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It is not true that "basically every field uses its concepts and notation to some degree at this point." In particular this is false for mainstream combinatoric

It is not true that "basically every field uses its concepts and notation to some degree at this point." In particular this is false for mainstream combinatorics, PDE, and probability theory, to give a few examples. In fact, I would suggest that most mathematicians don't care about category theory at all.

Koshkin 6y ago
> most mathematicians Agree, but only on the level on which they do not care about the abstract math (algebra, topology, etc.) in general. As soon as you step into the territory of the abstract math, especially where different disciplines blend, such as homology and cohomology, category theory (and its diagram language) helps a lot to clarify things. (Incidentally, a lot of this stuff is now part of the "applied math" as well, having found its way into theoretical physics, for example.)
- spekcular 6y ago
  I'm not sure I'm comfortable characterizing the fields where category theory is useful as "abstract math." Modern PDE is plenty abstract, for example. Probably it's better to say that the usefulness of category theory is proportional to the problem's distance from algebraic topology and algebraic geometry. I also am reluctant to characterize theoretical physics as "applied math." I haven't seen anyone who calls themselves an applied mathematician use category theory in a substantive way (where here I am thinking about numerical computing, mathematical biology, and so on).
bitdizzy 6y ago
This all hinges on "mainstream". For example, in combinatorics, combinatorial species are a vast organization of the all-important concept of generating function. They were developed by category theorists and are most tidily organized along categorical lines. If you don't think this is close enough to mainstream, I can't dispute that. It's a value judgment. There is often an undercurrent of category theory within a subject that maybe most people are not privy to. Anything to do with sheaves or cohomology (which I know factors into some approaches to PDEs) are using categorical ideas. Every generation, it seems, has some contingent of serious mathematicians who consider category theory marginal in their field of interest. But every generation, that contingent grows smaller as more mathematics as practiced is brought into the fold. Maybe they're coming for you next :)
- spekcular 6y ago
  Respectfully, I disagree. The question of what's mainstream and valued by the community is empirical and can be answered by looking at what's published in the leading combinatorics journals. And anyone can check those out and see that categories are basically absent. So as a sociological fact, I maintain it's far from the mainstream. Whether combinatorialists ought to elevate certain work is of course of a question of value, but it's also a different question. Also, in no way are sheaves or (co)homology essentially category-theoretic ideas. It's possible to develop and use these ideas without mentioning categories at all (and e.g. Hatcher's introductory textbook does just this, although he mentions in an appendix the categorical perspective later). In general I think it's good to remember that homological algebra and category theory are not the same subject. Sure, I can develop a theory of chain complexes over an arbitrary abelian category, but most of the time you just need Hom and Tor over a ring. (Again, see Hatcher.) Finally, I'm not sure there has been a serious uptake in category theory in the mainstream of some field of mathematics since, I don't know, at least 50 years ago? We've understood for a while now what it's good and not good for. This hasn't stopped people from trying to inject it in fields where it doesn't do any good (e.g. probability), but for that reason those attempts are mostly ignored.
  - bitdizzy 6y ago
    > Respectfully, I disagree. The question of what's mainstream and valued by the community is empirical and can be answered by looking at what's published in the leading combinatorics journals. And anyone can check those out and see that categories are basically absent. So as a sociological fact, I maintain it's far from the mainstream. I think it is also a reasonable interpretation to take "mainstream" as "pertaining to the main subject matter of the field". Anyway, I think it is the case that the mainstream of combinatorics or probability is yet so big that a particular researcher or even group of researchers can be comfortably in the mainstream and yet have never cared for or even heard of some other line of research that is also mainstream. The founding paper of combinatorial species [1] has hundreds of citations including many in what I gather are top journals in combinatorics, and even some in the Annals of Probability. So, what are we to make of that? Some people who are serious enough about combinatorics or probability to get published in serious journals have read, perhaps understood, and maybe even taken seriously some of these categorical ideas? In any case, I respect your viewpoint. In my youth I was a bit category-crazy, trying to use it to organize all of my mathematical knowledge. I'm much more prudent about it these days but I'm still an optimist that we will find more unifying ideas in mathematics through it. [1] https://www.sciencedirect.com/science/article/pii/0001870881900529 https://www.sciencedirect.com/science/article/pii/0001870881...

Koshkin 6y ago

> most mathematicians Agree, but only on the level on which they do not care about the abstract math (algebra, topology, etc.) in general. As soon as you step into the territory of the abstract math, especially where different disciplines blend, such as homology and cohomology, category theory (and its diagram language) helps a lot to clarify things. (Incidentally, a lot of this stuff is now part of the "applied math" as well, having found its way into theoretical physics, for example.)

spekcular 6y ago
I'm not sure I'm comfortable characterizing the fields where category theory is useful as "abstract math." Modern PDE is plenty abstract, for example. Probably it's better to say that the usefulness of category theory is proportional to the problem's distance from algebraic topology and algebraic geometry. I also am reluctant to characterize theoretical physics as "applied math." I haven't seen anyone who calls themselves an applied mathematician use category theory in a substantive way (where here I am thinking about numerical computing, mathematical biology, and so on).

bitdizzy 6y ago

This all hinges on "mainstream". For example, in combinatorics, combinatorial species are a vast organization of the all-important concept of generating function. They were developed by category theorists and are most tidily organized along categorical lines. If you don't think this is close enough to mainstream, I can't dispute that. It's a value judgment. There is often an undercurrent of category theory within a subject that maybe most people are not privy to. Anything to do with sheaves or cohomology (which I know factors into some approaches to PDEs) are using categorical ideas. Every generation, it seems, has some contingent of serious mathematicians who consider category theory marginal in their field of interest. But every generation, that contingent grows smaller as more mathematics as practiced is brought into the fold. Maybe they're coming for you next :)

spekcular 6y ago
Respectfully, I disagree. The question of what's mainstream and valued by the community is empirical and can be answered by looking at what's published in the leading combinatorics journals. And anyone can check those out and see that categories are basically absent. So as a sociological fact, I maintain it's far from the mainstream. Whether combinatorialists ought to elevate certain work is of course of a question of value, but it's also a different question. Also, in no way are sheaves or (co)homology essentially category-theoretic ideas. It's possible to develop and use these ideas without mentioning categories at all (and e.g. Hatcher's introductory textbook does just this, although he mentions in an appendix the categorical perspective later). In general I think it's good to remember that homological algebra and category theory are not the same subject. Sure, I can develop a theory of chain complexes over an arbitrary abelian category, but most of the time you just need Hom and Tor over a ring. (Again, see Hatcher.) Finally, I'm not sure there has been a serious uptake in category theory in the mainstream of some field of mathematics since, I don't know, at least 50 years ago? We've understood for a while now what it's good and not good for. This hasn't stopped people from trying to inject it in fields where it doesn't do any good (e.g. probability), but for that reason those attempts are mostly ignored.
- bitdizzy 6y ago
  > Respectfully, I disagree. The question of what's mainstream and valued by the community is empirical and can be answered by looking at what's published in the leading combinatorics journals. And anyone can check those out and see that categories are basically absent. So as a sociological fact, I maintain it's far from the mainstream. I think it is also a reasonable interpretation to take "mainstream" as "pertaining to the main subject matter of the field". Anyway, I think it is the case that the mainstream of combinatorics or probability is yet so big that a particular researcher or even group of researchers can be comfortably in the mainstream and yet have never cared for or even heard of some other line of research that is also mainstream. The founding paper of combinatorial species [1] has hundreds of citations including many in what I gather are top journals in combinatorics, and even some in the Annals of Probability. So, what are we to make of that? Some people who are serious enough about combinatorics or probability to get published in serious journals have read, perhaps understood, and maybe even taken seriously some of these categorical ideas? In any case, I respect your viewpoint. In my youth I was a bit category-crazy, trying to use it to organize all of my mathematical knowledge. I'm much more prudent about it these days but I'm still an optimist that we will find more unifying ideas in mathematics through it. [1] https://www.sciencedirect.com/science/article/pii/0001870881900529 https://www.sciencedirect.com/science/article/pii/0001870881...