[khazhoux]

> It sounds like category theory likely has no direct impact on your life. So you can move on.

Very dismissive statement that misses the person's point.

> It is a beautiful theory with many useful results.

But what are those results? Besides Yoneda, are there insightful, surprising, delightful results? I personally gave up on my CT study after seeing that it was just chapter after chapter of definitions and nothing else.

I always compare it to abstract algebra. AA can be studied without any connection whatsoever to the physical world or even to numbers -- as "abstract" as math can get. And yet from the first chapter you are hit with surprising theorems, and they continue non-stop, challenging your brain at every turn. I fail to see this in CT.

[jiggawatts]

This is what turned me off also. Category theory is all promises of potential benefits, but none seem to have materialised.

The closest example to something useful I’ve seen is a CT-based explanation for why Automatic Differentiation is formulated the way it is.

However, AD was invented before CT, and the explanation didn’t add any value that I could see. It didn’t result in a “better” AD, it simply attached esoteric labels to existing things.

[sleepyams]

One interesting result which you may find interesting: https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem

I also highly recommend this survey paper by John Baez and Mike Stay: https://math.ucr.edu/home/baez/rosetta.pdf

There are plenty of interesting results in category theory, in fact your comparison to abstract algebra is apt. There is only so much you can say about an arbitrary group in general, or an arbitrary topological space, just like there is only so much you can say about an arbitrary category.

[Twisol]

> Very dismissive statement that misses the person's point.

I read it as acknowledging that you shouldn't feel like you have to spend time on things that provide you no value. That seems to directly acknowledge the point I took from the earlier comment, which is that they keep spending time on it and coming away with no idea what they even should be getting from it, much less getting anything specifically.

(I like category theory, but it's a reasonable reaction for most people. I'd love for more people to engage with it on its merits, but also, people have finite time and may rather spend it on things they derive joy from.)

[maxiepoo]

Basic results that are useful on a day-to-day basis: 1. Yoneda's Lemma, and the corollary that objects defined by universal property are unique up to unique isomorphism 2. Right adjoints preserve limits and more generally, Kan extensions

The "big" theorem of basic category theory is probably the adjoint functor theorems which, once you realize that so many constructions in math are adjoint functors, gives very useful technical conditions to construct such an adjoint (ctrl-s for "applications" here for examples: https://math.stackexchange.com/questions/844131/adjoint-func...).

The most delightful surprising results to me come from categorical logic because that is what I am most familiar with. Here is one: any elementary topos is a model of intuitionistic higher-order logic, and also extensional dependent type theory. This vastly expands the domain of applications of logic if you are used to thinking only in terms of set-theoretic models. It also gives us applications of constructivism that are entirely independent from any philosophical debate about the nature of truth. This means that you can take many mathematical fields such as differential geometry, algebraic geometry, topology, as embodied in some category C, embed them into a sheaf topos and then use intuitionistic logic to do constructions and theorems in this area that are vastly simpler than the usual formulations. For instance you can do this with differential geometry and get an intuitionistic logic where you can work explicitly with infinitesimal numbers to calculate derivatives in a completely rigorous fashion. Ingo Blechschmidt has written some expository material in this vein (his main work being in using this in algebraic geometry): https://arxiv.org/abs/2204.00948

On the "surprising" side, I think the most surprising things for me where seeing how existing mathematical structures were examples of generalized categories: metric spaces are a kind of enriched category and topological spaces are generalized multicategories for the ultrafilter monad.