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The Euler Characteristic For Mathematical Objects

By Gabe Necoechea, Mathematics ‘16

Over the summer, I worked on a research project with Dr. Alex Gonzalez of Kansas State University and an undergraduate from Truman State University. Having never been exposed to many of the areas of mathematics involved in our project, we undergraduates had to undertake significant background reading. Furthermore, the project challenged us to make abstract concepts clear and concrete, for the sake of both our own problem-solving and the exposition of our work. Our research focused on defining the Euler characteristic for mathematical objects called higher categories. The Euler characteristic is an example of a topological invariant, which can be thought of as a label which lets us distinguish between (not obviously) different shapes. The shapes in question are not necessarily the ones we draw in geometry classes, and their complex nature means that mathematicians like to work with simpler objects that share the same sort of properties. For this reason, mathematicians have found all sorts of “simple” objects to approximate more complicated spaces, and higher categories are one such example.

Our work specifically built on the work of Tom Leinster at The University of Edinburgh. Leinster’s approach to defining the Euler characteristic for finite categories was the basis for most of our work in the case of finite higher categories. There are special higher categories that are called strict. Generally speaking, strict higher categories are easier to work with than higher categories than not-necessarily-strict, or weak, higher categories. We were able to present a method for calculating the Euler characteristic of weak higher categories, which ended up being a slight modification of an approach taken by Leinster to define Euler characteristic for strict higher categories. Our work can be extended further to creatures called “infinity categories,” but we did not achieve this during our time working on the project.