Emily Riehl is one of the best category theory writers in the business. Lurie’s opus was basically unreadable for me until I found her notes on (inf, 1)-categories and enrichment.
More recently, she wrote https://arxiv.org/abs/2510.15795 on how univalence drives some approaches to synthetic topology/homotopy.
One of the things I liked about her interview was how she candidly says her strengths are less in opening up new areas or proving new theorems and more reworking and clarifying existing areas (i.e. Lurie’s work) with cleaner approaches and new proofs to make them more accessible and therefore more useful.
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?
> I would like to learn category theory properly one day, at least to that kind of "advance undergraduate" level she mentions.
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.
Agreed. In addition to yours, notions like limits/colimits, equalisers/coequalisers, kernels/cokernels, epi/monic will be very hard to grasp a motivation for without a breadth of mathematical experience in other areas.
Like learning a language by strictly the grammar and having 0 vocabulary.
I should have mentioned in my post that I have an applied math masters and a solid amount of analysis and linear algebra with some group theory, set theory, and a smattering of topology (although no algebraic topology). So, I'm not coming to this with nothing, although I don't have the very deep well of abstract algebra training that a pure mathematician coming to category theory would have.
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.?
You could take a look at Topology: A Categorical Approach by Bradley, Bryson and Terilla.
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.
You might also find the work of David I. Spivak (no relation to the _Calculus on Manifolds_ Spivak) helpful in this endeavor.
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.
Oh thanks, I will take a look. I’ve read some of John Baez’s things but mostly on mathematical physics, which was my undergrad. I didn’t know he’d written on category theory.
I think he and Joan Baez are actually first cousins!
I'm maybe too close to the problem to evaluate well (studied foundational math) but I know that Lawvere and Schanuel's book "Conceptual Mathematics" has been fairly well-regarded as a path into category theory.
> it is just so hard to read unfamiliar mathematics
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.
Timothy Chow has a wonderful phrase for this - he describes one of his papers (on forcing) as solving an "open expository problem": https://timothychow.net/forcing.pdf
This title is a bit ironic when you consider the fact that one of the motivations of inventing category theory is to provide a foundation for many branches of mathematics
Can you elaborate what's ironic (what is "this" - higher CT)?
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.
Yes I should point out that I am a noob in this area, so you might be right in calling me out. My understanding is that CT was invented in part to provide a robust foundation for algebraic geometry, so it is quite ironic that people are now involved in trying to rework the foundation of the foundation.
Not really. For many years mathematics rested on traditional first order logic and traditional naive set theory. That was revisited at the begining of the twentieth century.
"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."
Emily Riehl is one of the best category theory writers in the business. Lurie’s opus was basically unreadable for me until I found her notes on (inf, 1)-categories and enrichment.
More recently, she wrote https://arxiv.org/abs/2510.15795 on how univalence drives some approaches to synthetic topology/homotopy.
One of the things I liked about her interview was how she candidly says her strengths are less in opening up new areas or proving new theorems and more reworking and clarifying existing areas (i.e. Lurie’s work) with cleaner approaches and new proofs to make them more accessible and therefore more useful.
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?
> I would like to learn category theory properly one day, at least to that kind of "advance undergraduate" level she mentions.
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.
Agreed. In addition to yours, notions like limits/colimits, equalisers/coequalisers, kernels/cokernels, epi/monic will be very hard to grasp a motivation for without a breadth of mathematical experience in other areas.
Like learning a language by strictly the grammar and having 0 vocabulary.
I should have mentioned in my post that I have an applied math masters and a solid amount of analysis and linear algebra with some group theory, set theory, and a smattering of topology (although no algebraic topology). So, I'm not coming to this with nothing, although I don't have the very deep well of abstract algebra training that a pure mathematician coming to category theory would have.
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.?
You could take a look at Topology: A Categorical Approach by Bradley, Bryson and Terilla.
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:
https://topology.mitpress.mit.edu/
Actually this is a better source for it because it includes a pdf of the table of contents and links to supplementary videos
https://jterilla.github.io/TopologyBook/
You might also find the work of David I. Spivak (no relation to the _Calculus on Manifolds_ Spivak) helpful in this endeavor.
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.
Oh thanks, I will take a look. I’ve read some of John Baez’s things but mostly on mathematical physics, which was my undergrad. I didn’t know he’d written on category theory.
I think he and Joan Baez are actually first cousins!
I'm maybe too close to the problem to evaluate well (studied foundational math) but I know that Lawvere and Schanuel's book "Conceptual Mathematics" has been fairly well-regarded as a path into category theory.
> it is just so hard to read unfamiliar mathematics
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.
It appears that they have.
Can we have ours back then? I wish I still remembered enough math to tackle one of them.
Timothy Chow has a wonderful phrase for this - he describes one of his papers (on forcing) as solving an "open expository problem": https://timothychow.net/forcing.pdf
This title is a bit ironic when you consider the fact that one of the motivations of inventing category theory is to provide a foundation for many branches of mathematics
Can you elaborate what's ironic (what is "this" - higher CT)?
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.
Yes I should point out that I am a noob in this area, so you might be right in calling me out. My understanding is that CT was invented in part to provide a robust foundation for algebraic geometry, so it is quite ironic that people are now involved in trying to rework the foundation of the foundation.
Not really. For many years mathematics rested on traditional first order logic and traditional naive set theory. That was revisited at the begining of the twentieth century.
>2021
Was
I guess the ceremony was programed for 2021, but the winner was anounced in 2020. (Like the Nobel, not like the Oscar.)
Information about this prize: https://awm-math.org/awards/awm-birman-research-prize/
"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.)