9 ms·

Category Theory Illustrated

191 points by signa11 4y ago

superb-owl 4y ago
I'm enjoying the article, but this line is comically false: > noone bashed string theory for failing to make useful predictions
- diyseguy 4y ago
  not to mention the "moist interesting category-theoretical concepts"
  - Koshkin 4y ago
    Speaking of those - they are not even mentioned in the text (yet), but without them any presentation of CT will seem necessarily dry: natural transformations and equivalence; adjunctions; and, of course, monads (them being monoids in the category of endofunctors notwithstanding).
timmg 4y ago
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.
- dr_kiszonka 4y ago
  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."
- adamnemecek 4y ago
  It's a general scaffolding that can be shared between branches of math.
  - resource0x 4y ago
    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.
    - gryn 4y ago
      > Why it's being constantly pushed on HN is beyond me. the answer is haskell
      - dimitrios1 4y ago
        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. These are good things.
        epgui 4y ago
        Exactly, and it's supposed to be what this forum is all about: intellectual curiosity and (to some degree) interdisciplinary knowledge-sharing! It drives me nuts that the anti-intellectual stance is so prevalent.
        resource0x 4y ago
        I actually put some effort in learning it, and found it intellectually empty. If you find it enlightening - good for you. :-)
- wisnesky 4y ago
  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.
- School-Cotton 4y ago
  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.
  - francogt 4y ago
    > 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.
    - School-Cotton 4y ago
      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.
  - proc0 4y ago
    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.
    - School-Cotton 4y ago
      Monoids do not come from category theory, they come from abstract algebra. How does category theory “describe type theory” ?
      - 4y ago
- pohl 4y ago
  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.
- deleted 4y ago
  [deleted]
- hermitcrab 4y ago
  >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. You might be interested to read Spolsky's essay on 'architecture astronauts': https://www.joelonsoftware.com/2001/04/21/dont-let-architecture-astronauts-scare-you/ https://www.joelonsoftware.com/2001/04/21/dont-let-architect...
  - nuancebydefault 4y ago
    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
    - hermitcrab 4y ago
      It was written in 2001. But the architecture astronauts are still with us.
- l__l 4y ago
  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)
  - skrtskrt 4y ago
    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
- bmacho 4y ago
  > 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.)
- maxiepoo 4y ago
  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.
  - dimitrios1 4y ago
    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.
    - cfiggers 4y ago
      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).
    - deterministic 4y ago
      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.
- epgui 4y ago
  Why are we not all downvoting anti-intellectualist comments like this? The odds that a whole discipline of mathematics is "of little use" or "nothing more than an inappropriate abstraction" is basically nil. I see little difference between this comment, and a comment that calls sociology or philosophy useless. No sympathy from me!
  - thfuran 4y ago
    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.
    - epgui 4y ago
      > 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".
  - timmg 4y ago
    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.
- agentultra 4y ago
  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!
- mirekrusin 4y ago
  They just want to articulate levels of abstractions for abstractions.
- proc0 4y ago
  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.
- HervalFreire 4y ago
  It just hasn't clicked for you yet. I wouldn't call it "another level of abstraction" 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.
- deterministic 4y ago
  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.
- mrkeen 4y ago
  > 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. Have a read of https://www.haskellforall.com/2014/04/scalable-program-architectures.html https://www.haskellforall.com/2014/04/scalable-program-archi...
dang 4y ago
Related: Category Theory Illustrated – Functors - https://news.ycombinator.com/item?id=35173632 https://news.ycombinator.com/item?id=35173632 - March 2023 (3 comments) Category Theory Illustrated – Logic - https://news.ycombinator.com/item?id=28660157 https://news.ycombinator.com/item?id=28660157 - Sept 2021 (112 comments) Category Theory: Orders - https://news.ycombinator.com/item?id=26658111 https://news.ycombinator.com/item?id=26658111 - April 2021 (52 comments)
civilized 4y ago
I'm a sympathetic mathematician but I found this whole passage bizarrely off-base: > mathematics is in a weird and, I’d say, unique position of always having to defend what they do with respect to it’s value for other disciplines. I again stress that this is something that would be considered absurd when it comes to any other discipline. > People don’t expect any return on investment from physical theories - noone bashed string theory for failing to make useful predictions. Huh? Tons of people bash string theory for failing to make useful predictions! There was a whole book criticizing it called "Not Even Wrong", meaning it didn't even reach the point of being testable. A better argument for advanced math: it's not always clear in advance what math will be useful. A popular example is Riemannian geometry, which had no clear real-world value when developed, but ended up being used for General Relativity a few decades later. I also support the argument that math is intrinsically and aesthetically valuable, but for those who don't agree, the argument above may make some headway.
- tylerhannan 4y ago
  Even in the late 90s when I was studying Topology there was a bit of furor over the lack of application... Until, suddenly, it was quite applicable in evolutionary biology and knot theory in context of D?NA enzymes.
- canjobear 4y ago
  When I think of string theory, the first thing that pops into my head is "mathematically sophisticated theory that sucked in a generation of physicists but is dead now because it doesn't make any testable predictions".
  - civilized 4y ago
    But some of the fancy math in this area is turning out to be helpful for QFT calculations, and it's becoming clear that we need more powerful math to even formulate proper quantum field theories. Check this article out for a briefing: https://www.quantamagazine.org/the-mystery-at-the-heart-of-physics-that-only-math-can-solve-20210610/ https://www.quantamagazine.org/the-mystery-at-the-heart-of-p...
mjmsmith 4y ago
“The set of queens of England is a singleton set.” Hmmm.
- yazzku 4y ago
  It's for sure a bad set of examples. There have been several queens of England. At the very least you should know Bloody Mary, if not for the Tudor's legacy of a golden age with Queen Elizabeth (another queen!), then for the goddamn alcoholic drink. And I'm not even British. "The set of all red balls contains just one ball." There are many reds too... It's just unintuitive to try to force "intuitive" examples into a mathematical framework... Might as well just do it the right, formal way.
  - FrustratedMonky 4y ago
    I think it is referring to the concept of one ruler at a time 'The King is dead, long live the king' means there is only one king, one is gone, and in same sentence the new one is in place. There is only ever one.
jbottoms 4y ago
"...and mathematics their" ?
pstoll 4y ago
> People don’t expect any return on investment from physical theories - noone bashed string theory for failing to make useful predictions. Uhhh… heck yeah they did. It’s the primary and most widely challenged aspect of the theory. This does not help your paper be more credible.
Koshkin 4y ago
> Illustrated Isn’t what diagrams were for?
akdor1154 4y ago
Not sure if the author is here, but I would happily fling a few dollars his way for an epub/pdf. This looks great.
- Myrmornis 4y ago
  Me too
  - boris_m 4y ago
    Especially for you: https://abuseofnotation.github.io/category-theory-illustrated/print/ https://abuseofnotation.github.io/category-theory-illustrate...
lincpa 4y ago
[dead]
teleforce 4y ago
This looks like a very good book on Category Theory with a lot of illustrations. Anyone can compare between this book and The Joy of Abstraction by Eugenia Cheng for learning Category Theory beginner? https://golem.ph.utexas.edu/category/2023/02/the_joy_of_abstraction.html https://golem.ph.utexas.edu/category/2023/02/the_joy_of_abst...
yazzku 4y ago
> In other words, we can solve problems without any advanced math, or with no math at all, as evidenced by the fact that the agyptians were able to build the pyramids without even knowing Euclidian geometry. Egyptians. Capital E regardless too. Also, the Egyptians knew a fair amount of geometry: https://en.wikipedia.org/wiki/Egyptian_geometry https://en.wikipedia.org/wiki/Egyptian_geometry https://en.wikipedia.org/wiki/History_of_geometry https://en.wikipedia.org/wiki/History_of_geometry https://www.britannica.com/science/mathematics/Mathematics-in-ancient-Egypt https://www.britannica.com/science/mathematics/Mathematics-i...
- boris_m 4y ago
  "...without even knowing Euclidian geometry."
  - yazzku 4y ago
    What exactly about Euclidian geometry is it that the Egyptians needed to build the pyramids? Clearly nothing. My point is that this part of the text is just fluff. Engineers have been building shit without having a formal description of that shit for millennia.
yazzku 4y ago
Sorry, it's a very low-quality "book". See the other comments for reasons why. This is going to need a lot of improvement before it can be recommended. I'd love to see a casual/friendly introduction to the topic, though.
sriram_malhar 4y ago
Whenever I have tried any text about category theory, it seems to me that it is a catalogue of one definition after another. I am yet to come across (or perhaps missed it entirely) an example use of any of these definitions in practical terms, other than their use in the next definition. It is quite possible that one requires a lot of set up before any it can be used for practical purposes (e.g explaining the basis for all of Haskell's prelude), but I have never made it that far. But I would like to ... I'd appreciate hearing from those of you who are programmers at heart and have hung in there long enough to gain some insight. What clicked for you?
- dirkt 4y ago
  I've also seen this in many books. When I learned it at university, we had lots of examples, but you need to be already familiar with a lot of somewhat abstract mathematics (monoids, groups, rings, sets, homomorphisms, ...) before you see how it unifies those. Category theory is called "general abstract nonsense" for a reason. You can think of it as "expressing other mathematics in terms of arrows and commuting diagrams".
FrustratedMonky 4y ago
It isn't first time there has been long threads about Category Theory with so much controversy. How does a subject like this get people so worked up? Maybe it is just all cross functional subjects, where a lot of people know a little bit of some aspect and make some sweeping assumptions. Kind of like all posts about AI are controversial now, because everyone is arguing from some thin slice of their perception on the subject.
- bruce343434 4y ago
  Frustration? A CT is a huge topic yet people struggle to really "get" it. That creates cognitive dissonance.
Myrmornis 4y ago
This looks excellent. Are more chapters planned and is there a way to buy a PDF?
- boris_m 4y ago
  Especially for you: https://abuseofnotation.github.io/category-theory-illustrated/print/ https://abuseofnotation.github.io/category-theory-illustrate...
  - Myrmornis 4y ago
    Beautiful, thanks very much!
galaxyLogic 4y ago
I wonder, SQL is based on, and implements the calculational aspects of Sets. Could we similarly think of a programming language that was based on and implemented the basic concepts of Category Theory? What might such a thing be useful for?

191 points by signa11 4y ago

I'm enjoying the article, but this line is comically false: > noone bashed string theory for failing to make useful predictions

diyseguy 4y ago
not to mention the "moist interesting category-theoretical concepts"
- Koshkin 4y ago
  Speaking of those - they are not even mentioned in the text (yet), but without them any presentation of CT will seem necessarily dry: natural transformations and equivalence; adjunctions; and, of course, monads (them being monoids in the category of endofunctors notwithstanding).

timmg 4y ago

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.

dr_kiszonka 4y ago
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."
adamnemecek 4y ago
It's a general scaffolding that can be shared between branches of math.
- resource0x 4y ago
  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.
  - gryn 4y ago
    > Why it's being constantly pushed on HN is beyond me. the answer is haskell
    - dimitrios1 4y ago
      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. These are good things.
      - epgui 4y ago
        Exactly, and it's supposed to be what this forum is all about: intellectual curiosity and (to some degree) interdisciplinary knowledge-sharing! It drives me nuts that the anti-intellectual stance is so prevalent.
        resource0x 4y ago
        I actually put some effort in learning it, and found it intellectually empty. If you find it enlightening - good for you. :-)
wisnesky 4y ago
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.
School-Cotton 4y ago
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.
- francogt 4y ago
  > 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.
  - School-Cotton 4y ago
    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.
- proc0 4y ago
  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.
  - School-Cotton 4y ago
    Monoids do not come from category theory, they come from abstract algebra. How does category theory “describe type theory” ?
    - 4y ago
pohl 4y ago
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.
deleted 4y ago
[deleted]
hermitcrab 4y ago
>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. You might be interested to read Spolsky's essay on 'architecture astronauts': https://www.joelonsoftware.com/2001/04/21/dont-let-architecture-astronauts-scare-you/ https://www.joelonsoftware.com/2001/04/21/dont-let-architect...
- nuancebydefault 4y ago
  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
  - hermitcrab 4y ago
    It was written in 2001. But the architecture astronauts are still with us.
l__l 4y ago
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)
- skrtskrt 4y ago
  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
bmacho 4y ago
> 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.)
maxiepoo 4y ago
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.
- dimitrios1 4y ago
  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.
  - cfiggers 4y ago
    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).
  - deterministic 4y ago
    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.
epgui 4y ago
Why are we not all downvoting anti-intellectualist comments like this? The odds that a whole discipline of mathematics is "of little use" or "nothing more than an inappropriate abstraction" is basically nil. I see little difference between this comment, and a comment that calls sociology or philosophy useless. No sympathy from me!
- thfuran 4y ago
  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.
  - epgui 4y ago
    > 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".
- timmg 4y ago
  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.
agentultra 4y ago
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!
mirekrusin 4y ago
They just want to articulate levels of abstractions for abstractions.
proc0 4y ago
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.
HervalFreire 4y ago
It just hasn't clicked for you yet. I wouldn't call it "another level of abstraction" 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.
deterministic 4y ago
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.
mrkeen 4y ago
> 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. Have a read of https://www.haskellforall.com/2014/04/scalable-program-architectures.html https://www.haskellforall.com/2014/04/scalable-program-archi...

dang 4y ago

Related: Category Theory Illustrated – Functors - https://news.ycombinator.com/item?id=35173632 https://news.ycombinator.com/item?id=35173632 - March 2023 (3 comments) Category Theory Illustrated – Logic - https://news.ycombinator.com/item?id=28660157 https://news.ycombinator.com/item?id=28660157 - Sept 2021 (112 comments) Category Theory: Orders - https://news.ycombinator.com/item?id=26658111 https://news.ycombinator.com/item?id=26658111 - April 2021 (52 comments)

civilized 4y ago

I'm a sympathetic mathematician but I found this whole passage bizarrely off-base: > mathematics is in a weird and, I’d say, unique position of always having to defend what they do with respect to it’s value for other disciplines. I again stress that this is something that would be considered absurd when it comes to any other discipline. > People don’t expect any return on investment from physical theories - noone bashed string theory for failing to make useful predictions. Huh? Tons of people bash string theory for failing to make useful predictions! There was a whole book criticizing it called "Not Even Wrong", meaning it didn't even reach the point of being testable. A better argument for advanced math: it's not always clear in advance what math will be useful. A popular example is Riemannian geometry, which had no clear real-world value when developed, but ended up being used for General Relativity a few decades later. I also support the argument that math is intrinsically and aesthetically valuable, but for those who don't agree, the argument above may make some headway.

tylerhannan 4y ago
Even in the late 90s when I was studying Topology there was a bit of furor over the lack of application... Until, suddenly, it was quite applicable in evolutionary biology and knot theory in context of D?NA enzymes.
canjobear 4y ago
When I think of string theory, the first thing that pops into my head is "mathematically sophisticated theory that sucked in a generation of physicists but is dead now because it doesn't make any testable predictions".
- civilized 4y ago
  But some of the fancy math in this area is turning out to be helpful for QFT calculations, and it's becoming clear that we need more powerful math to even formulate proper quantum field theories. Check this article out for a briefing: https://www.quantamagazine.org/the-mystery-at-the-heart-of-physics-that-only-math-can-solve-20210610/ https://www.quantamagazine.org/the-mystery-at-the-heart-of-p...

mjmsmith 4y ago

“The set of queens of England is a singleton set.” Hmmm.

yazzku 4y ago
It's for sure a bad set of examples. There have been several queens of England. At the very least you should know Bloody Mary, if not for the Tudor's legacy of a golden age with Queen Elizabeth (another queen!), then for the goddamn alcoholic drink. And I'm not even British. "The set of all red balls contains just one ball." There are many reds too... It's just unintuitive to try to force "intuitive" examples into a mathematical framework... Might as well just do it the right, formal way.
- FrustratedMonky 4y ago
  I think it is referring to the concept of one ruler at a time 'The King is dead, long live the king' means there is only one king, one is gone, and in same sentence the new one is in place. There is only ever one.

jbottoms 4y ago

"...and mathematics their" ?

pstoll 4y ago

> People don’t expect any return on investment from physical theories - noone bashed string theory for failing to make useful predictions. Uhhh… heck yeah they did. It’s the primary and most widely challenged aspect of the theory. This does not help your paper be more credible.

Koshkin 4y ago

> Illustrated Isn’t what diagrams were for?

akdor1154 4y ago

Not sure if the author is here, but I would happily fling a few dollars his way for an epub/pdf. This looks great.

Myrmornis 4y ago
Me too
- boris_m 4y ago
  Especially for you: https://abuseofnotation.github.io/category-theory-illustrated/print/ https://abuseofnotation.github.io/category-theory-illustrate...

lincpa 4y ago

[dead]

teleforce 4y ago

This looks like a very good book on Category Theory with a lot of illustrations. Anyone can compare between this book and The Joy of Abstraction by Eugenia Cheng for learning Category Theory beginner? https://golem.ph.utexas.edu/category/2023/02/the_joy_of_abstraction.html https://golem.ph.utexas.edu/category/2023/02/the_joy_of_abst...

yazzku 4y ago

> In other words, we can solve problems without any advanced math, or with no math at all, as evidenced by the fact that the agyptians were able to build the pyramids without even knowing Euclidian geometry. Egyptians. Capital E regardless too. Also, the Egyptians knew a fair amount of geometry: https://en.wikipedia.org/wiki/Egyptian_geometry https://en.wikipedia.org/wiki/Egyptian_geometry https://en.wikipedia.org/wiki/History_of_geometry https://en.wikipedia.org/wiki/History_of_geometry https://www.britannica.com/science/mathematics/Mathematics-in-ancient-Egypt https://www.britannica.com/science/mathematics/Mathematics-i...

boris_m 4y ago
"...without even knowing Euclidian geometry."
- yazzku 4y ago
  What exactly about Euclidian geometry is it that the Egyptians needed to build the pyramids? Clearly nothing. My point is that this part of the text is just fluff. Engineers have been building shit without having a formal description of that shit for millennia.

yazzku 4y ago

Sorry, it's a very low-quality "book". See the other comments for reasons why. This is going to need a lot of improvement before it can be recommended. I'd love to see a casual/friendly introduction to the topic, though.

sriram_malhar 4y ago

Whenever I have tried any text about category theory, it seems to me that it is a catalogue of one definition after another. I am yet to come across (or perhaps missed it entirely) an example use of any of these definitions in practical terms, other than their use in the next definition. It is quite possible that one requires a lot of set up before any it can be used for practical purposes (e.g explaining the basis for all of Haskell's prelude), but I have never made it that far. But I would like to ... I'd appreciate hearing from those of you who are programmers at heart and have hung in there long enough to gain some insight. What clicked for you?

dirkt 4y ago
I've also seen this in many books. When I learned it at university, we had lots of examples, but you need to be already familiar with a lot of somewhat abstract mathematics (monoids, groups, rings, sets, homomorphisms, ...) before you see how it unifies those. Category theory is called "general abstract nonsense" for a reason. You can think of it as "expressing other mathematics in terms of arrows and commuting diagrams".

FrustratedMonky 4y ago

It isn't first time there has been long threads about Category Theory with so much controversy. How does a subject like this get people so worked up? Maybe it is just all cross functional subjects, where a lot of people know a little bit of some aspect and make some sweeping assumptions. Kind of like all posts about AI are controversial now, because everyone is arguing from some thin slice of their perception on the subject.

bruce343434 4y ago
Frustration? A CT is a huge topic yet people struggle to really "get" it. That creates cognitive dissonance.

Myrmornis 4y ago

This looks excellent. Are more chapters planned and is there a way to buy a PDF?

boris_m 4y ago
Especially for you: https://abuseofnotation.github.io/category-theory-illustrated/print/ https://abuseofnotation.github.io/category-theory-illustrate...
- Myrmornis 4y ago
  Beautiful, thanks very much!

galaxyLogic 4y ago

I wonder, SQL is based on, and implements the calculational aspects of Sets. Could we similarly think of a programming language that was based on and implemented the basic concepts of Category Theory? What might such a thing be useful for?