More recently, she wrote https://arxiv.org/abs/2510.15795 on how univalence drives some approaches to synthetic topology/homotopy.
This seems to me to be admirable, and perhaps under-appreciated. Although it is probably much more valued in mathematics than most other fields, perhaps because mathematicians place more value than other fields on simplicity and clarity of exposition for its own sake, and because it is just so hard to read unfamiliar mathematics. Her north star goal of making her field accessible to mathematics undergraduates was a nice one.
I would like to learn category theory properly one day, at least to that kind of "advance undergraduate" level she mentions. It's always seemed to me when dipping into it that it should be easier to understand than it is, if that makes sense - like the terminology and notation and abstraction are forbidding, but the core of "objects with arrows between them" also has the feeling of something that a (very smart) child could understand. Time to take another crack at it, perhaps?
John Baez (who is distantly related to Joan Baez, if memory serves) has also written a lot of introductory category theory and applied category theory.
I think he and Joan Baez are actually first cousins!
As someone who tried to learn category theory, and then did a mathematics degree, I think anyone who wants to properly learn category theory would benefit greatly from learning the surrounding mathematics first. The nontrivial examples in category theory come from group theory, ring theory, linear algebra, algebraic topology, etc.
For example, Set/Group/Ring have initial and final objects, but Field does not. Why? Really understanding requires at least some knowledge of ring/field theory.
What is an example of a nontrivial functor? The fundamental group is one. But appreciating the fundamental group requires ~3 semesters of math (analysis, topology, group theory, algebraic topology).
Why are opposite categories useful? They can greatly simplify arguments. For example, in linear algebra, it is easier to show that the row rank and column rank of a matrix are equal by showing that the dual/transpose operator is a functor from the opposite category.
Like learning a language by strictly the grammar and having 0 vocabulary.
Although, it feels like category theory _ought_ to be approachable without all those years of advanced training in those other areas of math. Set theory is, up to a point. But maybe that isn't true and you're restricted to trivial examples unless you know groups and rings and fields etc.?
It's a crisp, slim book, presenting topology categorically (so the title is appropriate). It both deepens the undergraduate-level understanding of topology and serves as an extended example of how category theory is actually used to clarify the conceptual structure of a mathematical field, so it's a way to see how the flesh is put on the bare bones of the categorical concepts.
It's also available for free online:
I have completely given up on trying to learn anything about math from Wikipedia. It’s been overrun by mathematicians apparently catering to other mathematicians and that’s not the point of an encyclopedia.
It’s hostile and pointless. If you want a technically correct site make your own.
A note on the motivations - CT was not originally intended as a foundations. This is clear from both the name (General Theory of Natural Equivalences) and construction (based on set theory, which is was and still is the foundation for most of mathematics). There was indeed work in the foundational direction and there are relevant aspects, but I don't think that's even today the core aspect of it.
moralestapia•3mo ago
Was
gus_massa•3mo ago
madcaptenor•3mo ago
"Nomination Period: April 1 through May 15 of an even numbered year. The prize will be awarded the January after nominations close, which falls in an odd year."
See for example the dates on the various announcement notices as given in the notes in the Wikipedia article: https://en.wikipedia.org/wiki/Joan_%26_Joseph_Birman_Researc...
(Note also that 2020 may have been unusual because pandemic.)