I am a mathematician who is using category theory all the time in my work in algebraic geometry, so I am exactly the wrong audience for this write-up!
I think that talking about “bad definitions” and “confusing presentation” is needlessly confrontational. I would rather say that the traditional presentation of category theory is perfectly adapted to its original purpose, which is to organise and to clarify complicated structures (algebraic, topological, geometric, …) in pure mathematics. There the basic examples of categories are things like the category of groups, rings, vector spaces, topological spaces, manifolds, schemes, etc. and the notion of morphism, i.e. “structure-preserving map”, is completely natural.
As category theory is applied more broadly in computer science and the theory of networks and processes, it is great that new perspectives on the basic concepts are developed, but I think they should be thought of as complementary to the traditional view, which is extremely powerful in its domain of application.
FWIW, I like John’s description above (and probably object much less than baseline to humorously confrontational language in research contexts :). I agree that for most math contexts, using the standard definitions with morphism sets and composition mappings is easier to prove things with, but I think the intuition described here is great and often in better agreement with how mathematicians intuit about category-theoretic constructions than the explicit formalism.
the traditional presentation of category theory is perfectly adapted to its original purpose
I think this is too generous. The traditional way of conceptualizing a given math subject is usually just a minor modification of the original conceptualization. There’s a good reason for this, which is that updating the already known conceptualization across a community is a really hard coordination problem—but this also means that the presentation of subjects has very little optimization pressure towards being more usable.
This phenomenon exists, but is strongly context-dependent. Areas of math adjacent to abstract algebra are actually extremely good at updating conceptualizations when new and better ones arrive. This is for a combination of two related reasons: first, abstract algebra is significantly concerned about finding “conceptual local optima” of ways of presenting standard formal constructions, and these are inherently stable and require changing infrequently; second, when a new and better formalism is found, it tends to be so powerfully useful that papers that use the old formalism (in concepts where the new formalism is more natural) quickly become outdated—this happened twice in living memory, once with the formalism of schemes replacing other points of view in algebraic geometry and once with higher category theory replacing clunkier conceptualizations of homological algebra and other homotopical methods in algebra. This is different from fields like AI or neuroscience, where oftentimes using more compute, or finding a more carefully taylored subproblem is competitive or better than “using optimal formalism”. That said, niceness of conceptualizations depends on context and taste, and there do exist contexts where “more classical” or “less universal” characterizations are preferable to the “consensus conceptual optimum”.
I am a mathematician who is using category theory all the time in my work in algebraic geometry, so I am exactly the wrong audience for this write-up!
I think that talking about “bad definitions” and “confusing presentation” is needlessly confrontational. I would rather say that the traditional presentation of category theory is perfectly adapted to its original purpose, which is to organise and to clarify complicated structures (algebraic, topological, geometric, …) in pure mathematics. There the basic examples of categories are things like the category of groups, rings, vector spaces, topological spaces, manifolds, schemes, etc. and the notion of morphism, i.e. “structure-preserving map”, is completely natural.
As category theory is applied more broadly in computer science and the theory of networks and processes, it is great that new perspectives on the basic concepts are developed, but I think they should be thought of as complementary to the traditional view, which is extremely powerful in its domain of application.
FWIW, I like John’s description above (and probably object much less than baseline to humorously confrontational language in research contexts :). I agree that for most math contexts, using the standard definitions with morphism sets and composition mappings is easier to prove things with, but I think the intuition described here is great and often in better agreement with how mathematicians intuit about category-theoretic constructions than the explicit formalism.
I think this is too generous. The traditional way of conceptualizing a given math subject is usually just a minor modification of the original conceptualization. There’s a good reason for this, which is that updating the already known conceptualization across a community is a really hard coordination problem—but this also means that the presentation of subjects has very little optimization pressure towards being more usable.
This phenomenon exists, but is strongly context-dependent. Areas of math adjacent to abstract algebra are actually extremely good at updating conceptualizations when new and better ones arrive. This is for a combination of two related reasons: first, abstract algebra is significantly concerned about finding “conceptual local optima” of ways of presenting standard formal constructions, and these are inherently stable and require changing infrequently; second, when a new and better formalism is found, it tends to be so powerfully useful that papers that use the old formalism (in concepts where the new formalism is more natural) quickly become outdated—this happened twice in living memory, once with the formalism of schemes replacing other points of view in algebraic geometry and once with higher category theory replacing clunkier conceptualizations of homological algebra and other homotopical methods in algebra. This is different from fields like AI or neuroscience, where oftentimes using more compute, or finding a more carefully taylored subproblem is competitive or better than “using optimal formalism”. That said, niceness of conceptualizations depends on context and taste, and there do exist contexts where “more classical” or “less universal” characterizations are preferable to the “consensus conceptual optimum”.