An introduction to category theory, especially geared towards advanced undergraduates transitioning into graduate-level coursework.

Description

MUSA 174 was taught using Prof. Emily Riehl’s (excellent) Category Theory in Context, which I would emphatically recommend to anyone else looking for a good introduction to the subject. Both in Spring 2022 and Spring 2023 the course closely followed chapters 1 through 4, covering the following topics:

• Categories (duality, functors, natural transformations)
• Universal properties (representable functors, the Yoneda lemma)
• Limits and colimits (universal cones, continuity, functoriality of limits)
• Adjunctions (adjoint functors, units and counits, adjunction calculus)

The final few weeks of each class was devoted to surveying a handful of special topics, which included:

• Monads and algebras
• Topoi and their logics
• (∞,1)-categories

Background

I started this course for a few reasons: first, and probably most importantly, I fell for category theory’s allure as soon as we met first at the bar, and teaching a class on the subject felt like a good way to test my understanding of the subject once I felt comfortable; and second, in my grad-level coursework I found that categorical maturity was often tacitly taken for granted, while the technical details would be swept under the rug. I found the latter especially unsatisfying, and figured I was not the only one who felt this way, so I set out to spread the good word of our lord and savior Saunders Mac Lane to future acolytes. The class ended up being way more popular than I had expected, and teaching it was some of the most fun I had during my undergrad career. There was enough demand after the pilot run in Spring 2022 that I was compelled to offer MUSA 174 again in Spring 2023.

The name of the course, “MUSA 174”, originates from its lower division analog MUSA 74 (another class I’ve taught).

Legacy

It is my understanding that this course is still being taught at UC Berkeley, by my grand-alumni (alumni of my alumni, to whom I initially passed the torch). I am genuinely so stoked that this class has survived. While it would be really cool to see the university officially adopt the course into their mathematics curriculum, part of me believes that it is better left in the hands of the students as a kind of informal seminar, without the burden of exams or the threat of final grades.

Syllabus

The course syllabus from Spring 2023 can be found here.