Category theory is a branch of mathematics that emerged in the 1950s to understand how the different areas of mathematics relate to each other. Today, its influence pervades nearly all of mathematics, in addition to being an important area of study in its own right. This course will cover the basic concepts of category theory, with an emphasis on intuition and examples.
Brief history of category theory, followed by a discussion of what it is and how it is meant to be used. Discussed syllabus and format.
- Background: Monoids
Background lecture on monoids. Introduced the concepts of monoid and monoid morphism conceptually, then worked out how to implement these ideas rigorously.
- Categories, Functors, and Natural Transformations
Defined categories, functors, and natural transformations, with plenty of examples.
- Basic Constructions and Ideas
Defined some basic constructions on categories: duality, products, and comma categories. Also briefly discussed 2-categories, enriched categories, and large categories.
- Internal Features of Categories
Following the previous lecture, which discussed operations between categories, introduced some interesting features within a single categorically. Specifically, looked at categorical generalizations / analogues of the injectivity, surjectivity, and bijectivity, as well as (co-)products and pullbacks. Hinted at Yoneda and (co-)limits, to be covered in the next few lectures.
- Representable Functors
Discussion of representable functors and universality, as well as multiple perspectives on how they relate. Proved the Yoneda lemma, and explained why it's awesome. I tried to give more worked examples this time.
Some familiarity with higher mathematics. Topology is a huge plus, but not necessary. I think the following as a minimum:
set theory: know what products, coproducts (disjoint unions), etc. are, know the paradoxes, know what posets are, know what injections/surjections/bijections are and maybe a bit about cardinals. This is pretty much just language/history and can be picked up in an afternoon or two on Wikipedia.
algebra: know what vector spaces/linear transformations are, ideally know what groups/monoids are. Knowledge of module theory definitely a plus. Know a few kinds of homomorphisms (linear transformations, ring homomorphisms, group homomorphisms, etc.), and understand the concept of isomorphism
mathematical maturity: probably the most important; basically remain calm and carry on when all the mathematics you've ever known and loved gets abstracted away into dots and arrows.
Dates: July 2 - .
Lectures will be posted to YouTube, assignments will probably be on Scribd
The course will cover all the major ideas in basic category theory, roughly equivalent to the first 5-7 chapters of Mac Lane.
- categories, functors, natural transformations
- operations on categories; duality
- representable functors and Yoneda
- limits and colimits
- Kan extensions
And some possible bonus topics:
- model categories
- generalizations of categories (higher categories, enriched categories, and so forth)
I'm very big on intuition, so expect a lot of rambling "big picture" digressions. I'll also try to mention the computer sciencey aspects.
Studied category theory for 2.5 years (started when I was 17)
Taken graduate courses in algebra, algebraic topology, and homological/homotopical algebra
Attended workshops/conferences on applications of category theory