nLab
groupoid principal bundle

Contents

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Where a principal bundle with structure group GG (typically at least a topological group) is a fiber bundle with typical fiber GG and with action/transition functions given by multiplication in GG, the notion of a groupoid principal bundle generalizes this to structure groupoids 𝒢\mathcal{G} (typically at least topological groupoids) and typical fibers being the source-fibers of 𝒢\mathcal{G} (the subspaces of morphisms all emanating from a given object), with action/transition functions given by composition of morphisms in 𝒢\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 GG a group object in some (∞,1)-topos H\mathbf{H} (for instance H=\mathbf{H} = Smooth∞Groupoids 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 PXP \to X 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

Lie groupoid principal bundles

For H\mathbf{H} = LieGrpd\infty LieGrpd and G:=G 1G 0G:=G_1 \Rightarrow G_0 a Lie groupoid, a GG-principal 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 target fiber over an object x ix_i, or equivalently of the form U i× G 0G 1U_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 1G 0G=G_1 \Rightarrow G_0, a GG-principal bundle over a manifold MM (with right GG action) is a surjective submersion PπMP\xrightarrow{\pi} M together with a moment map PρG 0P\xrightarrow{\rho} G_0 (sometimes also called an anchor), such that there is a (right) GG-action on PP, that is, there is map Φ:P× G 0G 1P\Phi: P\times_{G_0} G_1 \to P, such that

  • π(Φ(p,g))=π(p)\pi(\Phi(p, g))=\pi(p),

  • ρ(Φ(p,g))=s(g) \rho(\Phi(p,g))=s(g),

  • Φ(Φ(p,g 1),g 2)=Φ(p,g 1g 2)\Phi(\Phi(p, g_1), g_2)=\Phi(p, g_1 g_2).

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

  • [free] P× G 0G 1id×ΦP×PP\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times P is injective,

  • [proper]P× G 0G 1id×ΦP×PP\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× G 0G 1id×ΦP× MP P\times_{G_0} G_1 \xrightarrow{id\times \Phi} P\times_M P

is an isomorphism.

Applications

References

Last revised on April 21, 2021 at 03:26:40. See the history of this page for a list of all contributions to it.