nLab
principal 2-bundle

Context

Bundles

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

A principal 2-bundle is the generalization of a GG-principal bundle over a group GG to a principal structure over a 2-group. It is a special case of a principal ∞-bundle.

For G=AUT(H)G = AUT(H) the automorphism 2-group of a group HH, GG-principal bundles are equivalent to HH-gerbe (see gerbe (general idea) for more background.). An HH-nonabelian bundle gerbe is a model for the total space of an AUT(H)AUT(H)-principal 2-bundle.

An expository introduction to the concepts is at infinity-Chern-Weil theory introduction.

Definition

For GG a topological of Lie 2-group, a topological or smooth GG-principal 2-bundle PXP \to X is a topological or Lie groupoid that arises as the homotopy fiber of a cocycle XBGX \to \mathbf{B}G in ETop∞Grpd or Smooth∞Grpd, respectively, i.e. as an (∞,1)-pullback of the form

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

By the general rules of homotopy pullbacks, this may be modeled by an ordinary pullback of topological or Lie 2-groupoids of the form

P EG C(U) BG X, \array{ P &\to& \mathbf{E}G \\ \downarrow && \downarrow \\ C(U) &\to& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X } \,,

where C(U)C(U) is the Cech nerve of a good open cover UXU \to X and where EG\mathbf{E}G is the universal principal 2-bundle (RS). This says that principal 2-bundles are classified by Cech cohomology with coefficients in deloopings of (sheaves of) 2-groups.

Properties

Classification

Theorem

Let GG be a well pointed topological 2-group. Write BG:=W¯G\mathbf{B}G := \bar W G for its delooping simplicial topological space and

BG:=BGTop B G := \vert \mathbf{B}G\vert \in Top

for the corresponding geometric realization of simplicial topological spaces. Then BGB G is a classifying space for topological GG-principal 2-bundles: for XX a (sufficiently nice…) topological space we have that the nonabelian Cech cohomology on XX with coefficients in BG\mathbf{B}G is naturally in bijections with the set of homotopy classes of continuous functions XBGX \to B G

H Top(X,BG)[X,BG]. H_{Top}(X, \mathbf{B}G) \simeq [X, B G] \,.

This appears as (BaezStevenson, theorem 1). It is also a special case of the analogous theorem for topological principal infinity-bundles in (RobertsStevenson).

Theorem

Let GG be a Lie 2-group with the property that π 0G\pi_0 G is a smooth manifold and the projection G 0π 0GG_0 \to \pi_0 G is a submersion. Then equivalence classes of smooth GG-principal bundles on a smooth manifold XX are in natural bijection with equivalence classes of topological GG-principal 2-bundles (regarding GG as a topological 2-group)

H Top(X,BG)H smooth(X,BG) H_{Top}(X, \mathbf{B}G) \simeq H_{smooth}(X, \mathbf{B}G)

induced by the natural forgetful functor SmoothMfd \to Top.

This appears as (NikolausWaldorf, prop. 4.1).

Corollary

For Lie 2-groups with the above property, also smooth GG-principal 2-bundles have classifying space BG=BGB G = \vert \mathbf{B}G\vert.

Connections

An ordinary principal bundle may be equipped with a connection by refining the cocycle

XBG X \to \mathbf{B} G

to a cocycle

P 1(X)BG P_1(X) \to \mathbf{B} G

where P 1(X)P_1(X) is the path groupoid of XX.

Similarly, 2-bundles may be equipped with connections by refining their cocycles XBHX \to \mathbf{B}H to cocycles out of a higher path groupoid. Details on this are at differential cohomology in a cohesive topos.

References

The general description of higher bundles internal to generalized spaces modeled as ∞-stacks is discussed in

  • Jardine, Cocycle categories (pdf) .

The above situation of ordinary GG-principal bundles is section 2.1 Torsors for sheaves of groups in that article. The generalization to principal 2-bundles and principal ∞-bundles is then briefly indicated in section 2.2, Diagrams and torsors .

The point is that in the (∞,1)-topos of topological or smooth or whatever ∞-groupoids (i.e. in the (∞,1)-category of ∞-stacks on our category of test spaces) the above situation generalizes straightforwardly:

For GG a 2-group, a GG-principal 22-bundle is a fibration of groupoids PXP \to X that arises as the homotopy fiber of a classifying morphism XBGX \to \mathbf{B}G (a 22-anafunctor)

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

This may be modeled by the pullback of the universal principal 2-bundle as described in

As ordinary principal bundles, the gadgets obtained this way may be described from various points of view, using anafunctor cocycles g:XYBHg : X \stackrel{\simeq}{\to}\leftarrow Y \to \mathbf{B}H in nonabelian cohomology, or the corresponding total spaces being 2-torsors equipped with 2-group action, or certain variants of this.

Maybe the earliest explicit description of a principal \infty-bundle using a geometric definition of higher category is

  • P. Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25, 1982, no. 1, 33- 105.

This describes torsors over ∞-groupoids in terms of the corresponding \infty-action groupoids.

This theory of higher bundles and gerbes was made to look manifestly like a systematic categorification of the familiar description of ordinary principal bundles in terms of cocycles and local trivializations in

  • Luca Mauri, PhD thesis, 1998 (pdf);

An abridged version is

  • L. Mauri, M. Tierney, Two-descent, two-torsors and local equivalence , J. Pure Appl. Algebra 143 (1999), 313–327.

The first article in the differential-geometric context was

One should notice that if one uses categories internal to diffeological spaces, then these are (under their nerve) in particular simplicial presheaves, and that the anafunctors used as morphisms between these simplicial presheaves represent precisely the morphisms the corresponding (∞,1)-category of (∞,1)-sheaves using the model structure on simplicial presheaves or, more lightweight, the structure of a Brown category of fibrant objects on \infty-groupoid valued sheaves.

A description of higher principal bundles (see also principal ∞-bundle) in terms of the model structure on simplicial presheaves appears in

  • Jardine, Luo, Higher order principal bundles (pdf)

The relation of such 2-categorical constructions of 2-bundles to the one of simplicially modeled \infty-bundles by Glenn was established in

Still more explicit descriptions of these constructions are given in

  • Christoph Wockel, A global perspective to gerbes and their gauge stacks (arXiv) .

These constructions either work internal to Diff or internal to some topos.

More generally, a principal 2-bundle is a (2-truncated principal ∞-bundle) in a (∞,1)-topos of ∞-stacks over some site.

This is for instance in

Notice that torsor is just another word for (internal) principal bundle.

Classification results of principal 2-bundles are in

An extensive discussion of various models of principal 2-bundles is in

For a comprehensive account in the general context of principal infinity-bundles see

For more references see at principal 2-connection.

Revised on October 4, 2013 03:21:29 by Urs Schreiber (188.200.54.65)