entry under construction. See blog discussion here
For a 2-group, a -principal 2-bundle is the first step in the generalization of principal bundles to principal ∞-bundles.
for the delooping of an abelian group , -principal 2-bundles are equivalent to -bundle gerbes;
for the automorphism 2-group of an ordinary group , -principal bundles are equivalent to -gerbes.
But 2-bundles are a bit more general than these examples. For instance for the string Lie 2-group, -principal 2-bundles are smooth realizations of String structures on a space, in analogy to how ordinary -principal bundles are realizations of Spin structures. Part of the interest in principal 2-bundles derives from this example, which is believed to be an ingredient in geometric models for elliptic cohomology.
Principal 2-bundles had also been the testing ground for definitions of higher generalizations of the notion of connection on a bundle in the nonabelian situation. Proposals for definitions of the notion of connections on general 2-bundles or at least gerbes were put forward in BrMe and AsJu . But it remained unclear how to define parallel transport for these. In BaSc it was suggested that parallel transport in a 2-bundle should be a refinement of the cocycle of the bundle to a cocylce on the path 2-groupoid of . It was shown in BaSc, ScWaiII that this is equivalent to the data proposed by BrMe and AsJu – except that the 2-form curvature? appearing was constrained to vanish.
This led to some puzzlement, which in turn eventually led to the idea put forward here – at theory of differential nonabelian cohomology – , that a fully general differential cocycle is not an ordinary cocycle on a path n-groupoid, but instead a cocycle in twisted cohomology measuring the obstruction to having a cocycle on the full path ∞-groupoid .
Below we will first recall the ordinary cocycles on the path 2-groupoid from BaSc and ScWaIII . Then we unwrap the full machinery of differential nonabelian cohomology and show how this induces a notion of connectoin data on principal 2-bundles which has no constraints on the curvature? forms and is essentially the data proposed by BrMe and AsJu .
Context and notation
Throughout this page we fix the following choices and notation
For most of the development we can assume, for simplicity and definiteness, that is presented by the model structure on simplicial presheaves on Diff – and even simpler and even more definite, all of the objects that we will encounter are strict 2-groupoids or at worst Gray-groupoids (semistrict 3-groupoids) internal to diffeological spaces.
Recall from differential cohomology - nonabelian case? that a differential refinement in of a cocycle of a principal 2-bundle for some cover is a diagram
where is the cone on and its suspension.
Expressing this in terms of ∞-Lie algebroid valued differential forms by extracting the corresponding Cartan-Ehresmann ∞-connection data yields a diagram of dg-algebras
To express the explicit data encoded by a connection on a principal 2-bundle, we will in the following describe the data encoded by the horizontal morphism and then characterize the constraints imposed on it by the requirement that it sits in a diagram as shown.
strict structure 2-group
We concentrate first on the special case that the structure group is a strict 2-group.
group- and Lie-theoretic data
Let is a strict structure 2-group, i.e. is a strict ∞-Lie groupoid. This is equivalently encoded in a crossed module
in diffeological spaces.
The corresponding Lie 2-algebra is equivalent to a differential crossed module
and is characterized by its sheaf of ∞-Lie algebroid valued differential forms as
the cone structure group
A useful realization of the cone object as a Gray-groupoid was given in [RoSc].
Its Lie 3-algebra is with .
This Lie 3-algebra is such that
a truncation of the path -groupoid
Using the Gray-groupoid model for we may without loss generality replace morphisms out of the path ∞-groupoid by morphisms out of a Gray-groupoid truncation . An explicit such path 3-groupoid has been described in [MaPi], we obtain from this by identifying 3-morphisms that are given by homotopic maps.
Also described explicitly there is how Lie 3-algebra valued form data such as our integrates, using the above discussion, to morphisms .
the cover of the path -groupoid
Half of the work of extracting the explicit data of a differential cocycle is done by exhibiting a generators-and-relations presentation of the cover of the path ∞-groupoid that is induced from the given cover .
For this is done in great detail in [BaSc, ScWaIII] and the description of is entirely analogous. In fact, since by the very nature of the diagrams that define our cocycles all degree 3-generators of that are not given by 3-dimensional paths in will be mapped to the 2-groupoid and hence to identies, we can ignore all of them. The remaining generators and relations are precisely those of together with the 3-dimensional paths in .
Here is a list of the generators
for each point in a triple overlap a 2-morphism
for each path in a double overlap a 2-morphism
for each 2-path in a single patch a 2-morphism
and some others…
We shall assume now for definiteness that the object covering our based space by a weak equivalence is the Cech nerve
of a Cech cover . We could also consider more general covers and in particular could consider hypercovers, but this only makes the notation less transparent without yielding new insights for our purpose of extracting the nature of differential cocycle data on a principal 2-bundle.
So then let
be cocycle on for some -principal 2-bundle.
vanishing 2-form curvature
Before describing the general case, it is useful to recall the cocycle data of connections on principal 2-bundles for which the 2-form curvature? vanishes. Writing for the path 2-groupoid, these are given by extensions of the cocycle of the form
Detailed description of this is in [BaSc, ScWaIII].
We shall now demonstrate that the cocycle data for a general connection on a principal 2-bundle, in the sense of a general element in is given by the same data as above, the only difference being that the 2-form curvature? is not required to vanish.
The cocycle data of an element in is like that above, except that the 2-form is not required to vanish.
On the generators of cominmg from paths in the morphism comes from differential form data
with unconstrained curvature? form
The first -Ehresmann condition says that there is a commuting square
be one of the generating 2-cells in . Then this condition says that evaluated on this 2-cell has to reproduce the cocycle , with values in .
The second Ehresmann condition says that all curvature characteristic forms? obtained from invariant polynomials by descent to forms on .
This implies that for any 2-morphism generator coming from a surface in a double overlap with 2-morphism generators coming from the corresponding paths in double overlaps
we have that . By the very definintion of this is the case if the transformation relating with is given by transition functions in the image of the inclusion .
This transformation is given by conjugating with the value of on 2-cells of the form
So the second Ehresmann condition says that that also colors these 2-cells with elements in the image of the inclusion
weak structure 2-group