Homology, Homotopy and Applications
Volume 13 (2011)
Smooth functors vs. differential forms
Pages: 143 – 203
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