nLab
groupoid principal bundle

Idea

For $𝒢$ a groupoid object a $𝒢$-principal bunde is a morphism $P\to X$ with an principal action of $𝒢$ on $P$.

Definition

For $G$ a group object in some (∞,1)-topos $H$ (for instance $H=$ ∞LieGrpd for smooth Lie groupoid-bundles), and $BG$ the corresponding delooping object, $G$-principal bundles are the (∞,1)-pullbacks of the form

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X& \to & BG\end{array}.$$\array{ P &\to& * \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G } \,.

One equivalently (with non-negligible but conventional chance of confusion of terminology) calls such $P\to G$ a $BG$-groupoid principal bundle.

So more generally, for $𝒢$ any groupoid object with collection ${𝒢}_0$ of objects, the $(\mathrm{\infty },1)$-pullbacks

$$\begin{array}{ccc}P& \to & {𝒢}_0\\ \downarrow & & \downarrow \\ X& \to & 𝒢\end{array}$$\array{ P &\to& \mathcal{G}_0 \\ \downarrow && \downarrow \\ X &\to& \mathcal{G} }

are groupoid principal bundles .

Examples

For $H$ = $\mathrm{\infty }\mathrm{LieGrpd}$ and $𝒢$ a Lie groupoid, a $𝒢$-prinipal bundle is locally of the form

$$U_i\times {𝒢}_{x_i}$$U_i \times \mathcal{G}_{x_i}

for ${𝒢}_{x_i}$ the source fiber over an object $x_i$.

Applications

• Regarded as groupoid bibundles, groupoid principal bundles serve as models for Morita morphisms between Lie groupoids.

Related concepts

• principal bundle / torsor / groupoid principal bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / bundle 2-gerbe

• principal ∞-bundle

• higher differential geometry

References

• Ieke Moerdijk, J. Mrčun, Introduction to foliations and Lie groupoids Bulletin (New Series) of the AMS, Volume 42, Number 1, Pages 105–111