---
tag: WSZ40H5E4T
kind: concept
name: Classifying space
msc: [55R35, 55R15]
sources:
  - url: https://en.wikipedia.org/wiki/Classifying_space
same_as:
  wikidata: Q5128445
  wikipedia: Classifying space
---

## Summary

The classifying space $BG$ of a topological group $G$ is the base of a [[universal bundle|1HDH716Z24]] $EG \to BG$.
Principal $G$-bundles over a CW complex $X$ correspond to homotopy classes of maps $X \to BG$.
Cohomology classes of $BG$ give the [[characteristic classes|7HACJC0JE2]] of such bundles.
For a discrete group $G$, the space $BG$ is the [[classifying space of the group|WA796ZCNRX]] in the sense of an Eilenberg–MacLane space $K(G, 1)$.

## Explanations

### L1 human

A classifying space is a catalogue of all bundles with a given structure group.
Each map from $X$ into the catalogue picks out a bundle over $X$, and deforming the map does not change the bundle.
For real line bundles the catalogue is the infinite projective space $\R P^\infty$, the space of lines through the origin in $\R^\infty$.
Mapping a point of $X$ to a line tells which line sits over that point.