Parallel transport and functors
type: Item
instance of: scholarly article
title: Parallel transport and functors
zbMATH Open document ID: 1189.53026
author: Urs Schreiber
author: Konrad Waldorf
published in: Journal of Homotopy and Related Structures
publication date: 2010-02-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/0705.0452
full work available at URL: abstract.htm
review text: Higher dimensional parallel transport generalizes parallel transport along curves to parallel transport along higher dimensional objects. Several mathematical objects have been used to define higher dimensional parallel transport, among them classes in Deligne cohomology, bundle gerbes with connection and curving, or 2-bundles with 2 connections. Bundle gerbes with connection can be obtained as a generalization of principal bundles with connection. However, in the case of both bundle gerbes and Deligne classes one encounters the obstruction that the structure group has to be abelian. Hence it is desirable to find a reformulation of fiber bundles with connection, that brings along a natural generalization for arbitrary structure group. A functor from the path groupoid of the base manifold \(M\) into a category that describes the fibers of the bundle arises from parallel transport of a connection in a smooth fiber bundle. The authors define two notions of local trivializations and smooth descent data to characterize the functor. These characterizations allow to substitute categories of functors with categories of smooth fiber bundles with connection. The authors point out that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 55R10
Mathematics Subject Classification ID: 53C08
zbMATH DE Number: 5673022
zbMATH Keywords: higher dimensional parallel transport
zbMATH Keywords: functors and local trivialization
zbMATH Keywords: transpaort functor
zbMATH Keywords: descent data
zbMATH Keywords: groupoid bundle with connection
MaRDI profile type: Publication
arXiv ID: 0705.0452
description: scientific article