Given a Lie $2$-group $G$ does every principal $G$ $2$-bundle admit a $2$-connection?

Question

The statement is true for Lie groups and principal bundles, with every principal bundle admitting a connection and I see no reason for the analogue result not to hold in the Lie $2$-group case but I can't find the precise statement anywhere. As a bonus question, given a Lie $n$-group $G$, do principal $G$ $n$-bundles always admit an $n$-connection?

What I am mainly concerned about is the case of the Lie $2$-group $G=\operatorname{String}(k)$ (bonus: the other $n$-groups appearing in the Whitehead tower of the orthogonal group).

Answer 1

It depends on what version of connection on principal 2-bundles you consider. There are at least four versions:

1. Fake-flat connections, these are the ones that have a well-defined 2-dimensional parallel transport

2. Regular connections

3. Adjusted connections (this requires additional structure on your Lie 2-group)

4. General connections

Versions 1, 2, and 4 are discussed in my paper "A global perspective to connections on principal 2-bundles". Version 3 is discussed, e.g., in "Explicit Non-Abelian Gerbes with Connections" by Rist, Saemann, and Wolf.

EDIT: the underlying structure in all four versions is a Lie 2-algebra-valued 1-form $\Omega$ on the total space of the principal 2-bundle (which is a Lie groupoid). Note that a 1-form involves (because a Lie 2-algebra is considered to live in degrees $-1$ and $0$) two 1-forms and one 2-form. The 1-form $\Omega$ satisfies a condition completely analogous to the condition imposed on connection 1-forms on ordinary principal bundles. Without further conditions, this is version 4. The other versions impose conditions on the curvature 2-form.

For version 1 it is clear that one cannot expect the existence of connections, since there is a quadratic equation involved. The space of fake-flat connections on a trivial bundle is not contractible.

For version 2 the existence of connections is not clear to me.

For version 3 it seems that the question has not yet been investigated, but it could be true that every principal 2-bundle admits adjusted connections. At least this is true in certain examples of adjusted 2-groups.

For version 4, there is an existence theorem with a mild additional assumption (Theorem 5.2.14) in my above-mentioned paper. A particular weird problem in this context is that connections on 2-bundles cannot easily be pulled back along morphisms of 2-bundles. The morphism itself has first to be equipped with a kind of connection (called a "pullback" in my paper), and the existence thereof is also obstructed.

Better results are of course possible when restricting to 2-groups of the form $BA$, where $A$ is an ordinary abelian Lie group. Then, connections of versions 1 and 2 coincide, form a contractible space, and always exist.

For the String-2-group, every String-2-bundle admits a string connection in the sense developed in my paper "String Connections and Chern-Simons Theory", and the space of such string connections is contractible. However, as far as I know, it is not worked out to which of the above versions of connections on principal 2-bundles these string connections correspond. Probably, it is version 3.

Answer 2

For a solution that does not involve direct constructions using partitions of unity, we can deploy Theorem 1.1 in arXiv:1912.10544, which provides an explicit formula for the classifying space of an ∞-sheaf $F$ (valued in spaces or any algebraic ∞-category) on the site of smooth manifolds.

The classifying space is $$\def\hocolim{\mathop{\rm hocolim}}\def\op{{\rm op}}\def\gs{{\bf Δ}}\def\B{{\sf B}_\smallint}\B F=\hocolim_{n∈Δ^\op}F(\gs^n),$$ where $\gs^n$ denotes $n$-simplex considered as a smooth manifold.

In our case, $F(M)$ can be taken to be the 2-groupoid (more generally: $n$-groupoid) of principal $G$-bundles with connection over $M$. Thus, concordance classes of principal $G$-bundles with connection over $M$ are in bijection with elements of the set $[M,\B F]$.

The ∞-sheaf $F$ admits a forgetful map $F→L$, where $L$ is defined in the same way as $F$, but without connection. Thus, concordance classes of principal $G$-bundles over $M$ are in bijection with elements of the set $[M,\B L]$.

The map $F→L$ induces a map of classifying spaces $$\B F=\hocolim_{n∈Δ^\op}F(\gs^n)→\B L=\hocolim_{n∈Δ^\op}L(\gs^n).$$

In our case $π_0(L(\gs^n))$ is a singleton set and concordant sections of $L(\gs^n)$ are isomorphic, and this map is a weak equivalence if and only if for every section $p∈F(S^{n-1})$ whose image in $L(S^{n-1})$ extends along the map $S^{n-1}→D^n$, the section $p$ itself extends along the same map.

This boils down to saying that any connection on the trivial bundle over $S^{n-1}$ extends to a connection on the trivial bundle over $D^n$.

For example, if $G$ is an ordinary Lie group, then connections on the trivial $G$-bundle are given by Lie algebra-valued differential 1-forms, which do possess the extension property from spheres to disks because such differential forms are sections of a certain vector bundle. Thus, we have a bijection of sets $$[M,\B F]→[M,\B L]$$ induced by the map $F→L$. Hence, any principal $G$-bundle over $M$ admits a connection.

For the case when $G$ is a Lie 2-group, the argument has the same structure, but first one needs to decide on the notion of a connection to use (see Konrad Waldorf's answer), and then see whether the above extension property is satisfied.