### 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.)

Posted at August 21, 2006 8:01 PM UTC
## Re: On n-Transport, Part II

I have now uploaded a new version of the above file, polished and refined:

The discussion is somewhat of a blend of the ideas of Synthetic Transitions, but without really using synthetic differential geometry.

Instead, the setup is internal to smooth spaces (as we always had it here), and infinitesimal neighbourhood is replaced by derivatives along 1-parameter flows on categories.

(Even though the latter aspect is not made all that explicit yet.)