When I tell you my project is concerned with ‘the category of dessins d’enfants’, it is likely that three questions come to mind:
1. What are dessins d’enfants?
2. What is a category?
3. How can we have a category of dessins d’enfants?
To answer your first question, dessins d’enfants is French for ‘children’s drawings’. Indeed, dessins d’enfants are graphs which look a lot like how a toddler depicts the human form (circles and sticks). And there are deep connections between these graphs and other objects in mathematics.
A graph in this context is not a chart or the graph of an equation, like what a parabola is to a quadratic equation. Rather, a graph is a mathematical structure studied in graph theory, consisting of a set of points (called nodes or vertices) and a collection of edges between those points. A dessin d’enfant is a graph which satisfies the following properties: Each vertex is coloured either black or white. Each edge connects a black vertex to a white vertex. The connection of the edges to each vertex follows a cyclic ordering. And lastly, between any two vertices, there is a path along the edges.
The second question is more difficult to answer. ‘Category’ pertains to a topic of mathematics called category theory. In your previous studies of mathematics, you’ve probably come to associate mathematics with equations, functions, calculus, etc. But in pure mathematics there is an almost omnipresent concept: structure. An example of structure is when we have functions between sets of numbers. The idea of structure and the relationships which preserve aspects of structure from one object to another, or even from one structure to another, became very pervasive over the last two centuries. Consequently, in the 1940s mathematicians were compelled to formulate a theory which specialises in formalising and understanding ‘structure’ in mathematics. The abstractedness of this topic is quite unlike other areas of pure mathematics, it is powerful, far-reaching, and potentially ground-breaking in how the fundamentals of mathematics are understood. And its name is – you guessed it – category theory!
The notion of a category is how category theorists formalise mathematical structure. It is simply a collection of objects and the maps between those objects which satisfy some natural conditions. Now we can answer the third question; the category of dessins d’enfants is the category where the objects are dessins d’enfants and the maps are the ways to turn one dessin d’enfant into another. To formulate this category, we need to understand how to characterise the structure of a dessin d’enfant and how we can preserve this structure when we go from one dessin d’enfant to another.
University of Adelaide