nLab
universal principal infinity-bundle

Idea

A universal principal ∞-bundle over an ∞-group-object in an (∞,1)-topos $H$ is a morphism $EG\to BG$ in a 1-categorical model $C$ for $H$ (a homotopical category) such that every $G$-principal ∞-bundle $P\to X$ in $H$ is modeled in $C$ by an (ordinary) pullback of $EG\to BG$.

Notice that in the proper (∞,1)-topos-context the universal $G$-principal ∞-bundle for an ∞-group $G$ is nothing but the point inclusion $*\to BG$ into the delooping of $G$: every $G$-principal $\mathrm{\infty }$-bundle $P\to X$ is the (∞,1)-pullback

$$\begin{array}{ccc}P& \to & *\\ \downarrow & {}^{\simeq }⇙& \downarrow \\ X& \stackrel{}{\to }& BG\end{array}$$\array{ P &\to& * \\ \downarrow &{}^{\mathllap{\simeq}}\swArrow& \downarrow \\ X &\stackrel{}{\to}& \mathbf{B}G }

of the point in $H$, namely the homotopy kernel of its classifying map $g$. In other words, in a full $(\mathrm{\infty },1)$-categorical context the notion of universal bundle disappears. It is a notion genuinely associated with 1-categorical models for $H$.

Standard models

Assume that we have a homotopical category model $C$ for $H$ that has the structure of a category of fibrant objects. Notably this can be the full subcategory on fibrant objects of a model structure on simplicial presheaves.

By fibrations

By standard results on homotopy pullbacks every morphism $EG\to B\prime G$ that

1. is a fibration

2. fits into a diagram

   $$\begin{array}{ccc}EG& \stackrel{\simeq }{\to }& *\\ \downarrow & & \downarrow \\ B\prime G& \stackrel{\simeq }{\to }& BG\end{array}$$\array{ \mathbf{E}G &\stackrel{\simeq}{\to}& * \\ \downarrow && \downarrow \\ \mathbf{B}'G &\stackrel{\simeq}{\to}& \mathbf{B}G }

   with the horizontal morphisms being weak equivalences;

is a model for the universal $G$-principal $\mathrm{\infty }$-bundle.

By path fibrations

A standard construction of a fibration $EG\to BG$ is above is obtained as follows:

by standard results on homotopy pullbacks, we have that the bundle $P\to X$ classified by a morphism $X\stackrel{\simeq }{\leftarrow }\hat{X}\to BG$ is given by the limit

$$\begin{array}{ccccc}P& \to & & \to & *\\ \downarrow & & & & \downarrow \\ & & (BG{)}^I& \stackrel{\simeq }{\to }& BG\\ \downarrow & & {\downarrow }^{\simeq }\\ *& \to & BG\end{array},$$\array{ P &\to& &\to& * \\ \downarrow && && \downarrow \\ && (\mathbf{B}G)^I &\stackrel{\simeq}{\to}& \mathbf{B}G \\ \downarrow && \downarrow^{\mathrlap{\simeq}} \\ * &\to& \mathbf{B}G } \,,

where $(BG{)}^I$ is a path space object for $BG$.

This limit may be computed as two consecutive pullbacks

$$\begin{array}{ccccc}P& \to & EG& \to & *\\ \downarrow & & & & \downarrow \\ & & (BG{)}^I& \to & BG\\ \downarrow & & \downarrow \\ *& \to & BG\end{array}.$$\array{ P &\to& \mathbf{E}G &\to& * \\ \downarrow && && \downarrow \\ && (\mathbf{B}G)^I &\to& \mathbf{B}G \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G } \,.

The intermediate pullback

$$EG:=(BG{)}^I{\times }_{BG}*$$\mathbf{E}G := (\mathbf{B}G)^I \times_{\mathbf{B}G} *

is the path fibration over $BG$. By the factorization lemma we have that the projecton $EG\to BG$ is indeed a fibration and by the fact that the acyclic fibration $(BG{)}^I\to BG$ is preserved under pullback that indeed $EG\to *$ is a weak equivalence.

By decalage

For $X$ a Kan complex with a single vertex, the decalage construction $\mathrm{Dec}X\to X$ is a Kan fibration that fits into a diagram

$$\begin{array}{ccc}\mathrm{Dec}X& \stackrel{\simeq }{\to }& *\\ \downarrow & & \downarrow \\ X& \stackrel{=}{\mathrm{to}}& X\end{array}.$$\array{ Dec X &\stackrel{\simeq}{\to}& * \\ \downarrow && \downarrow \\ X &\stackrel{=}{to}& X } \,.

For $G$ a simplicial group the standard simplicial model for the delooping of $G$ in $H=$∞Grpd is denoted $\overline{W}G$. This is a Kan complex with a single vertex and $\mathrm{Dec}\overline{W}G$ is the standard model for the universal simplicial principal bundle, traditionally written $WG$.

$$\mathrm{Dec}\overline{W}G=WG\to \overline{W}G.$$Dec \bar W G = W G \to \bar W G \,.

These constructions are functorial and hence extend to models for (∞,1)-toposes by a model structure on simplicial presheaves.

The model $WG$ for the universal $G$-principal bundle has the special property that it is a groupal model for universal principal ∞-bundles.

Related concepts

• universal principal bundle

• principal ∞-bundle

• universal principal $\mathrm{\infty }$-bundle , groupal model for universal principal ∞-bundles

• universal connection

• classifying space, classifying stack

• moduli space, moduli stack, derived moduli space

• classifying topos

• subobject classifier

• classifying morphism