nLab principal 2-bundle

Idea

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

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

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

Definition

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

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

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

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

where $C(U)$ is the Cech nerve of a good open cover $U \to X$ and where $\mathbf{E}G$ 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 $G$ be a well pointed topological 2-group. Write $\mathbf{B}G := \bar W G$ for its delooping simplicial topological space and

$$B G := \vert \mathbf{B}G\vert \in Top$$

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

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

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 $G$ be a Lie 2-group with the property that $\pi_0 G$ is a smooth manifold and the projection $G_0 \to \pi_0 G$ is a submersion. Then equivalence classes of smooth $G$-principal bundles on a smooth manifold $X$ are in natural bijection with equivalence classes of topological $G$-principal 2-bundles (regarding $G$ as a topological 2-group)

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

induced by the natural forgetful functor SmoothMfd $\to$ Top.

This appears as (Nikolaus-Waldorf 11, prop. 4.1).

Corollary

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

Connections

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

$$X \to \mathbf{B} G$$

to a cocycle

$$P_1(X) \to \mathbf{B} G$$

where $P_1(X)$ is the path groupoid of $X$.

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

Examples

T-Folds

Principal 2-bundles for the T-duality 2-group serve to model T-folds in string theory (Nikolaus-Waldorf 18)

Related concepts

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

  • principal 2-connection

  • central extension of groupoids

  • line 2-bundle

• principal 3-bundle / 2-gerbe / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle / ∞-gerbe

  • circle n-bundle with connection

References

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

• J. F. Jardine, Cocycle categories (pdf) .

The above situation of ordinary $G$-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 $G$ a 2-group, a $G$-principal $2$-bundle is a fibration of groupoids $P \to X$ that arises as the homotopy fiber of a classifying morphism $X \to \mathbf{B}G$ (a $2$-anafunctor)

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

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

• Urs Schreiber, David Roberts, The inner automorphism 3-group of a strict 2-group (arXiv)

As ordinary principal bundles, the gadgets obtained this way may be described from various points of view, using anafunctor cocycles $g : X \stackrel{\simeq}{\to}\leftarrow Y \to \mathbf{B}H$ 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

• Paul Glenn, Realization of cohomology classes in arbitrary exact categories, J. Pure Appl. Algebra 25, 1982, no. 1, 33- 105 (doi:10.1016/0022-4049(82)90094-9)

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

• Toby Bartels, 2-Bundles (arXiv, web)

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

• J. F. Jardine, Z. Luo, Higher principal bundles, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 140, Issue 2 March 2006 , pp. 221-243 (pdf, doi:10.1017/S0305004105008911)

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

• Igor Bakovic, Bigroupoid 2-torsors (pdf).

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

• Behrang Noohi, E. Aldrovandi, Butterflies II: Torsors for 2-group stacks,arXiv

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

Classification results of principal 2-bundles are in

• John Baez, Danny Stevenson, The classifying space of a topological 2-group Algebraic Topology

  Abel Symposia, 2009, Volume 4, 1-31 (arXiv:0801.3843)

• David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces (web)

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

• Thomas Nikolaus, Konrad Waldorf, Four Equivalent Versions of Non-Abelian Gerbes, Pacific J. Math., 264-2 (2013), 355-420 (arXiv:1103.4815, DOI: 10.2140/pjm.2013.264.355)

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

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal ∞-bundles – theory, presentations and applications (arXiv:1207.0248, arXiv:1207.0249)

For more references see at principal 2-connection.

The example for structure group the T-duality 2-group is discussed, as a formalization of T-folds, in

• Thomas Nikolaus, Konrad Waldorf, Higher geometry for non-geometric T-duals (arXiv:1804.00677