Homology, Homotopy and Applications

Volume 13 (2011)

Number 1

Smooth functors vs. differential forms

Pages: 143 – 203

DOI: https://dx.doi.org/10.4310/HHA.2011.v13.n1.a7


Urs Schreiber (Mathematisch Instituut, Universiteit Utrecht, The Netherlands)

Konrad Waldorf (Fakultät für Mathematik, Universität Regensburg, Germany)


We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as derivatives of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.


connection; gerbe; 2-group; path 2-groupoid; parallel transport

2010 Mathematics Subject Classification

18F15, 53C05, 55R65

Published 12 July 2011