nLab
principal 2-bundle

Idea

A principal 2-bundle is a categorification of a principal bundle. It is closely related to and in fact a generalization (in the case of a single group rather than a sheaf of groups) of the concept gerbe (although that concept could easily be further generalised to match). See gerbe (general idea) for more background.

Definition

Recall first the situation for ordinary principal bundles:

As essentially originally due to an an old observation by Segal, recalled in the context of interest here in

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

for $G$ a group and $BG$ the corresponding one-object groupoid, a $G$-principal bundle $P\to X$ is a bundle of “spaces” (in whatever context of generalized space one works) that arises as the homotopy pullback of the point (see generalized universal bundle) along a morphism $X\to BG$ in the right higher category of groupoids (for instance given by an anafunctor)

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

This is a very general construction that is to be interpreted and makes sense with all objects here appearing as generalized spaces called ∞-stacks, in practice in particular presented using the model structure on simplicial presheaves. The general idea used here is described at motivation for sheaves, cohomology and higher stacks and at gerbe (general idea).

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

• 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 $H$ a 2-group, an $H$-principal $2$-bundle is a fibration of groupoids $P\to X$ that arises as the homotopy fiber of a classifying morphism $X\to BH$ (a $2$-anafunctor?)

$$\begin{array}{ccc}P& \to & *\\ \downarrow & & \downarrow \\ X& \to & BH\end{array}.$$\array{ P &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}H } \,.

As ordinary principal bundles, these gadgets may be described from various points of view, using anafunctor cocycles $g:X\stackrel{\simeq }{\to }\leftarrow Y\to BH$ 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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-bundles by Glenn was established in

• Igor Baković, 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.

Connections

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

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

to a cocycle

$$P_1(X)\to BG$$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 $X\to BH$ to cocycles out of a higher path groupoid. Details on this are at Differential Nonabelian Cohomology.