Mr Patrick Coleman

Honours thesis

The presheaf category of Dessins d'Enfants

A dessin des enfants is a connected bipartite graph with a cyclic ordering of the edges around each vertex. Made famous by Alexander Grothendieck, dessins are used to represent ?nite branched coverings from a Riemann surface to the Riemann sphere, with at most three critical values, though they can be expressed more simply in terms of two permutations of a set and their orbits. A category of generalised dessins is explored, shown to be a presheaf category, hence an elementary topos, and properties of their morphisms are examined.