The String Coffee Table

A fiber bundle with connection is essentially a parallel transport functor from paths in base space to the transport groupoid of the bundle.

As a categorification of this fact, locally trivialized 2-transport 2-functors from 2-paths (surfaces) to certain 2-groupoids were shown in [9,10] to encode the same cocycle data as fake flat nonabelian gerbes with connection and curving [7].

It turns out that a 2-transport 2-functor can be locally trivialized in two steps. Performing only one of these steps trivializes the 2-functor locally, but does not trivialize the 2-functor 1-morphisms which give the transitions between the local trivializations. This first step shall hence be called a pre-trivialization of the 2-transport 2-functor.

It was shown in [3] that pre-trivializations of 2-functors with values in $\mathbf{Vect}_1$, the monoidal category of 1-dimensional vector spaces (when regarded as a weak 2-category with a single object), are in bijection with abelian bundle gerbes [1] with connection and curving. A bundle gerbe with connection is a realization of the information contained in a proper gerbe with connection in terms of trivializable transition bundles with connection on double intersections of a good covering of base space. These transition bundles with connection are in bijection with pseudonatural transformations of local trivial transport 2-functors.

Here this result is generalized to nonabelian bundle gerbes (NABGs), which were defined in [2] as a generalization of the concept of an abelian bundle gerbe.

While it is straightforward to define a nonabelian bundle gerbe without connection and curving, the proper notion of connection in an NABG turns out to be non-obvious. The solution found in [2] justifies itself mainly in that it leads to the same cocylce data as found in [7].

We show that 2-transport 2-functors from 2-paths to the monoidal category $\mathbf{BiTor}(H)$ of bitorsors of a group $H$ (when regarded as a 2-category with a single object), have pre-trivializations that are in bijection with ‘fake flat’ NABGs with connection and curving.

In particular, it is shown that 2-functor 1-morphisms between local trivial 2-transport 2-functors are in bijection with $H$-bibundles that are equipped with a generalized notion of connection which reproduces precisely the definition found in [2].

Hence we find and investigate a purely arrow-theoretic description of bibundle connections on $H$-bibundles. All the curious properties which distinguish a bibundle connection from an ordinary connection are shown to be results of the fact that bibundle connections are not related to transport functors (like ordinary connections are) but to pseudonatural transformations between 2-transport 2-functors. A pseudonatural transformation has properties similar to but different from those of a proper functor.

The translation between the arrow-theory which we use and the differential-form description of bibundle connections used in [2] is done using synthetic differential calculus as described in [4,5].

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The main part of this paper is section 2, which investigates the arrow-theoretic description of bibundle connections.

In order to introduce our language, ordinary bundles with connection are described in that fashion in section 2.1, mainly reviewing material presented by A. Kock. Bibundle connections are then defined arrow-theoretically in section 2.2 and it is shown how their synthetic differential version is equivalent to the structures defined in [2].

The remaining part 3 introduces 2-transport 2-functors with values in $\mathbf{BiTor}(H)$ and establishes the theorem which relates their 1-morphism to bibundles with bibundle connection. In a sequel to this paper pre-trivializations of such 2-functors will be defined and will be shown to be in bijection with fake flat nonabelian bundle gerbes.

Section 3 makes essential use of pseudonatural transformations between 2-functors, the details of whose definition can be found in the appendix of [3].