6
votes
Accepted
Delooping of a group object as a one object groupoid
To provide a constructive answer, suppose $C$ is a category with finite products, and $G$ is a group object in $C$. Then the one-object groupoid object $G\rightrightarrows \ast$ in $C$ really is the d …
- 35.5k
10
votes
Classifying space as the geometric realization of the nerve of $G$ viewed as a small category
You should ignore simplicial objects at first, and just consider groupoids. In the following, you can let $G$ be a topological group such that $e\hookrightarrow G$ is a closed cofibration. All groupoi …
- 35.5k
1
vote
Classifying space of large category?
I will ignore the distinction here between the classifying space as a topological space or as a simplicial set. Fix for once and for all the adjunction $sSet \leftrightarrows Top$. Then giving the cla …
- 35.5k
11
votes
How are the classifying space of $E_8$ and $K(\mathbb{Z},4)$ related?
This answer is only a bit of a sketch for (2), because it's been a while since I thought/heard about this. And I interpret 'physical considerations' loosely, since this all comes from string theory, w …
- 35.5k
2
votes
Classifying Space of a Group Extension
A quick and dirty answer is that if we regard the groups as one-object groupoids, they again form a short exact sequence of groupoids, and the geometric realisation of this sequence is the $BH$ bundle …
- 35.5k