bundles

cohomology

# Contents

## Idea

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

## Definition

For $G$ a group object in some (∞,1)-topos $\mathbf{H}$ (for instance $\mathbf{H} =$ ∞LieGrpd 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 $P \to G$ 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

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

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

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

## References

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

Revised on April 2, 2013 21:37:59 by Urs Schreiber (131.174.41.18)