Math 161: Category Theory in Context

An introduction to categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and other topics as time permits with the aim of revisiting a broad range of mathematical examples from the categorical perspective. I hope to explain how to calculate the free product of groups, rings, and other algebraic objects as well as why “all concepts are Kan extensions.”

PREREQUISITES

Math 123 (may be taken concurrently) and Math 131, or permission of the instructor. All the category theory we will study will be developed from scratch. The prerequisites are designed to guarantee a level of mathematical sophistication sufficient to appreciate the examples that will be discussed. In the absence of this, extraordinary enthusiasm and motivation will likely suffice.

LOGISTICS

The course will meet on Tuesdays and Thursdays from 11:30-1 in Science Center 304. The syllabus can be found here.

Sections will be held on Fridays at 3pm in Science Center 310.

My office hours will be held on Mondays from 3-4pm, immediately after class, or by appointment. Michael Fountaine, our course assistant, will hold office hours on Mondays at 7:30pm in the 4th floor math department common room.

TEXT

The primary text will be comprised of lecture notes, which may be found here (last update: 7/23). These notes are a work in progress. Please don't hesitate to get in touch with comments, to make corrections, or to point out typos!

FURTHER REFERENCES

For those seeking supplemental reading, I recommend Basic Category Theory by Tom Leinster. Copies will be available at the Coop.

The “discursive and hopefully fairly accessible” Notes on Category Theory is in the process of being written by Peter Smith, based on lectures given in Part III of the maths tripos at Cambridge.

The handout On the construction of new topological spaces from existing ones, written for Math 131, explores the theory of limits and colimits in the category of topological spaces from a categorical but categorically naive perspective.

PROBLEM SETS

• Problem Set 1, due February 10.
• Problem Set 2, due February 24.
• Problem Set 3, due March 3.
• Problem Set 4, due March 10.
• Problem Set 5, due March 24.
• Problem Set 6, due April 7.
• Problem Set 7, due April 14.
• Problem Set 8, due April 21.
• Problem Set 9, due April 28.

• Oral 1: State the definition of a natural transformation and explain how a morphism in a locally small category gives rise to a natural transformation between representable functors. (I get to pick whether they are co- or contravariant.)
• Oral 2: Define a mathematical object of your choosing by means of a universal property. Explain both what functor it represents and in what category it defines an initial or terminal object.

If you have any questions about the course, please get in touch. My contact info can be found on my personal website.