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I had a brush with category theory more than forty years ago. I got a master’s degree in math at the University of Chicago in 1980, Saunders Mac Lane was on the

I had a brush with category theory more than forty years ago. I got a master’s degree in math at the University of Chicago in 1980, Saunders Mac Lane was on the faculty, a few grad students were doing category theory, and I worked through part of Mac Lane’s Categories for the Working Mathematician on my own. I left mathematics soon after—with some regret—and haven’t kept up with the field. The interview with Emily Riehl and the introduction (which I read just now) to her book Category Theory in Context suggest that the field has made a lot of advances in the years since. I have a question for people who are familiar with recent developments. Forty years ago, the opinions of other math grad students about category theory were divided. Some thought it had the potential to yield great breakthroughs and solve previously unsolved problems in many branches of mathematics. Others thought it was pretty and useful for identifying similar structures in different fields but wasn’t much use for making significant new mathematical discoveries. A sentence in Emily Riehl’s book—“The category-theoretic perspective can function as a simplifying abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline”—seems to align with the latter opinion. What has in fact happened in recent decades? Has category theory turned out to be mainly a “simplifying abstraction” as well as an interesting branch of mathematics in its own right? Or has it been used to prove meaty new results in other fields as well?

bopbeepboop 6y ago
I’m not familiar with category theory in depth, but homotopy type theory uses category theory for its semantic model and HoTT’s a big deal in the automated reasoning world. > Others thought it was pretty and useful for identifying similar structures in different fields but wasn’t much use for making significant new mathematical discoveries. A sentence in Emily Riehl’s book—“The category-theoretic perspective can function as a simplifying abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline”—seems to align with the latter opinion. Two points: 1. Being able to “lift and shift” techniques is a big deal in mathematics — so the two cases are the same. A lot of recent innovations are along those lines, where techniques in one area were applied to a new area. In that sense, category theory is wonderful because it gives us a map for how to move techniques around. You could even go so far as to say the algebra-geometry correspondence is category theory’s first “big win”, as it came about as a way to formalize that body of work. 2. Category theory is the native language of data fusion, while categorical syntax is easy to represent in an image-completion kind of way... so category theory is useful for labs working on using machine learning to fuse data and extract semantically meaningful information. Or labs working on an AI model which can suggest improvements to itself.
andi999 6y ago
I would also like to know. At least here 20 years ago the opinion was: nice, general framework, but (naturally) no hard theorems.
spekcular 6y ago
> What has in fact happened in recent decades? Has category theory turned out to be mainly a “simplifying abstraction” [...]? In short: yes, with some exceptions. The majority of mathematicians are working on problems where category theory is of no importance. A quick list of some of the major exceptions I'm aware of: the general cluster of ideas surrounding the Fukaya category, derived categories, and homological mirror symmetry [0]; geometric Langlands [1]; and the bulk of modern algebraic topology (I have in mind homotopy theory especially). It's also incredibly useful, practically speaking, for modern algebraic geometry and number theory, although my impression is that you have to get pretty deep in the weeds before it becomes more than a linguistic convenience. For a taste of the latter, see the discussion at the beginning of the category section of this algebraic geometry text (which is in principle accessible to someone who has taken an undergraduate course in abstract algebra – though I feel in reality a significant amount of mathematical maturity is required): Section 1.1 of http://math.stanford.edu/~vakil/216blog/ http://math.stanford.edu/~vakil/216blog/. Also, it has become somewhat of a meme here to talk about Homotopy Type Theory. I think this is overhyped; the experts in foundations don't seem to think highly of it. Many arguments for its utility or theoretical importance are, in my opinion, misguided – for example, the claim that set theory proves "junk theorems" such as "pi" being an element of "4." [0]: https://en.wikipedia.org/wiki/Homological_mirror_symmetry https://en.wikipedia.org/wiki/Homological_mirror_symmetry [1]: https://en.wikipedia.org/wiki/Geometric_Langlands_correspondence https://en.wikipedia.org/wiki/Geometric_Langlands_correspond...
- chas 6y ago
  While I'm not an expert in any of this, I'm primarily interested in Homotopy Type Theory (HoTT) because so many tools originally developed for algebraic geometry and algebraic topology have been successfully applied to logic and programming languages (e.g. categories, functors, natural transformations, monads, comonads, topoi) but homology and homotopy--core tools in algebraic geometry and algebraic topology--haven't been investigated to nearly the same degree in logic and computing. So even if HoTT isn't particularly compelling from the standpoint of pure foundations, I'm still excited for the work they are doing exploring these core tools in geometry in the context of logic and programming languages.

bopbeepboop 6y ago

I’m not familiar with category theory in depth, but homotopy type theory uses category theory for its semantic model and HoTT’s a big deal in the automated reasoning world. > Others thought it was pretty and useful for identifying similar structures in different fields but wasn’t much use for making significant new mathematical discoveries. A sentence in Emily Riehl’s book—“The category-theoretic perspective can function as a simplifying abstraction, isolating propositions that hold for formal reasons from those whose proofs require techniques particular to a given mathematical discipline”—seems to align with the latter opinion. Two points: 1. Being able to “lift and shift” techniques is a big deal in mathematics — so the two cases are the same. A lot of recent innovations are along those lines, where techniques in one area were applied to a new area. In that sense, category theory is wonderful because it gives us a map for how to move techniques around. You could even go so far as to say the algebra-geometry correspondence is category theory’s first “big win”, as it came about as a way to formalize that body of work. 2. Category theory is the native language of data fusion, while categorical syntax is easy to represent in an image-completion kind of way... so category theory is useful for labs working on using machine learning to fuse data and extract semantically meaningful information. Or labs working on an AI model which can suggest improvements to itself.

andi999 6y ago

I would also like to know. At least here 20 years ago the opinion was: nice, general framework, but (naturally) no hard theorems.

spekcular 6y ago

> What has in fact happened in recent decades? Has category theory turned out to be mainly a “simplifying abstraction” [...]? In short: yes, with some exceptions. The majority of mathematicians are working on problems where category theory is of no importance. A quick list of some of the major exceptions I'm aware of: the general cluster of ideas surrounding the Fukaya category, derived categories, and homological mirror symmetry [0]; geometric Langlands [1]; and the bulk of modern algebraic topology (I have in mind homotopy theory especially). It's also incredibly useful, practically speaking, for modern algebraic geometry and number theory, although my impression is that you have to get pretty deep in the weeds before it becomes more than a linguistic convenience. For a taste of the latter, see the discussion at the beginning of the category section of this algebraic geometry text (which is in principle accessible to someone who has taken an undergraduate course in abstract algebra – though I feel in reality a significant amount of mathematical maturity is required): Section 1.1 of http://math.stanford.edu/~vakil/216blog/ http://math.stanford.edu/~vakil/216blog/. Also, it has become somewhat of a meme here to talk about Homotopy Type Theory. I think this is overhyped; the experts in foundations don't seem to think highly of it. Many arguments for its utility or theoretical importance are, in my opinion, misguided – for example, the claim that set theory proves "junk theorems" such as "pi" being an element of "4." [0]: https://en.wikipedia.org/wiki/Homological_mirror_symmetry https://en.wikipedia.org/wiki/Homological_mirror_symmetry [1]: https://en.wikipedia.org/wiki/Geometric_Langlands_correspondence https://en.wikipedia.org/wiki/Geometric_Langlands_correspond...

chas 6y ago
While I'm not an expert in any of this, I'm primarily interested in Homotopy Type Theory (HoTT) because so many tools originally developed for algebraic geometry and algebraic topology have been successfully applied to logic and programming languages (e.g. categories, functors, natural transformations, monads, comonads, topoi) but homology and homotopy--core tools in algebraic geometry and algebraic topology--haven't been investigated to nearly the same degree in logic and computing. So even if HoTT isn't particularly compelling from the standpoint of pure foundations, I'm still excited for the work they are doing exploring these core tools in geometry in the context of logic and programming languages.