Smooth Functors vs. Differential Forms

Details Smooth Functors vs. Differential Forms Smooth Functors vs. Differential Forms

2008-02-05

arxiv.org/abs/0802.0663v4

Homology, Homotopy Appl., 13(1), 143-203 (2011)

Mathematics

Differential Geometry

75 pages, 1 figure; v2 with only minor changes; v3 has a layout improvement; v4 is the published version, with small improveme

Scientific paper

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 curvatures 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.

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