nLab universal principal infinity-bundle

Idea

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

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

$$\array{ P &\to& * \\ \downarrow &{}^{\mathllap{\simeq}}\swArrow& \downarrow \\ X &\stackrel{}{\to}& \mathbf{B}G }$$

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

Standard models

Assume that we have a homotopical category model $C$ for $\mathbf{H}$ 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 $\mathbf{E}G \to \mathbf{B}'G$ that

1. is a fibration

2. fits into a diagram

   $$\array{ \mathbf{E}G &\stackrel{\simeq}{\to}& * \\ \downarrow && \downarrow \\ \mathbf{B}'G &\stackrel{\simeq}{\to}& \mathbf{B}G }$$

   with the horizontal morphisms being weak equivalences;

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

By path fibrations

A standard construction of a fibration $\mathbf{E}G \to \mathbf{B}G$ is above is obtained as follows:

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

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

where $(\mathbf{B}G)^I$ is a path space object for $\mathbf{B}G$.

This limit may be computed as two consecutive pullbacks

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

The intermediate pullback

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

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

By decalage

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

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

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

$$Dec \bar W G = W G \to \bar W G \,.$$

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

The model $W G$ 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 $\infty$-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