Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 127Further generalization is performed with string theory. Just as pointparticles naturally interact with a 1−form A, strings naturally interactwith a 2−form B, such that [Baez (2002)]∫( ∫ )action = B, and phase = exp i B .ΣΣThis 2−form connection B has a 3−form curvature G = dB, which satisfiesMaxwell–like equations, i.e., implies Bianchi–like relation dG = 0, but doesnot imply the dual, current relation ∗d ∗ G = J, with the current 2−formJ.In this way, the higher Yang–Mills theory assigns holonomies to pathsand also to paths of paths, so that we have a 3−morphismγ 1xα 1γ 2 ✲◆y∨α 2 ✍∨γ 3allowing us to ask not merely whether holonomies along paths are equal,but whether and how they are isomorphic.This generalization actually proposes categorification of the basic geometricalconcepts of manifold, group, Lie group and Lie algebra [Baez(2002)]. Replacing the words set and function by smooth manifold andsmooth map we get the concept of smooth category. Replacing themby group and homomorphism we get the concept of 2−group. Replacingthem by Lie group and smooth homomorphism we get the concept of Lie2−group. Replacing them by Lie algebra and Lie algebra homomorphismwe get the concept of Lie 2−algebra. Examples of the smooth categoriesare the following:(1) A smooth category with only identity morphisms is a smooth manifold.(2) A smooth category with one object and all morphisms invertible is aLie group.(3) Any Lie groupoid gives a smooth category with all morphisms invertible.(4) A generalization of a vector bundle (E, M, π), where E and M aresmooth manifolds and projection π : E → M is a smooth map, gives avector 2−bundle (E, M, π) where E and M are smooth categories andprojection π : E → M is a smooth functor.