This entry discusses in the general context of differential nonabelian cohomology the special case of $G$-principal 2-bundles with connection: differential cocycles in $\bar \mathbf{H}(X, \mathbf{B}G)$ for $G$ a 2-group.
entry under construction. See blog discussion here
For $G$ a 2-group, a $G$-principal 2-bundle is the first step in the generalization of principal bundles to principal ∞-bundles.
For instance
for $G = \mathbf{B}A$ the delooping of an abelian group $A$, $G$-principal 2-bundles are equivalent to $A$-bundle gerbes;
for $G = AUT(H)$ the automorphism 2-group of an ordinary group $H$, $G$-principal bundles are equivalent to $H$-gerbes.
But 2-bundles are a bit more general than these examples. For instance for $G$ the string Lie 2-group, $G$-principal 2-bundles are smooth realizations of String structures on a space, in analogy to how ordinary $Spin(n)$-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 $X \to \mathbf{B}G$ of the bundle to a cocylce $\P_2(X) \to \mathbf{B}G$ on the path 2-groupoid of $X$. 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 $\Pi(X)$.
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 .
Throughout this page we fix the following choices and notation
a context $\mathbf{H}$ for differential cohomology in wich we have a notion of ∞-Lie theory, as described there:
a 2-group $G$, i.e. an object $G \in \mathbf{H}$ which has a delooping ∞-Lie groupoid $\mathbf{B}G$ which is 2-truncated;
the corresponding ∞-Lie algebroid $\mathfrak{g} = Lie(\mathbf{B}G)$ – which is a Lie 2-algebra.
For most of the development we can assume, for simplicity and definiteness, that $\mathbf{H}$ 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 $\bar \mathbf{H}(X,\mathbf{B}G)$ of a cocycle $g : Y \to \mathbf{B}G$ of a principal 2-bundle for some cover $Y \stackrel{\simeq}{\to} X$ is a diagram
where $\epsilon \mathbf{B}G$ is the cone on $\mathbf{B}G$ and $\Sigma \mathbf{B}G$ 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 $\Pi(Y) \to \epsilon \mathfrak{g}$ and then characterize the constraints imposed on it by the requirement that it sits in a diagram as shown.
We concentrate first on the special case that the structure group $G$ is a strict 2-group.
Let $G$ is a strict structure 2-group, i.e. $\mathbf{B}G$ is a strict ∞-Lie groupoid. This is equivalently encoded in a crossed module
The corresponding Lie 2-algebra $\mathfrak{g}$ is equivalent to a differential crossed module
and is characterized by its sheaf of ∞-Lie algebroid valued differential forms as
A useful realization of the cone object $\epsilon \mathbf{B}G$ as a Gray-groupoid $\mathbf{B}INN(G)$ was given in [RoSc].
Its Lie 3-algebra is $inn(\mathfrak{g})$ with $CE(inn(\mathfrak{g})) = W(\mathfrak{g})$.
This Lie 3-algebra is such that
Using the Gray-groupoid model $\mathbf{B}INN(G)$ for $cone(\mathbf{B}G)$ we may without loss generality replace morphisms $\Pi(Y) \to \mathbf{B}INN(G)$ out of the path ∞-groupoid by morphisms out of a Gray-groupoid truncation $\Pi_3(Y)$. An explicit such path 3-groupoid $P_3(Y)$ has been described in [MaPi], we obtain $\Pi_3(Y)$ 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 $\Omega^\bullet(Y) \leftarrow CE(inn(\mathfrak{g}))$ integrates, using the above discussion, to morphisms $\Pi_3(Y) \to \mathbf{B}INN(G)$.
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 $\Pi(Y) \stackrel{\simeq}{\to} \Pi(X)$ of the path ∞-groupoid $\Pi(X)$ that is induced from the given cover $Y \stackrel{\simeq}{\to} X$.
For $P_2(Y) \to P_2(X)$ this is done in great detail in [BaSc, ScWaIII] and the description of $\Pi_3(Y)$ is entirely analogous. In fact, since by the very nature of the diagrams that define our cocycles all degree 3-generators of $\Pi_3(Y)$ that are not given by 3-dimensional paths in $U$ will be mapped to the 2-groupoid $\mathbf{B}G$ and hence to identies, we can ignore all of them. The remaining generators and relations are precisely those of $P_2(Y)$ together with the 3-dimensional paths in $U$.
Here is a list of the generators
for each point $(x,i,j,k) \in U_{i} \cap U_j \cap U_k$ in a triple overlap a 2-morphism
for each path $(\gamma,i,j) \in P (U_i \cap U_j)$ in a double overlap a 2-morphism
for each 2-path $(\Sigma, i) \in P^2 U_i$ in a single patch a 2-morphism
and some others
We shall assume now for definiteness that the object $Y$ covering our based space $X$ by a weak equivalence $Y \stackrel{\simeq}{\to} X$ is the Cech nerve
of a Cech cover $U = \coprod_i U_i \to X$. 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 $Y$ for some $G$-principal 2-bundle.
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 $P_2(Y)$ for the path 2-groupoid, these are given by extensions of the cocycle $g$ of the form
Detailed description of this is in [BaSc, ScWaIII].
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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 $\bar \mathbf{H}(X,\mathbf{B}G)$ 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 $\bar \mathbf{H}(X,\mathbf{B}G)$ is like that above, except that the 2-form $\beta$ is not required to vanish.
On the generators of $\Pi_3(Y)$ cominmg from paths in $U$ the morphism $tra_{(A,B),g} : \Pi_3(Y) \to \mathbf{B}INN(G)$ comes from differential form data
with unconstrained curvature? form
The first $\infty$-Ehresmann condition says that there is a commuting square
let
be one of the generating 2-cells in $P_3(Y)$. Then this condition says that evaluated on this 2-cell $tra_{(A,B),g}$ has to reproduce the cocycle $g$, with values in $\mathbf{B}G$.
The second Ehresmann condition says that all curvature characteristic forms? obtained from invariant polynomials $P \in inv(\mathcal{g})$ by $P(\beta,H) \in \Omega^\bullet(U)$ descent to forms on $X$.
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 $P(\beta_i,H_i) = P(\beta_j, H_j)$. By the very definintion of $ing(\mathcal{g}) \hookrightarrow W(\mathcal{g})$ this is the case if the transformation relating $(A_i, B_i)$ with $(A_j, B_j)$ is given by transition functions in the image of the inclusion $G \hookrightarrow INN(G)$.
This transformation is given by conjugating with the value of $\nabla_{(A,B)}^{g}$ on 2-cells of the form
So the second Ehresmann condition says that that $\nabla_{(A,B)}^g$ also colors these 2-cells with elements in the image of the inclusion $\mathbf{B}G \hookrightarrow \mathbf{B} INN(G)$
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