So you want to learn category theory - where should you start? I’ve spent the past six months trying to get a background in the essentials, and so it’s a good time to write up a reading list to point others along the same path. This is more of a record of my journey than an exhaustive guide, although I’ll try to mention all the popular resources I know of.
Physics, Topology, Logic and Computation: A Rosetta Stone — John Baez and Michael Stay
If you’re like me, you’ll want to get excited about the subject before wading through a couple hundred pages of textbook. This paper is perfect for that — it describes how category theory draws beautiful analogies between different areas of physics, maths and computer science, all assuming no prerequisites.
The paper is available from John Baez’s website.
Category Theory for Programmers — Bartosz Milewski
The first part of this is one of the best gentle introductions to category theory that I know of. He tries to avoid too much mathematical notation and technical detail, and instead gives intuition with cute drawings and keeps things grounded with snippets of Haskell code. If you have a formal training in mathematics you might prefer the next item on the list; on the other hand if you want to learn category theory in order to understand Haskell then this might be the only thing you need to read.
The posts are online here — that page also links to a pdf version, a hardcover book and a series of YouTube videos.
Category Theory in Context — Emily Riehl
This was my choice of introductory category theory textbook. It’s accessible to a later year maths undergrad without needing to know an excessive amount of abstract algebra or other prerequisites, and the writing is clear and of a consistently high quality. The content is evidently very well thought through: every time I skipped something I would find that it became important later, and need to go back and learn it properly! It’s a fairly dense book but not needlessly so — all the key intuition and examples are there.
You can read the book for free online, and the paperback version is inexpensive.