Smooth Functors vs. Differential Forms

Abstract

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.

75 pages, 1 figure; v2 with only minor changes; v3 has a layout improvement; v4 is the published version, with small improvements and a better proof of Lemma 2.6

References in corpus (2)

• Parallel Transport and Functors
• The inner automorphism 3-group of a strict 2-group

Cited by in corpus (9)

• An Invitation to Higher Gauge Theory
• Twisted differential String and Fivebrane structures
• On unitary 2-representations of finite groups and topological quantum field theory
• AQFT from n-functorial QFT
• Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps
• The inner automorphism 3-group of a strict 2-group
• Connections on non-abelian Gerbes and their Holonomy
• A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces
• On higher holonomy invariants in higher gauge theory II