Abstract. Graphs are easily understood mathematical objects, comprised of dots and lines between them. Category theory is a way of looking at mathematical structures and their relationships. My research focused on how graphs can be viewed through the lens of category theory, and interesting results which followed. This blog is a beginner friendly way of introducing these ideas.
My research over the summer introduced me to two interesting concepts: graphs and categories. This blog gives a bit of explanation of each of these, and how they are related.
To a mathematician, a graph means something different from the usual interpretation. Most people will typically think of a diagram with some axes, and some curve showing a relationship. However, to us, a graph is a collection of dots (called ‘vertices’ or ‘nodes’) with some lines between those dots (called ‘edges’); in some other areas of science these are called networks. Graphs are frequently used in mathematical modelling in many different settings, such as optimisation and computer science.
Now, in particular, I focused on graphs where all of the edges must have no given direction, so they really are just dots and lines. These types of graphs are usually called undirected graphs, and I was very lenient as to what counts as one. Namely, an edge going from a vertex to the same vertex (called a ‘loop’) is allowed, and vertices may have many edges between them.
These undirected graphs form something called a category. Categories are a very general way of viewing many kinds of mathematical structures. A category is a collection of some objects that are related to each other, with the relationships between objects called ‘arrows’. The lens of category theory allows the unification of potentially disparate areas of mathematics. This is one of the reasons, after being introduced in the mid-20th century, category theory has become ubiquitous in theoretical mathematics.
As mentioned, undirected graphs actually form a category of their own. I called this category U, where the objects of U are undirected graphs, and the arrows are mappings between graphs. A mapping from some graph A to graph B is a way of assigning each vertex in A to a vertex in B, and each edge in A to an edge in B. Additionally, these mappings must preserve edge structure, meaning an edge connecting vertices in A must connect the corresponding vertices in B after being mapped.
Continuing this, my goal was to work with the category of undirected graphs and determine if it satisfied certain technical properties. In particular, I wanted to see if U satisfied the appropriate properties to be considered similar to another important category called Set (comprised of sets of numbers and functions between them). Spoiler alert: U is not “similar enough” to Set!
Alex Marciano
The University of Adelaide
