Nonabelian multiplicative integration on curves is a classical idea. This quantity is ready the 2-dimensional case, that is even more tricky. In our development, the setup is a Lie crossed module: there's a Lie team H, including an motion on it by way of one other Lie staff G. The multiplicative necessary is a component of H, and it's the restrict of Riemann items. each one Riemann product includes a fractal decomposition of the skin into kites (triangles with strings connecting them to the bottom point). there's a twisting of the integrand, that comes from a 1-dimensional multiplicative critical alongside the strings, with values within the staff G.
The major results of this paintings is the three-d nonabelian Stokes theorem. This result's new; just a specified case of it was once envisioned (without facts) in papers in mathematical physics. Our structures and proofs are of a simple nature. there are many illustrations to elucidate the geometric constructions.
Our quantity touches on a number of the principal matters (e.g., descent for nonabelian gerbes) in an surprisingly down-to-earth demeanour, related to research, differential geometry, combinatorics and Lie conception — rather than the 2-categories and 2-functors that different authors prefer.