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(=_=) (Mar 31 2020 at 03:14):@Bobby T said:
Hi, my name is Bobby Tosswill. I did my undergrad in linguistics at the University of Massachusetts Amherst, and in the last two years I've been trying to dig into category theory, to mixed effect, haha. I found out about this group (and Zulip, if I'm being honest) from a Slack group on category theory in functional programming and figured I might be able to learn a thing or two. I guess I should say ahead of time that just about all of my questions are liable to be stupid ones, haha
Hi Bobby! More linguists, yay! Have you checked out the work done by Bob Coecke on DisCoCat? There's a thread here about that as well: #applied category theory > DisCoCat. Warning: controversial opinions there haha.
All my questions here have been stupid, so you're all good. :sweat_smile:
(=_=) (Mar 31 2020 at 03:23):Michael Zargham said:
i've been working on compositions of invariant preserving mechanisms and configuration spaces for economic systems.
Hi Michael! Hmm... what are "configuration spaces for economic systems"? In any case, @Jules Hedges is interested in applying CT to economics, but you probably knew that?
We should have an economics thread!
Vinay Madhusudanan (Mar 31 2020 at 04:27):Hi Gabo! What sort of undergraduate degree is seven years long… on the average? And if it has an inflexible curriculum, which aspect makes the time vary about the average?
@Gabriel Goren
By the way, I too got an undergraduate degree in physics and then switched to mathematics. I'd almost forgotten that… -_-
(=_=) (Mar 31 2020 at 04:56):Vinay Madhusudanan said:
Hi Gabo! What sort of undergraduate degree is seven years long… on the average? And if it has an inflexible curriculum, which aspect makes the time vary about the average?
From Wikipedia:
- University level: 4- to 6-years of professional education taught at universities offering many different degrees, such as Licentiate, Engineer's degree, Medic Title, Attorney Title, Professorships, Translation degrees, etc.
Basically, this was the continental system until the Bologna Process, which introduced the shorter Bachelor's and Master's degrees in continental Europe to harmonise with the Anglo-Saxon system.
(=_=) (Mar 31 2020 at 05:17):@Gabriel Goren said:
Hi, I'm Gabo, an undergraduate student from Buenos Aires, Argentina who wants to do a PhD in math, maybe Applied Category Theory if fate permits.
Hi Gabo!
I feel like I've already specialized in something I didn't want to, which... Sucks?
Welcome to academic life! :sweat_smile: :rainbow:
It makes me feel that, given my true interests, I "should already know" so many things that I don't...
Everybody wishes they should have already known so many things that they didn't. It's normal. Ignorance is a fundamental human condition. Just keep learning.
Also no one knows what "Applied Category Theory" is around here
Because that name was only invented a couple of years ago. The idea was to apply category theory to the sciences and engineering disciplines, given the success it has had in computer science.
(=_=) (Mar 31 2020 at 05:30):@Tarun Raheja said:
For my bachelor's thesis I made a switch to computational condensed matter, out of sheer curiosity towards Physics, and have been hooked since. I am now at Prof. Franco Nori's group (Theoretical Quantum Physics Group) in RIKEN, Japan as a funded research assistant for the last six months, and I will spend the rest of this year here as well.
Hi Tarun! Wow, that's a very interesting group! My PhD advisor (this was a long time ago) did some mathematical work around the quantum Hall effect, so he's worked with operator algebras, which I picked up a bit while I was doing my PhD. Some operator algebraists are now working with more advanced category theory, so there's potential there.
My introduction to Category Theory was through following the work of Prof. Bob Coecke in the Foundations, Structures and Quantum group at Oxford (which I feel is doing some phenomenal work at the intersection of theoretical CS and physics).
And natural language processing (NLP) too! It's awesome.
John Baez (Mar 31 2020 at 05:33):There's a wonderful field of studies that connects operator algebras and physics to higher category theory, which I believe is far from exhausted.
Bobby T (Mar 31 2020 at 09:49):@Rongmin Lu Haha, it's a great club to be in! Yeah, it's actually kind of a funny story - one of my big interests is diagrammatic reasoning, and so unrelated to linguistics or category theory, I found his work with quantum physics diagrams. I kind of put it to the side, and then months later, I was having trouble at work (AI startup) trying to find a model for natural language that we could implement alongside our vector-based approach, and suddenly there he pops up again haha. I've applied for the Masters of Logic at the ILLC at UvA in Amsterdam, and Martha Lewis, who worked with him at Oxford is there, so something tells me I'm gonna keep running into Bob Coecke's work haha.
Btw, how do you all do these nice formatting things? With the grey boxes and stuff?
Also, good to hear I'm not the only one with the basic questions ahah. I find this stuff fascinating, but I didn't have a strong maths background before I got into it, so I've had a lot of catching up to do - lots of reading a book, understanding just about nothing, and then coming back months later and going "OHHHHH! That's what a pullback is!" As I'm sure you understand, this group is SUPER intimidating
Bobby T (Mar 31 2020 at 09:53):@Gabriel Goren I understand how you feel, I think. I did my undergrad in linguistics, and while I liked what I learned, coming into category theory from there can really give you the feeling of "why don't I understand this already? Why isn't this coming together quicker!"
Also, I also got interested in category theory partly from an interest in complex systems! We should set up a stream if there isn't one already!
Fabrizio Genovese (Mar 31 2020 at 10:09):Bobby T said:
Also, good to hear I'm not the only one with the basic questions ahah. I find this stuff fascinating, but I didn't have a strong maths background before I got into it, so I've had a lot of catching up to do - lots of reading a book, understanding just about nothing, and then coming back months later and going "OHHHHH! That's what a pullback is!" As I'm sure you understand, this group is SUPER intimidating
Yes, traditionally category theory has been considered quite an intimidating subject, but I can assure you many people in this group are doing everything they can to disprove this. In the end category theory is about abstracting stuff, but as such, each abstraction models things you can experience and observe in "everyday life". This is especially evident in applied category theory. I hope that with time we will be able to show everyone that it's not as hard as many people think!
Matteo Capucci (he/him) (Mar 31 2020 at 10:25):Fabrizio Genovese said:
Yes, traditionally category theory has been considered quite an intimidating subject, but I can assure you many people in this group are doing everything they can to disprove this. In the end category theory is about abstracting stuff, but as such, each abstraction models things you can experience and observe in "everyday life". This is especially evident in applied category theory. I hope that with time we will be able to show everyone that it's not as hard as many people think!
I'm very convinced a great deal of CT complexity is not really there, it's just 'cultural'. Many mathematical topics which are considered more elementary than CT are actually more sophisticated in terms of ideas and reasoning (i.e. calculus is harder than basic CT, imho). Basic CT barely has any theorems!
To be fair, this could be said of many areas of mathematics, albeit few have such a gentle learning curve as CT imho. Maybe there's a big Dirac delta at the beginning because of the diagrammatic reasoning, but after then it isn't steep at all (while, e.g., graph theory starts super-easy and then blows up)
Bobby T (Mar 31 2020 at 10:38):As someone who's come into this field with a pretty limited maths background, I think that there are probably two main reasons why category theory can get a bad rap:
Because it has so many broad applications in higher mathematics, it's often discussed using examples from a wide range of these fields. Maybe if the learner isn't as comfortable with all of them, it can be easy to get lost. As in, let's say I'm comfortable talking about group theory, but know nearly nothing about topology as such. If I'm being taught about limits and the first example is from group theory, but the second is about topological spaces, I might just not catch the pattern, and then it becomes harder to follow everything. If that happens enough, you might just throw your hands up and say "it's abstract nonsense!"
I think that the flexibility of CT can make it hard to keep track of as well. You start by talking about things in 1-categories, but then to introduce new gadgets, you might need to implicitly take advantage of 2-categories, without having been told what they are. The domains and codomains of things can also be flexible, which can rattle your understanding: I was just getting used to the idea that a functor brings objects to objects and morphisms to morphisms when suddenly I hear about hom-functors, and now we're having objects mapped to sets of morphisms. Just earlier in the #basic questions chat, someone was asking about cospans, and @John Baez says that you can define them as the morphisms in another category - keeping track of what remains steady and what can vary in a theory as flexible as CT can make your head spin.
But again, I'm new to all of this, so take this all with a grain of salt.
Jules Hedges (Mar 31 2020 at 10:38):In my opinion category theory is genuinely difficult because abstraction is genuinely difficult, especially if you're coming through the traditional route via algebra and geometry, which are already at a pretty high level of abstraction
Jules Hedges (Mar 31 2020 at 10:40):Reasoning with points is definitely partly cultural, but I have a strong feeling that it's also partly innate as well - reasoning without points and only with morphisms feels like an "unnatural" thing that needs to be learned and practiced, to me
Fabrizio Genovese (Mar 31 2020 at 10:53):Bobby T said:
As someone who's come into this field with a pretty limited maths background, I think that there are probably two main reasons why category theory can get a bad rap:
- I think that the flexibility of CT can make it hard to keep track of as well. You start by talking about things in 1-categories, but then to introduce new gadgets, you might need to implicitly take advantage of 2-categories, without having been told what they are. The domains and codomains of things can also be flexible, which can rattle your understanding: I was just getting used to the idea that a functor brings objects to objects and morphisms to morphisms when suddenly I hear about hom-functors, and now we're having objects mapped to sets of morphisms. Just earlier in the #basic questions chat, someone was asking about cospans, and John Baez says that you can define them as the morphisms in another category - keeping track of what remains steady and what can vary in a theory as flexible as CT can make your head spin.
So, there's a saying: In category theory everything is everything else. You can often describe the same thing using limits, ends, adjunctions, Kan extensions. What really changes is the point of view, not what you describe.
About the hom-functor, it is a functor! The hom functor on is a functor : Now you see, on the left you have , so objects (resp. morphisms) are pairs of objects (resp. morphisms) of . So each pair of objects goes to a set (the set of morphisms between those objects in ), that is an object in . Pairs of morphisms act by pre- and post-composition.
In any case, this is not the right stream to discuss about this, let's move to #basic questions :slight_smile:
Matteo Capucci (he/him) (Mar 31 2020 at 11:02):Bobby T said:
- Because it has so many broad applications in higher mathematics, it's often discussed using examples from a wide range of these fields. Maybe if the learner isn't as comfortable with all of them, it can be easy to get lost. As in, let's say I'm comfortable talking about group theory, but know nearly nothing about topology as such. If I'm being taught about limits and the first example is from group theory, but the second is about topological spaces, I might just not catch the pattern, and then it becomes harder to follow everything. If that happens enough, you might just throw your hands up and say "it's abstract nonsense!"
I think this is the main reason, books like MacLane's speak in elfic if you're not an advanced math undegrad.
Matteo Capucci (he/him) (Mar 31 2020 at 11:04):I orbited around categories for quite some time before they clicked, and in retrospective it was because by that time I accumulated enough math knowledge to understand many examples. Then you get on the train, and goes smoothly.
This is the Dirac delta at the origin of the learning curve that I was referring to in my other message.
Matteo Capucci (he/him) (Mar 31 2020 at 11:06):But as @Fabrizio Genovese said before, many catsters (especially applied ones) are trying to make CT more and more accessible. Two great examples are Seven Sketches by Fong & Spivak, and Notes on CT by Perrone. If I had to suggest to a 'non-specialist' where to start, I'd point those two out @Bobby T .
Vinay Madhusudanan (Mar 31 2020 at 11:09):Not to mention, if you're into programming then there are great resources that introduce you to category theory with immediate examples or applications in programming. I can't recommend Milewski's Category Theory for Programmers enough. Just about halfway through that book and I'm understanding all this stuff like never before. Not without any struggle at all, of course — but unlike with resources I tried previously, it doesn't get to a point where I find that I have to go learn some other math first before I can understand the examples of these ultra-high level abstractions being discussed
Bobby T (Mar 31 2020 at 11:18):Yeah, I think I'm moving out of the "orbiting" stage just about now, which is a great feeling! Haha, I think maybe my lack of math knowledge might have interfered with my understanding the Dirac delta bit. And "elfic" is the perfect word haha!
As to the more accessible bit, I completely agree! Most of the newer books are much better about introducing things to a non-specialist audience (trust me, during my orbit, I've read a good number of them haha). I really like Seven Sketches, and I think it does a great job of showing the applications outside of pure math, but when I started it, I didn't have the math background, and unfortunately, I don't know if they do enough to cover the basic gadgets first, so I'm excited to reread it again after I feel more certain of the basics! I'll also have to check out Perrone's Notes, thanks!
@Vinay Madhusudanan , Milewski has a great talent for explanation, and I can't tell you how many times I've gotten confused with a topic and then went to the section in CTfP to see how he explained it, but unfortunately, I don't know any Haskell, so that's gotten in my way.
So far, I've found that some of the best resources for me have been Harold Simmons' An Introduction to Category Theory, Tai-Danae Bradley's Math3ma blog, and videos on Youtube by The Catsters, and this 16yo called "Fematika". Also, Barry Mazur's When is one thing equal to another? was really helpful for me to understand some of the motivations behind category theory. Btw, do you think we should make a discussion in #basic questions for this one?
Matteo Capucci (he/him) (Mar 31 2020 at 12:06):Bobby T said:
Haha, I think maybe my lack of math knowledge might have interfered with my understanding the Dirac delta bit.
:face_palm: I fell in the same pitfall I was pointing at
(=_=) (Mar 31 2020 at 12:21):Jules Hedges said:
Reasoning with points is definitely partly cultural, but I have a strong feeling that it's also partly innate as well - reasoning without points and only with morphisms feels like an "unnatural" thing that needs to be learned and practiced, to me
That's interesting to hear, because I was told sometime in my undergrad days that you don't refer to a function as "", as that refers to its evaluation at a point . So pointless reasoning had become second nature by the time I got my first degree.
Or is pointless reasoning the perspective that a point is a morphism ?
Nathanael Arkor (Mar 31 2020 at 12:25):it would be good to move this conversation about mathematical education to another topic :)
(=_=) (Mar 31 2020 at 12:28):Bobby T said:
Haha, it's a great club to be in!
I'm not actually a bona fide linguist, just a dabbler who got hooked after taking a linguistics elective in my undergrad days. But yeah, I'm all in when it comes to NLP.
something tells me I'm gonna keep running into Bob Coecke's work haha.
Sounds great!
Btw, how do you all do these nice formatting things? With the grey boxes and stuff?
You do this
```quote insert text here ```
That's not an apostrophe, by the way, that's the thing that points the other way. On my keyboard, it's under the tilde, but YMMV.
There's a guide to the markdown you can get to if you click on the "A" button next to the smiley icon on the bottom left of the box where you type in the text. At least that's what I see on desktop, not sure about the other devices. You can also hover over some message and see the source code by clicking on the clipboard on the top-right corner of the message next to the time stamp.
Matteo Capucci (he/him) (Mar 31 2020 at 12:35):Nathanael Arkor said:
it would be good to move this conversation about mathematical education to another topic :)
Right, I forked the topic to this from where the conversation started
Daniel Geisler (Apr 05 2020 at 00:54):What is the best collection of books to read to learn basic and advanced CT. Is reviewing This Week in Mathematical Physics still one of the most effective ways of learning CT.
John Baez (Apr 05 2020 at 00:56):This Week's Finds is mainly good if you want to learn all the coolest math and physics (up to when I quit writing it).
Fabrizio Genovese (Apr 05 2020 at 00:56):I think it really depends where you start from. For me the most insightful book as a beginner has been MacLane, but many people would disagree
John Baez (Apr 05 2020 at 00:57):For basic category theory I recommend the books at the bottom of this page:
Fabrizio Genovese (Apr 05 2020 at 00:58):I think Categories for the Working Mathematician is the perfect book to learn if you know some mathematics, but with some caveats, namely: Do not try to understand all of the examples, at least for me some examples where about stuff I know very little about, so I just ignored them. Also, do not focus too much on the exercises. They are notoriously very difficult, if you don't know this beforehand it can be a very frustrating experience. But I find MacLane writing style increadibly terse and insightful, hence I loved the book
Fabrizio Genovese (Apr 05 2020 at 01:00):Another book that completely blew my mind is Sheaves in geometry and Logic. It's quite hard and progress for me has been slow, but it really made a lot of things click. More importantly, for people like me that do not have any geometric intuition, it gave me for the first time a sense of how geometry works. Before going through it I could deal with toposes as logical entities, but all the things people were saying about their connection with topology etc made just no sense. After I finally understood! I cannot say I'm fluent in this, but at least the general intuition is there now!
Evan Patterson (Apr 05 2020 at 01:09):If you have some mathematical background, another really nice introduction to category theory is Emily Riehl's textbook. When learning a new subject, I like to see lots of examples, and she has collected many good ones from math and computer science. The selection of topics is great too.
Fabrizio Genovese (Apr 05 2020 at 01:40):Evan Patterson said:
If you have some mathematical background, another really nice introduction to category theory is Emily Riehl's textbook. When learning a new subject, I like to see lots of examples, and she has collected many good ones from math and computer science. The selection of topics is great too.
Yes, that's also in the list that John linked!
Evan Patterson (Apr 05 2020 at 02:20):OK, then I am seconding that recommendation :)
Paolo Perrone (Apr 05 2020 at 05:39):More thumbs up for Emily Riehl's book!
sarahzrf (Apr 05 2020 at 06:00):shes a riehl one
Thomas Read (Apr 05 2020 at 06:29):I wrote a blog post on this a little bit ago:
Getting Started with Category Theory
(definitely agree about Riehl!)
There are also some recommendations there if you have less of an undergrad maths / equivalent background.
Bobby T (Apr 06 2020 at 16:27):Thomas Read said:
I wrote a blog post on this a little bit ago:
Getting Started with Category Theory
(definitely agree about Riehl!)
There are also some recommendations there if you have less of an undergrad maths / equivalent background.
As someone who came into this with very little math background and tried hitting my head against it for quite a few months, haha, I've gotta say this is a great list - I think these are the resources that have been most helpful for me (along with Simmons's Introduction to Category Theory and a few others I'm probably forgetting). It's an intimidating topic to jump into, but I think as someone pointed out last week, that there's been a lot of work in the last few years to make things more accessible, and I'm very grateful for that
Matteo Capucci (he/him) (Apr 06 2020 at 20:35):It just popped into my mind that also Aluffi's "Algebra: Chapter 0" is a great start if you know some 'classical' algebra and need to pick up on the categorical thinking. One only needs to go through the first 2/3 chapters if they are only interested in CT
Faré (Apr 09 2020 at 07:41):Should we create a stream specifically for beginners? In any way, I enjoyed this introduction:
https://youtu.be/UwYLaGzhDb4
Beppe Metere (Apr 09 2020 at 12:15):T. Leinster "Intro to CT". An excellent source for beginners. It's easy, basic , recent and short. Very gentle introduction, with examples from many areas of maths.
RFC Walters. "Categories and computer science". Old fashioned approach by a master of CT. The book is easy to read, with many toy-examples, but with incursions in some deep subjects.
FW Lawvere and SH Schanuel "Conceptual mathematics: a first introduction to categories". At the roots of some basic ideas in CT, in maths, in science.
:)
Joe Moeller (Apr 09 2020 at 16:47):re: stream for beginners, that's the point of the stream #learning: basic questions
Beppe Metere (Apr 09 2020 at 17:35):@Todd Trimble cheers, I fixed it... (I was typing with my mobile, and 'lost' Schanuel in copypasting!)
:upside_down:
Christian Williams (Apr 09 2020 at 19:26):I'm definitely open to a stream just for beginners. could be a good place for quite a few people.
Joe Moeller (Apr 09 2020 at 20:56):What's wrong with #learning: basic questions ?
John Baez (Apr 09 2020 at 20:57):Indeed!
Simon Burton (Jul 09 2020 at 18:11):Sets & functions is a very good start.. Have you seen Lawvere and Schanuel's book on categories "Conceptual mathematics" ? Highly recommended.
James Wood (Jul 10 2020 at 12:27):I guess a funny point is that the objects of Set really do have elements in a category-theoretic sense. For a set X, the collection of morphisms f : 1 → X is in bijection with the elements of X (in the set-theoretic sense). This is a trick that comes up again, for example, in vector spaces, where given X a vector space over a field F, its “elements” are the linear maps of type F → X (with F seen as a vector space over itself). You can recover the classical notion of element by applying such a map to 1, and noticing that its action on arbitrary elements is determined by its action on 1. (Take f : F → X. Note f(x) = f(x × 1) = xf(1).) This notion is expanded fully at https://ncatlab.org/nlab/show/generalized+element. An interesting quick exercise is to take the category of sets and relations, and work out what its elements – relations of type 1 ⇸ X – are.
James Wood (Jul 10 2020 at 15:50):I've never heard of information and possibility talked about this way for category theory, but maybe it's related to the notion of property, structure, and stuff, which indeed is a useful intuition. You could say that how a property is constructed carries no information – the only information being whether it exists at all. To characterise via universal properties is a way to have structure-like things that are actually properties (because the universal property constrains them to be unique).
Julio Song (Jul 11 2020 at 00:12):Lee Bloomquist said:
But 'all' doesn't work in the fourth sentence. Why not? 'Function' is certainly a noun, at least in mathematics, that is in some sense or other collective by itself. In a function from one set to another, it is the elements that are mapped to elements in another set.
The problem here seems to be that the word "function" is a count noun and singular in form, which leads to a grammatical "disagreement" between it and the quantifier "all" -- agreement in natural language is usually syntactically/morphologically rather than semantically/encyclopedically determined, as far as I know. :-)
Julio Song (Jul 12 2020 at 06:47):Lee Bloomquist said:
Julio Song said:
Lee Bloomquist said:
But 'all' doesn't work in the fourth sentence. Why not? 'Function' is certainly a noun, at least in mathematics, that is in some sense or other collective by itself. In a function from one set to another, it is the elements that are mapped to elements in another set.
The problem here seems to be that the word "function" is a count noun and singular in form, which leads to a grammatical "disagreement" between it and the quantifier "all" -- agreement in natural language is usually syntactically/morphologically rather than semantically/encyclopedically determined, as far as I know. :-)
Thank you for your replay, Julio. Here's my question: Is there some type of functor from set functions to directed information channels, and from sets to situations with possibilities as constituents? Or vice versa? Say, for applications in linguistics or pedagogy ...
https://plato.stanford.edu/archives/sum2019/entries/situations-semantics/
You are welcome, Lee! Unfortunately I'm not familiar with such applications, so there might be some but I'm not aware of them.
Henri Tuhola (Dec 31 2022 at 13:48):I've finally understood Bartosz Milewski's video lectures, except the third series. I wonder where to look next.
John Baez (Dec 31 2022 at 23:31):Go to Mathstodon or Twitter and ask Bartosz.
Henri Tuhola (Jan 01 2023 at 03:13):I'll do that.
Henri Tuhola (Jan 01 2023 at 14:09):He pointed at https://github.com/BartoszMilewski/Publications/blob/master/TheDaoOfFP/DaoFP.pdf