nLab groupoid principal bundle

Idea

Where a principal bundle with structure group $G$ (typically at least a topological group) is a fiber bundle with typical fiber $G$ and with action/transition functions given by multiplication in $G$, the notion of a groupoid principal bundle generalizes this to structure groupoids $\mathcal{G}$ (typically at least topological groupoids) and typical fibers? being the source-fibers of $\mathcal{G}$ (the subspaces of morphisms all emanating from a given object), with action/transition functions given by composition of morphisms in $\mathcal{G}$.

Under the identification of groups with pointed connected groupoids (see at looping and delooping), groupoid principal bundles subsume ordinary principal bundles.

Conversely, a general groupoid principal bundle has typical fibers? that may (isomorphically) vary even over connected base spaces in a controlled way.

Definition

For $G$ a group object in some (∞,1)-topos $\mathbf{H}$ (for instance $\mathbf{H} =$ Smooth∞Groupoids for smooth Lie groupoid-bundles), and $\mathbf{B}G$ the corresponding delooping object, $G$-principal bundles are the (∞,1)-pullbacks of the form

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

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

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

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

are groupoid principal bundles .

Examples

Lie groupoid principal bundles

For $\mathbf{H}$ = $\infty LieGrpd$ and $G:=G_1 \Rightarrow G_0$ a Lie groupoid, a $G$-principal bundle is locally of the form

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

for $\mathcal{G}_{x_i}$ the target fiber over an object $x_i$, or equivalently of the form $U_i\times_{G_0} G_1$.

More precisely, we have the following sub-definition for Lie groupoid principal bundles: Given a Lie groupoid $G=G_1 \Rightarrow G_0$, a $G$-principal bundle over a manifold $M$ (with right $G$ action) is a surjective submersion $P\xrightarrow{\pi} M$ together with a moment map $P\xrightarrow{\rho} G_0$ (sometimes also called an anchor), such that there is a (right) $G$-action on $P$, that is, there is map $\Phi: P\times_{G_0} G_1 \to P$, such that

• $\pi(\Phi(p, g))=\pi(p)$,

• $\rho(\Phi(p,g))=s(g)$,

• $\Phi(\Phi(p, g_1), g_2)=\Phi(p, g_1 g_2)$.

Moreover, we require the $G$-action to be free and proper, that is,

• [free] $P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is injective,

• [proper]$P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P$ is a proper map , i.e. the preimage of a compact is compact.

These two conditions are equivalent to the fact that the shear map

$$P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times_M P$$

is an isomorphism.

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, Janez Mrcun Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003) (doi:10.1017/CBO9780511615450)