nLab
groupoid principal bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

Definition

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

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

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

So more generally, for 𝒢\mathcal{G} any groupoid object with collection 𝒢 0\mathcal{G}_0 of objects, the (,1)(\infty,1)-pullbacks

P 𝒢 0 X 𝒢 \array{ P &\to& \mathcal{G}_0 \\ \downarrow && \downarrow \\ X &\to& \mathcal{G} }

are groupoid principal bundles .

Examples

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

U i×𝒢 x i U_i \times \mathcal{G}_{x_i}

for 𝒢 x i\mathcal{G}_{x_i} the source fiber over an object x ix_i.

Applications

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)