[HN Gopher] Emily Riehl is rewriting the foundations of higher c...
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Emily Riehl is rewriting the foundations of higher category theory
(2020)
Author : perihelions
Score : 88 points
Date : 2025-10-22 22:34 UTC (6 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| moralestapia wrote:
| >2021
|
| Was
| gus_massa wrote:
| I guess the ceremony was programed for 2021, but the winner was
| anounced in 2020. (Like the Nobel, not like the Oscar.)
| madcaptenor wrote:
| 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.wi
| kipedia.org/wiki/Joan_%26_Joseph_Birman_Researc...
|
| (Note also that 2020 may have been unusual because pandemic.)
| TimorousBestie wrote:
| 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.
| libraryofbabel wrote:
| 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?
| TimorousBestie wrote:
| 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.
| libraryofbabel wrote:
| 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!
| jonah-archive wrote:
| 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.
| tylerhou wrote:
| > 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.
| lambdas wrote:
| Agreed. In addition 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.
| hinkley wrote:
| > 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.
| michaelcampbell wrote:
| It appears that they have.
| michael_nielsen wrote:
| 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
| ak_111 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
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