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Emily Riehl is rewriting the foundations of higher category theory (2020)

102 points by perihelions 1y ago

moralestapia 1y ago
>2021 Was
- deleted 1y ago
  [deleted]
- gus_massa 1y ago
  I guess the ceremony was programed for 2021, but the winner was anounced in 2020. (Like the Nobel, not like the Oscar.)
  - madcaptenor 1y ago
    Information about this prize: https://awm-math.org/awards/awm-birman-research-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_Research_Prize_in_Topology_and_Geometry https://en.wikipedia.org/wiki/Joan_%26_Joseph_Birman_Researc... (Note also that 2020 may have been unusual because pandemic.)
TimorousBestie 1y ago
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 https://arxiv.org/abs/2510.15795 on how univalence drives some approaches to synthetic topology/homotopy.
- libraryofbabel 1y ago
  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?
  - tylerhou 1y ago
    > 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.
    - 1y ago
ak_111 1y ago
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
- random3 1y ago
  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.
  - ak_111 1y ago
    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.
    - anon291 1y ago
      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.

102 points by perihelions 1y ago

moralestapia 1y ago

>2021 Was

deleted 1y ago
[deleted]
gus_massa 1y ago
I guess the ceremony was programed for 2021, but the winner was anounced in 2020. (Like the Nobel, not like the Oscar.)
- madcaptenor 1y ago
  Information about this prize: https://awm-math.org/awards/awm-birman-research-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_Research_Prize_in_Topology_and_Geometry https://en.wikipedia.org/wiki/Joan_%26_Joseph_Birman_Researc... (Note also that 2020 may have been unusual because pandemic.)

TimorousBestie 1y ago

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 https://arxiv.org/abs/2510.15795 on how univalence drives some approaches to synthetic topology/homotopy.

libraryofbabel 1y ago
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?
- tylerhou 1y ago
  > 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.
  - 1y ago

ak_111 1y ago

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

random3 1y ago
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.
- ak_111 1y ago
  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.
  - anon291 1y ago
    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.