Category theory #

Category theory is a branch of mathematics focussed on the study and formalization of general abstract mathematical structures and the relationships between them. It was “invented” in the 1940s by Saunders Mac Lane and Samuel Eilenberg.

The fundamental result is the Yoneda lemma, which can be thought of as an abstracted version of Cayley’s theorem for finite groups.

Links and resources #

• Categories for the Working Mathematician by Saunders Mac Lane, the canonical text on category theory. I cited it in my PhD thesis (“the strokes of the creator’s brush are on display”) where I used the language of category theory to define various fundamental objects of study.
• A nice article in Scientific American (i.e. fairly accessible to a lay audience) about $∞$-category theory.
• A great book on compositionality that I hope to get around to reading some day.
• A great online textbook called Kerodon which is Jacob Lurie’s attempts to digitise his work on \(\infty\) categories.
• A textbook on Higher Categories and Homological algebra.
• A very interesting paper proposing a string-diagram approach for handling calculations in category theory (instead of “diagram chasing”). It references this paper from the 90s proposing an algebraic notation for doing these calculations. Interesting stuff.