---
created: 2022-02-23T18:45
updated: 2023-04-02T13:47
title: classifying space
aliases: [classifying space, BG, EG, classifying, classify]
---


---

- Tags 
	- #homotopy #higher-algebra/stacks 
- Refs:
	- #todo/add-references
- Links:
	- [classifying stack](classifying%20stack.md)
	- [principal bundle](principal bundle.md)
	- [classifying space of a category](classifying space of a category.md)
	- [stack](Unsorted/stacks%20MOC.md)
	- [gerbe](gerbe.md)


---

# classifying space

A representation $V$ of $G$ is the same as [descent data](descent.md) to $\BG$ for $V$ (viewed as a vector bundle over a point).

![](attachments/Pasted%20image%2020220527234818.png)
![](attachments/Pasted%20image%2020220527234835.png)

![](attachments/Pasted%20image%2020220503124356.png)
![](attachments/Pasted%20image%2020220503124403.png)
![](attachments/Pasted%20image%2020220503124349.png)

![Pasted image 20211206201917.png](Pasted%20image%2020211206201917.png)

## Relation to characteristic classes

See [characteristic classes](Unsorted/characteristic%20class.md):
![](attachments/Pasted%20image%2020220503124434.png)

# Definitions

- An object of the category $\B G$ is a $G$-[torsor](torsor.md) $T$ (i.e. a non-empty $G$-set on which $G$ acts transitively and free). The morphisms are morphisms of $G$-sets.

:::{.definition title="?"}
The **classifying space** of a category is given by
$$
\B \cat{C} \da \realize{\nerve{\cat{C}}}
,$$
the [geometric realization](geometric realization.md) of the [nerve](nerve.md) of $\cat{C}$.
:::

![](attachments/Pasted%20image%2020220316195126.png)

- If $M$ is a monoid, then $\pi_1 \B M \cong M\complete{\gp}$ is the group completion of $M$.


:::{.definition title="?"}
Given $G\in \Top\Grp$, a **classifying space** for $G$, denoted $\B G$, is the base space of a universal [principal](principal%20bundle.md) $G\dash$ bundle
making $\B G$ a quotient of the contractible space $EG$ by a free $G\dash$ action, so $\B G \cong EG/G$. Call this the **the classifying bundle**:

\begin{tikzcd}
	G && EG \\
	\\
	&& BG
	\arrow["\pi", from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [https://q.uiver.app/?q=WzAsMyxbMCwwLCJHIl0sWzIsMCwiRUciXSxbMiwyLCJCRyJdLFsxLDIsIlxccGkiXSxbMCwxXV0=](https://q.uiver.app/?q=WzAsMyxbMCwwLCJHIl0sWzIsMCwiRUciXSxbMiwyLCJCRyJdLFsxLDIsIlxccGkiXSxbMCwxXV0=)

:::



Any other pullback of the classifying bundle along a map $X \to \B G$.

Let $I(G, X)$ denote the set of isomorphism classes of principal $G\dash$ bundles over a base space $X$, then
$$
I(G, X) \cong \hoTop(X, \B G)
$$
So in other words, isomorphism classes of principal $G\dash$ bundles over a base $X$ are equivalent to homotopy classes of maps from $X$ into the classifying space of $G$.

**Proposition**: 
There is a bijection for [vector bundles](Unsorted/vector%20bundles.md):
$$
\hoTop(X, \Gr_n(\RR)) \cong \theset{\text{rank $n$ $\RR\dash$vector bundles over $X$}} / \sim
$$
- Every such vector bundle is a pullback of the principal bundle
$$
\GL(n, \RR) \to V_n(\RR^\infty) \to \Gr(n, \RR)
$$

# Notes

- $\Sh(\BG; \Vect\slice k) \cong \Rep(G)\slice k$, and $\Sh(\B GX; \Vect\slice k)\cong \Sh^G(X; \Vect\slice k)$, the category of [equivariant sheaves](Unsorted/equivariant%20sheaves.md) for $G$ over $X$.
- $\B G \homotopic K(G, 1)$ when $G$ is discrete.
- $\pi_1(\B G) = G$ and $\pi_n(\B G) = \pi_n EG = 1$.

![attachments/Pasted image 20210505015056.png](attachments/Pasted%20image%2020210505015056.png)
![attachments/Pasted image 20210505015233.png](attachments/Pasted%20image%2020210505015233.png)

- $X/G$ may fail to be a nice space if points have nontrivial stabilizers.

- It is useful to think of $\B G$ as a space whose points are copies of $G$, so the classifying map $f\in \hoTop(X,\B G)$ assigns each $x \in X$ to the fiber above $x$, which is a copy of $G$.
- For a discrete group $G$, we have $\B G = K(G,1)$, so that $\pi_1(\B G) = G$ and $\pi_k(\B G) = 0$ for $k \neq 1$.
	- Follows from contractibility of $EG$ ?
- For $X\in \Top\Grp$, there is a weak equivalence $\Loop \B X \weakeq X$
	- How to prove: show they both represent the functor $\Prin_G(S^1 \smashprod (\wait)\addbase)$

![attachments/Pasted image 20210613124743.png](attachments/Pasted%20image%2020210613124743.png)

![Pasted image 20211104011349.png](Pasted%20image%2020211104011349.png)

## Constructions

- Standard procedure for constructing a classifying space for any group:
	- Construct a 2-complex with the given fundamental group, and then one inductively attaches higher dimensional cells to kill all higher homotopy groups. 
	- Each element $c\in \pi_n(X_{n−1})$ is represented by some continuous map $\gamma_c:S^n\to X_{n−1}$ with image in the $n\dash$-skeleton. 
	- Let $X_n$ be obtained from $X_{n−1}$ by attaching an $(n+1)\dash$cell along $\gamma_c$, for each $c\in π_n(X_{n−1})$.


![](attachments/Pasted%20image%2020220212184907.png)
![](attachments/Pasted%20image%2020220212185113.png)


Relation to [loop spaces](loop%20spaces) and [monoidal categories](Unsorted/monoidal%20category.md):
![](attachments/Pasted%20image%2020220316213400.png)

# Further Reading

- $\pi_{i+k}\B^k G = \pi_i G$.
	- Proof: If $G$ is a topological group, there is a universal principal $G\dash$bundle $EG \to BG$ which induces a LES in homotopy. 
	- Since $EG$ is contractible, $$\pi_i EG = \pi_{i+1}EG = 0\implies \pi_{i+1}BG \cong \pi_i G.$$ 
	- When $G$ is an $E_2$ space, $BG$ is a topological group, and so $$\pi_{i+2}(B^2G) = \pi_{i+2}(B(BG)) = \pi_{i+1}(BG) = \pi_i(G).$$
- Corollary: If $G$ is a discrete group, $\B^k G \homotopic K(G, n)$.
	- Proof: $\pi_0 G = G$ and $\pi_i G = 0$ for $i > 0$, so $\pi_k \B^k G = G$.

- One can take classifying spaces of [stacks](Unsorted/stacks%20MOC.md). 
	- There is a stack that classifies [connections](Unsorted/Riemannian%20Geometry.md), but it has issues: it is not a [representable](representable).

- $EG$ can be constructed as 
$$
EG \cong \bigcup_n G \ast G \ast \cdots \ast G
,$$ 
where $\ast$ is join of two spaces: the suspension of the smash product. For example, $G = \ZZ_2$ implies 
$$
EG \cong \bigcup_n \ZZ_2 \ast \cdots = \bigcup_n S^{n-1} = S^\infty
.$$

# Relation to Group Cohomology
![](attachments/Pasted%20image%2020220212233623.png)

Computing the torsion classes of $H^{*}(B \mathcal{G} ; \mathbb{Z})$ is an important problem; for example $H^{3}(B \mathcal{G} ; \mathbb{Z})$ classifies the set of [gerbes](Unsorted/gerbe.md).
# Unknown?

- What is $\pi_* \B G$?
	- What is the stable homotopy $\pi_* \Suspend^\infty \B G$?

- Conjecture: $\B (G \oplus H) = \B G \cross \B H$
	- Proof outline: $EG \cross EH$ is contractible, and $G \cross H$ acts freely on it with quotient equal to the RHS?
- Conjecture: $\B(G \ast H) = BG \vee BH$
- Conjecture: $\B(G \tensor_\ZZ H) = ?$ for $G, H\in \Ab$?
- Conjecture: $\B(G \semidirect_\phi H) = ?$

# Examples


![](attachments/Pasted%20image%2020220323175102.png)
![](attachments/Pasted%20image%2020220323175116.png)

![](attachments/Pasted%20image%2020220330195437.png)

![](attachments/Pasted%20image%2020220403181840.png)

# For topological groups

![](attachments/Pasted%20image%2020220403182000.png)

See [internal category](Unsorted/internal%20category.md)

![](attachments/Pasted%20image%2020220408001011.png)

# Homotopy construction

See [Blakers Massey](Unsorted/Blakers%20Massey.md)

![](attachments/Pasted%20image%2020220408001047.png)

# Pictures

![](attachments/2023-03-05BG.png)

Use bar labeling: ![](attachments/2023-03-05barlabel.png)
Can realize any element in $\BG$ as $[g_1 | \cdots | g_n]\in G\cartpower{n}$, so cross this with the $n\dash$simplex to get the collection of all $n\dash$simplices.

![](attachments/2023-03-05-1.png)