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August 21, 2006

On n-Transport, Part II

Posted by Urs Schreiber

As promised - or threatened - I want to go through all kinds of examples of nn-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 33-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 nn-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)(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

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1 Comment & 11 Trackbacks

Read the post n-Curvature
Weblog: The n-Category Café
Excerpt: Definition of n-curvature and the nature of "fake curvature".
Tracked: August 23, 2006 8:36 PM
Read the post n-Transport and Higher Schreier Theory
Weblog: The n-Category Café
Excerpt: Understanding n-transport in terms of Schreier theory for groupoids.
Tracked: September 5, 2006 3:25 PM
Read the post On n-Transport: 2-Vector Transport and Line Bundle Gerbes
Weblog: The n-Category Café
Excerpt: Associated 2-transport, 2-representations and bundle gerbes with connection.
Tracked: September 7, 2006 2:05 PM
Read the post Kock on 1-Transport
Weblog: The n-Category Café
Excerpt: A new preprint by Anders Kock on the synthetic formulation of the notion of parallel transport.
Tracked: September 8, 2006 5:51 PM
Read the post Bulk Fields and induced Bimodules
Weblog: The n-Category Café
Excerpt: Bulk field insertions in 2D CFT in terms of 2-transport: endomorphisms of 2-monoids.
Tracked: September 27, 2006 5:28 PM
Read the post Local Transition of Transport, Anafunctors and Descent of n-Functors
Weblog: The n-Category Café
Excerpt: Conceps and examples of what would be called transition data or descent data for n-functors.
Tracked: December 8, 2006 9:24 AM

Re: On n-Transport, Part II

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

Smooth 2-Functors and pp-Forms

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.)

Posted by: urs on February 9, 2007 2:07 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Gauge Theory Kinematics
Weblog: The n-Category Café
Excerpt: A review of Fleischhack's discussion of a Weyl algebra on spaces of generalized connections.
Tracked: March 1, 2007 2:58 PM
Read the post QFT of Charged n-Particle: Disk Path Integral for String in trivial KR Field
Weblog: The n-Category Café
Excerpt: The arrow-theoretic perspective on the path integral for the disk diagram of the open string.
Tracked: March 5, 2007 10:19 PM
Read the post The First Edge of the Cube
Weblog: The n-Category Café
Excerpt: The notion of smooth local i-trivialization of transport n-functors for n=1.
Tracked: May 5, 2007 2:19 AM
Read the post Smooth 2-Functors and Differential Forms
Weblog: The n-Category Café
Excerpt: An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
Tracked: February 6, 2008 11:09 AM
Read the post Connections on Nonabelian Gerbes and their Holonomy
Weblog: The n-Category Café
Excerpt: An article on transport 2-functors.
Tracked: August 15, 2008 7:56 PM

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