Investigating Monomorphism and Epimorphisms

INTRODUCTION
This project is an opportunity to study fundamental concepts of category theory through the lens of dessins d’enfants (dessins). These are hands-on mathematical objects with which I can formulate an understanding of category theory. A dessin is a bipartite graph embedded on a plane. As well as vertices, edges, and faces, the rotation system of the graph is also important for this project.

RESEARCH COMPONENT
The research entails studying the combinatorial aspects of dessins d’enfants to answer several research problems concerning:
– rigorously defining the class of dessins d’enfants,
– the definition of morphisms of dessins and their properties,
– describing monomorphisms and epimorphisms of dessins,
– investigating whether the category of dessins is a topos,
– and defining the Euler characteristic of a dessin.

There is room to answer further questions that I develop in the research process. These may concern conjugate dessins, functors, and Riemann surfaces.

COMPLETING THE PROJECT
Whilst there are many aspects to the above research component, the process will be sequential and my supervisors have assisted in establishing a timeline (see proposal) for the completion of the research. A distinguishing feature of this project is that it has been designed by my supervisors as a semi-group project, this is discussed in the proposal. We have submitted individual applications, will carry out our own research, and will each submit a unique report in the end. However, the three of us students will be building our research from the same set of
initial problems. Thus, we will have the opportunity to enrich or research journey through indepth discussions about are research. The collaborative nature of this process (and the implications on my findings) will be made clear in the statement of authorship on my report.

GETTING STARTED
I will spend some time at the beginning of the project, reading textbooks and reference books as an introduction to the research topic, these include Basic Category Theory by Thomas Leinster, Graph Theory by Reinhard Diestel, and the Handbook of Categorical Algebra Vol.1 Basic Category Theory by Francis Borceux.

Thomas Dee

University of Adelaide

Thomas Dee has recently completed his second year studying a Bachelor of Mathematical Sciences (Advanced) at the University of Adelaide. He intends to major in pure mathematics and pursue a career in research.

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