Working as a SWE for lots of years, I've worked with people that always feel the need to "add a level of abstraction" or "build a framework" for (and out of) everything. A lot of the time it doesn't make things better (and often makes things worse).
Every time I try to learn about Category Theory, I get to a certain point and just start wondering "why?"
I can't tell if I'm not smart enough to get it (I think that's the most likely answer). But it feels like some math people -- who have nothing better to do -- are just trying to create another level of abstraction.
Are there things that Category Theory does that other branches couldn't already do? I'm curious to hear other takes on this.
It sounds like category theory likely has no direct impact on your life. So you can move on.
But please do not let this bleed into a criticism of category theory as used in mathematics. Category theory from the very beginning was developed to help manage the complexity of modern mathematical fields like algebraic topology. It was then famously used by Grothendieck in algebraic geometry where some of the basic notions (schemes) were defined in terms of category theoretic concepts (functors). It's been applied to many other fields, since, including yes computer science, and mainly for the same purpose: giving precise terminology to common patterns (monads functors etc) and giving us the right concepts to design new functional programming languages. As someone who is an expert on these topics I find some of the popular sentiments about programmers using category theory to be a bit silly, but I'll admit it's overall probably good for my field because it cultivates a lot of interest in students. It probably plays a similar role to pop science/math in other fields: not very deep but fun and can be a gateway to "the real thing".
The popular perception of category theory is a bit bizarre to me though. It is a beautiful theory with many useful results. But you don't see the same excitement or resentment towards fields like order theory or abstract algebra, which are very closely related to, and just as abstract as, category theory and are used in similar ways.
> 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.
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.
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.
> 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.)
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.
What is your recommendation for learning? I've been recommended Bartosz Milewski enough times, and started to dive in. I am not interested in learning the totality of Category theory, just enough to be more effective in FP languages like Haskell.
Category Theory will not make you more efficient in FP languages like Haskell. I studied it enough to realise that it doesn’t give me any insights or thinking tools I can actually use to write better software. However I highly recommend reading “Software Foundations”. It is almost guaranteed to make you better at writing software in any language. Including Haskell. That’s what happened to me.
Have you been recommended his blog series/book only, or the playlists on YouTube as well? I really like his recorded lectures on YouTube and they cover similar ground in a similar order to the blog posts (from what I've read—I'm further into the YouTube series than I am his written material).
You don't need to learn category theory to be more effective in FP languages like Haskell. Category theory was used as inspiration for design of ML and Haskell, but it is not necessary for programming in them.
Weirdly, I don’t love or hate category theory but I get very excited about order theory and abstract algebra in software. It feels lonely exploring them compared to other things, which is funny to say. It’s not fully explored or the new hotness.
It's because the lack of practical applications. People can make broken abstractions without knowledge of the fundamentals like compositions or functors and still get their shit to work.
That matters more to engineers then true understanding. So a lot of engineers end up trying to understand it and when they fail they easily move on to other things because it's not required knowledge for their job.
category theory has a very hard "k" sound at the front and a g sound which is also hard, which makes people react more strongly to the name than order theory or abstract algebra.
So I come at this from a mathematical background --- graduate student in categorical algebra --- but I've done a couple years of SWE work, so I'm not unsympathetic to this point of view. The way I see it, when you want to reason about things like data processing (which is what a massive chunk of writing software is, moving data from form A to form B), category theory provides what is in some sense the "correct" language/framework in which to think. It's not just an abstraction circle-jerk, it's a genuinely useful perspective, particularly for guiding your mind to spot non-obvious connections between pieces of code.
The problem as I view it is that CT is first and foremost a discipline of maths. It was developed to help mathematicians, is very sophisticated, and very specialised; learning CT for SWE is taking a sledgehammer to a nut. I can't honestly recommend it as a field of study to someone who isn't interested in the problems it was developed to solve. If you don't have a solid grounding in set theory, logic, algebra, topology, etc., it's a very tough field to motivate. The vast majority of CT is of little to no relevance for SWE work. Adjunctions, for example, are absolutely fundamental to all of maths, but in truth are not really relevant for SWEs. As a result, you see people trying to teach concepts like monads without reference to them; this is slightly insane from where I'm standing...
Your question about whether there are things CT does that other branches didn't already do; one of the fundamental utilities of CT in pure mathematics is "making trivial things trivially trivial". That is to say, it makes it very clear which parts of your problem are local to your specific situation, and which are purely "structural" from the categorical constraints. The SWE analogy would be separating business logic from other layers. So at least for mathematicians, it absolutely does have novel utility, and has drawn links between a huge number of disparate studies that were not well-understood previously.
So if you _do_ care about posets and groups and cohomology theories, CT will genuinely open your eyes, and (albeit, this coming from someone with less working experience than yourself) it could give you a deeper, or at least different understanding of the code you're writing. Otherwise, I'm not sure it's worth putting yourself through it, tbqh.
(If you do decide to give it another go, please use a better resource than the linked post; after a quick scan it looks pretty weak)
I think this is a good perspective - it follows the general approach in software engineering that if you're going to introduce something with a massively different approach, learning curve, and cost of adoption, it needs to come with the associated real-world benefits.
For most software stuff it would be very hard the benefit of introducing this stuff vs. what the standard paradigms that everyone already knows
Category theory has virtually zero application to software engineering. Abstract algebra (knowing what a group, monoid, etc. are) has a bit more practical application. Knowing what a monad is (in the programming sense) has some as well. Neither of the latter two require learning category theory.
CT describes type theory really well, which is used in a lot of programming languages. Monoids and Monads come from CT, and their implementation doesn't really explain what the concept really is (it kind of does, but not really). The programmer might not need to learn CT, but it helps to have broader understanding of how the underlying rules of the language are working.
Nope. Type Theory (like Matin Luff TT) describes Type Theory really well. And it is way more useful for software development than CT is. Tools like Coq, LEAN, Agda, Idris etc. are built on TT not CT.
> Category theory has virtually zero application to software engineering.
> Knowing what a monad is (in the programming sense) has some [practical application]as well.
You’re contradicting yourself. You don’t need to know category theory to use practical abstractions like functors and monads. They are still however category theoretical concepts. If these category theoretical concepts have “some practical application” as you say, then category theory does have application to software engineering.
It had application in the sense that it provided inspiration for these concepts, yes. However learning anything else from category theory is irrelevant to understanding and using them.
If you have a symmetric system, you can use group theory to make predictions about it. Similarly, if you have a compositional system, you can use category theory to make predictions about it. As for whether those predictions are useful, or non-obvious, or worth it, etc, tends to very a lot depending on the application, and the other techniques available- it's not a matter of being smart enough to get it. FWIW, I think category theory is certainly indispensable in denotational programming language semantics.
I can't answer your question in terms of pragmatic applications. Based on skimming this book and similar resources posted on HN, my -- naive, I am sure -- understanding is that category theory unifies other theories and serves as a glue.
One example from the linked book, "Remember that we said that programming types (classes) are somewhat similar to sets, and programming methods are somewhat similar to functions between sets, but they are not exactly identical? A formal connection between the two can be made via category theory."
>Working as a SWE for lots of years, I've worked with people that always feel the need to "add a level of abstraction" or "build a framework" for (and out of) everything.
Very funny to read, especially because it feels outdated. "With this new architecture we cannot simply..." years after we can say we can!
Edit - "No, Sun, we’re not going to be able to analyze our corporate sales data “as simply as putting a DVD into your home theatre system.”" is what I am talking about
> Every time I try to learn about Category Theory, I get to a certain point and just start wondering "why?"
> I can't tell if I'm not smart enough to get it (I think that's the most likely answer). But it feels like some math people -- who have nothing better to do -- are just trying to create another level of abstraction.
> Are there things that Category Theory does that other branches couldn't already do? I'm curious to hear other takes on this.
No, category theory was not created for the sake of an another level of abstraction. It was created to talk about algebraic topology. I am not sure if you gain anything studying that. Probably you won't but who knows. (But I am thinking more and more that CT advocates are harmful, they trick people to dive into CT which just steals their time, and gives them nothing.)
> I've worked with people that always feel the need to "add a level of abstraction" or "build a framework" for (and out of) everything
This is my experience with the Java folk, and they're certainly not being inspired by category theory.
Indirection isn't abstraction. What I usually come across is someone wanting to make a REST POST, but they wrap it in a networkconnection in a networkclient in a networkworker in networkmanager, which can only be instantiated by a network factory.
CT-thinking (not that I've studied any) is more like: a POST is an arrow from request to response. Can we join such items head-to-tail? Is it ok to substitute createUser.then(createPassword()) for createUserAndPassword? Can we map over the arrow without unpacking it and repacking it, etc.
You haven’t taken the time to understand it, therefore it must be useless. Others who practice it must be doing useless things. This seems like a common solipsism among software developers.
The problem with your definition of abstraction is that you’re thinking of indirection.
Abstractions in mathematics create precise definitions with provable laws. When you have a proof of a theorem you can ignore the details underneath and think in terms of the abstraction.
I wouldn’t be so cavalier about dismissing category theory. It’s interesting for many reasons but it’s not here to serve you. You could find ways that it can improve your programs and how you think about them, as others have, or you can get on without it. And that’s fine too!
Abstraction is powerful. It lets you control more with less. As you keep abstracting things get complicated. Category Theory studies abstractions and lets you build extremely abstract systems, which can be leveraged to create reliable and also flexible systems. The downside is of course the learning curve, and because of this it isn't as friendly for business and teams to use. In an ideal tech industry (in my opinion), engineers are encouraged to persue the most complicated engineering, despite the rest of the company having very little idea of how it works. It would require a lot of trust, and also a different work ethic, but I digress.
You are most definitely smart enough. And your instincts are correct. Category Theory is not an efficient thinking tool for thinking about software. You can use it for that of course (monads were originally modelled in CT) but there are much better tools out there that will get you there faster (Abstract Algebra, Reduction Machines and State Machines for example).
There are a few (usually Haskell) people that love to make claims about the need for you to learn CT. They are wrong. It’s that simple.
Mathematics, at its foundations, loves to see how much they can get with the smallest number of concepts and assumptions. It’s good to learn what’s really necessary.
Between some branches of math. And you have to be familiar with these branches of (graduate level) math to appreciate it. Learning category theory in isolation will leave you baffled. Why it's being constantly pushed on HN is beyond me.
And that's a fine enough reason. Sometimes learning assembly leads people to learn about electrical engineering concepts, and boolean algebra. Sometimes writing algorithms that need to be efficient leads people to learn about number theory. Sometimes people trying to develop a really effective database schema get interested in set theory.
It's more of a "fundamental theory of abstraction". Once it clicks you can see it. And you will see all the design patterns and abstractions you see in books like the gof are the ones that are "arbitrary layers of abstraction"
That's why a lot of programmers want to incorporate the concepts into their languages. Because once it clicks you realize that all other abstractions (especially design patterns in oop) are flawed and wrong.
You start to see cracks and broken abstractions everywhere. This is not just another GoF design patterns thing. It's fundamental.
But the thing is it's so hard usually if you get it, you'll be living in a world where most people don't get it. So people will have trouble understanding you.
If you don't ever get it, I would view it as a layman would view quantum physics or general relativity. You know it's a more fundamental description of the universe but you just can't grasp it.
Sorry if I offended your sensibilities. Feel free to downvote my comment if you don’t think it adds to the conversation.
I would appreciate, though, if you didn’t misquote my comment when replying to it. Neither of the things you have quoted are things I wrote. Worse, they are a misrepresentation of my comment.
Why? For a lot of mathematicians, work being of practical use isn't even a consideration. I know math professors who would freely tell you that they were never aware of any application for their PhD work and never cared, because it was interesting. A lot of math does have a lot of useful applications, and there certainly are applied mathematicians, but it is not, on the face of it, absurd to suppose that there are areas of math with little to no known practical application.
> I know math professors who would freely tell you that they were never aware of any application for their PhD work and never cared, because it was interesting
Exactly, but that's the opposite of an anti-intellectual attitude. It's not anti-intellectual to engage with pure maths-- on the contrary lol! What's anti-intellectual is to say that if you can't see the value of maths, that it's not worth engaging with. You'll never see the value of anything if you don't engage with it, and also I'm pretty sure that by now people shouldn't need any convincing that maths is worth learning.
And when I say "engage with maths" I don't mean "I read an article or a book once but I didn't get it so I decided that it wasn't worth the time".
Every time I try to learn about Category Theory, I get to a certain point and just start wondering "why?"
I can't tell if I'm not smart enough to get it (I think that's the most likely answer). But it feels like some math people -- who have nothing better to do -- are just trying to create another level of abstraction.
Are there things that Category Theory does that other branches couldn't already do? I'm curious to hear other takes on this.