Combinatorial Stacks and the Four-Colour Theorem

Abstract

We interpret the number of good four-colourings of the faces of a trivalent, spherical polyhedron as the 2-holonomy of the 2-connection of a fibered category, phi, modeled on Rep(sl(2)) and defined over the dual triangulation, T. We also build an sl(2)-bundle with connection over T, that is a global, equivariant section of phi, and we prove that the four-colour theorem is equivalent to the fact that the connection of this sl(2)-bundle vanishes nowhere. This interpretation may be a first step toward a cohomological proof of the four-colour theorem.