nLab groupoid-principal infinity-bundle

Idea

The generalization of a $G$-principal ∞-bundle over an ∞-group $G$ as $G$ is generalized to a groupoid object in an (∞,1)-category.

Definition

For $\mathbf{H}$ an (∞,1)-topos and $\mathcal{G}_\bullet \in Grp_\infty(\mathbf{H})$ a groupoid object in an (∞,1)-category, a $\mathcal{G}_\bullet$-principal $\infty$-bundle over $X$ is

• a morphism $P \to X$

• equipped with an anchor $a \colon P \to \mathcal{G}_0$ and a groupoid ∞-action of $\mathcal{G}_\bullet$ on $(P,a)$ over $X$;

• such that $P \to X \simeq (P//\mathcal{G})$ is the corresponding quotient map.

Related concepts

• ∞-action, principal ∞-bundle

• bibundle