The n-Category Café

On n-Transport, Part II

Posted by Urs Schreiber

As promised - or threatened - I want to go through all kinds of examples of $n$-transport with trivialization and transition ($\to$).

We already needed some of these examples in the comment sections (e.g. here or here) and they are relevant for most of the stuff that I plan to discuss here. I’d like to have a repository of worked examples.

One of my aims is to, eventually, give a coherent description of the $3$-transport that describes 11D supergravity ($\to$).

Transport - the way it is defined ($\to$) - by itself is an integral - or finite - notion. But most of the familiar examples of $n$-transport are known in their differential formulation only, involving differential forms. Therefore an important first step is to develop a notion of smooth transport, which may be differentiated and re-expressed in terms of differential form data.

The techniques for doing so can nicely be discussed in the context of the archetypical example that shall interest us, namely the $(n=2)$-analog of a principal bundle with connection.

Hence the content of this entry shall be smooth nonabelian fake-flat differential 3-cocycles, classifying principal 2-bundles with fake-flat connection. I shall follow our original discussion ($\to$) but use a recent streamlined formulation of the proofs ($\to$).

In fact, I’d dare to say that you can find here the quickest and most transparent derivation of the full cocycle data of a (fake flat) nonabelian gerbe with connection:

$\;\;\;$ Nonabelian Differential Cocycles.

(The non-fake flat case will be discussed elsewhere.)