nLab groupoid principal bundle

Contents

Context

Bundles

bundles

Context

Classes of bundles

Universal bundles

Presentations

Examples

Constructions

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 07:26:40. See the history of this page for a list of all contributions to it.