Ivancevic_Applied-Diff-Geom

124 Applied Differential Geometry: A Modern Introductionand equations between functions by natural isomorphisms between functors,which in turn should satisfy certain equations of their own, called ‘coherencelaws’. Iterating this process requires a theory of n−categories.Categorification uses the following analogy between set theory and categorytheory [Crane and Frenkel (1994); Baez and Dolan (1998)]:Set Theoryelementsequationsbetween elementssetsfunctionsequationsbetween functionsCategory Theoryobjectsisomorphismsbetween objectscategoriesfunctorsnatural isomorphismsbetween functorsJust as sets have elements, categories have objects. Just as there arefunctions between sets, there are functors between categories. Now, theproper analog of an equation between elements is not an equation betweenobjects, but an isomorphism. Similarly, the analog of an equation betweenfunctions is a natural isomorphism between functors.2.3.9 Application: n−Categorical Framework for HigherGauge FieldsRecall that in the 19th Century, J.C. Maxwell unified Faraday’s electric andmagnetic fields. Maxwell’s theory led to Einstein’s special relativity wherethis unification becomes a spin–off of the unification of space end time inthe form of the Faraday tensor [Misner et al. (1973)]F = E ∧ dt + B,where F is electromagnetic 2−form on space–time, E is electric 1−form onspace, and B is magnetic 2−form on space. Gauge theory considers F assecondary object to a connection–potential 1−form A. This makes half ofthe Maxwell equations into tautologies [Baez (2002)], i.e.,F = dA =⇒ dF = 0 the Bianchi relation,but does not imply the dual Bianchi relation, which is a second half ofMaxwell’s equations,∗d ∗ F = J,