I want to work through category theory for the working mathematician by Mac Lane and the followup book Sheaves in Geometry and Logic by Mac Lane. I have a PhD in pure math but I'm somewhat slow and not the brightest. I was looking for someone who would be interested in working through these books with me or someone who already knows the subjects and would be able to check my answers to the solutions. I intend to try to solve every exercise.About me: I'm very friendly and silly and like anime
>>16904996How do you have PhD in pure math but not be at least of average intelligence
>>16905028I'm relatively smart and I'm a good problem solver, but I have a bad memory and other mathematicians learn like 1000x more theory than I do
What's your field?Sadly cats and types aren't exactly something I'd want to do a reading group on - I was on that trip a decade ago. It's nice, but I'm not completely aligned. That said, even if the board is a bit dead, you can keep a thread going (they usually live now 2+ days without any post).tbd in a sense I think LLMs killed communication also. Barely a need to get info from others now... It's more of a social need of us.
>>16905199>MUH AIkys
>>16904996Type theory or topos theory?I've gone through a decent chunk of MacLane's book on topoi, and have probably seen more or less everything in his category theory book at some point or another, so I'm happy to check your answers to exercises, or discuss any questions. I'll say that you shouldn't probably aim to go through every exercise unless you were trying to get a migraine. At a certain point, it'll get very tedious. Moreover, it is very difficult to learn category unless you have some kind of motivation; it is very dry on its own. That said, one person who manages to make it rather interesting as a standalone subject is Richard Southwell, whose youtube channel is a goldmine.
>>16905031Do you know or apply spaced repetition, active recall reading? I'm also like you (and I'm still working on it) but what I realise is that experts constantly (usually in their head) are always working examples of what they learnt even as their reading it and constantly after, which helps with memory. They also have a clear structure of concepts
>>16905199Measure theory/probability >>16905249Topos theory. Thanks! I'll definitely check out Southwell on yt! Add me on discord: urist1334I'm doing this because I want to go into research in category theory>>16905292I remember stuff temporarily, but I forget the theorems/math I learned a year ago unless I'm super actively using it.
>>16905306I don't have discord.
>>16905398My email is moleculedumal@gmail.com. shoot an email
>>16904996picrel book is outdated garbageJoy of Cats is much better
>>16905891They >>16905249 never emailed me. I'd appreciate anyone else that'd like to learn with me or could go over my proofs.>>16905900:DD
>>16905306>research in category theoryI ask Grok (pls don't hate me) about diving into CT and it's awesome. Things I've struggled with or neglected became clear within a short time. Now I throw CT at complex or mundane things, everything becomes immediately clear and obvious.
>>16905900True, I tried that reading it only to find out metacategory is an outdated term on nLab. I would suggest Category in Context by Emily Riehl instead.
>>16908005>oh no a term is outdated So? >Emily RiehlI always hear that dyke’s name everywhere like she’s the next Grothendieck but to my knowledge she hasn’t contributed a single major result to the field.I can predict the response btw. You’re going to drop some papers she coauthored with some simps who considered it their duty to put her as the coauthor despite her doing 1% of the work at most.I have had the misfortune of working under female professors and they’re the most turbocharged sycophants in existence. Their entire career revolves around finding simps to do their work for them.
Be honest, are you trans?
>>16908197>So?I have only skimmed through the first part of both books, but McLane's writing style and visualization of some concepts are also outdated. Emily's book is just more fleshed out and easier to read imo.>Blah blah blah...Didn't ask buddy.
Well, hi!To be honest, I'm an undergrad student, and not even from pure maths, but from CS. But I really would like to try to participate, even if it could be a little too hard for me to understand some concepts.I like pure maths, because of college I do this just as a hobby. But I never walked into "the ways of algebra", until now I was really attached to classical, constructionist and modal logic.To be straight, if you think my participation could be of some value, mail me (hft2yq8i2@mozmail.com, yes, it's an alias. But you can call me Colin).
>>16904996>Looking for someone to learn category theory and typos theory with>typos theoryLooks to me like you have already mastered typos.
>>16908445This is so obvious, I'm mad I didn't come up with this
>>16904996>someone>working mathematician>workingLearn to nurse.
>>16908388I emailed you but you didn't respond
>>16905891in the future it is advised to use "at" rather than "@" because of bots stealing data. For example: email at gmail.com
>>16911202I sent you a request on discord 1 or 2 days ago
so who are you really trying to fish for here sweetie?
>>16913447Send another one, sorry! I thought it was a solicitor adding me due to the common server so I declined it ;-;
>>16904996Do not waste time reading this book it is super old and outdated.Start with "An Invitation to Applied Category Theory" and go from there.
>>16914391I did resend it now but to me it had appeared as still pending so maybe youd declined someone else? My profile picture is orange.
>>16914575Yes there is free lectures too with the book/notes http://brendanfong.com/7sketches.htmlNot sure if OP cares but there is finally decent material on Dependent Type theory like this draft https://carloangiuli.com/papers/type-theory-book.pdf
>>16914575I might not even recommend it, but outdated seem a bit overboard.There's nothing in there that's superflous now, is there?Maybe it doesn't go with type theory trends etc., but like learning about ring theory, or number theory, a 70's book is fine for algebra.
>>16904996What was the topic for your doctoral dissertation?
>>16916540this
>>16907531>muh aikys>>16905199>ai da foochur>muh aikys>t. arithmetic geometer 140 IQ
>>16918891Unironically, how are you writing papers in arithmetic geometry with such a low IQ?
>>16909466Why?
>>16918891>arithmetic geometerSo you can count integers and draw rectangles?
Why not this bad boy?
>>16922168What do you like about it?
>>16918921hard work
>>16905306>Measure theory/probabilityAre you still doing research in those areas?
>>16928405geometry of random networks is used a lot nowadays
category theory seems kinda daunting. like i don't know when i'd be ready to approach it. i've only studied mathematics for fun and got up to some aspects of manifolds, but categories are so abstract that the notion of learning them for a particular purpose seems out of reach. what do you do with category theory? is it just a more efficient language to encode preexisting ideas?
It has its origins in the goal of finding a way to define "natural transformations".It also "explains" the reason for why many bijections and such exist in the first place.
>>16905900>>16908005>>16914575there's a rev2 that's from 2014. is it really that outdated? seems not that old relative to other references.
>>16928462Try it. Could be good. I skimmed through and it seemed very robust, but more like a technical manual and not very good for beginners.
I'm also really surprised nobody has suggested "The Rising Sea" by Vakil. I really liked this one.https://math.stanford.edu/~vakil/216blog/FOAGjul2724public.pdf
>>16928467fair point, i'll check out every book that was posted here and just jump between them if i encounter any form of impediment. hopefully that will suffice, otherwise i'll probably just jump back down a layer of abstraction to something more approachable.
>>16928430The algebraic topologists really wanted to start it up because set theory was starting to become too limiting, especially with its persistent weaknesses.
>>16928468Probably because the other posters were at least able to read the title of the thread.
>>16928562No, it's probably not because of that. Perhaps you're fucking retarded.
>>16928462Why did they keep the metacategory thing? Category in Context is still better imo and it also reads just like Category for the Working Mathematician.
>>16928746>Perhaps you're fucking retarded.I'm not the one suggesting an AG text in a thread about category theory and topos theory.
>>16928430Categories are not abstract as much as the trannies on this board want you to think. It’s literally just a generalization of functions between sets. You distill it down to algebraic properties of functions themselves and their composition rather than worrying about what an element is or how it’s mapped or whatnot.Most of mathematics doesn’t care about set theoretic elementwise definitions. Do you ever see someone prove that 2+2 =4 by interpreting both sides as sets and comparing their elements?
>>16928818but how do you define a function to have properties without looking how it affects elements? is that the notion of functors where you are showing how a function changes under particular transformations?
I've never seen a field of math that couldn't be mastered by watching a 30 minute YouTube video.
>>16928818This is true. It only gets interesting when not in isolation, but it wasn't meant to exist in isolation in the first place.
>>16928776Yes it is an AG book. However, you're fucking retarded if you don't think it's an excellent intro to category theory within the context of something you might actually use it for. Go fuck yourself you FUCKING NIGGER.HOLY SHIT you're a fucking prick dude. DIE. Fucking DIE.
>>16928776You dumb fucking nignog. What sorts of fields do you think category theory was developed for.
>>16928824You can prove that a (usual set-theoretic) function is injective if and only if it is left-cancellative (only one of the inverse equations holds) for example. You can “pick” an element from a set using maps from a singleton set. You can “count” the number of elements of a set by establishing bijections with other sets. This is the basic observation that lead to category theory. The advantage of this method is that your objects need not have elements and morphisms need not be literal functions. Partially ordered sets are self-contained categories with <= serving as the morphism.One of the corollaries of Yoneda lemma is that you can learn all there is about any given object (set, group, topological group, etc) via maps into it. This is a lot more appealing to abstract algebra for example as you may have learned from isomorphism theorems. It’s much easier to investigate structure of a group by mapping to/from other groups and looking at kernels.
>>16928818>You distill it down to algebraic properties of functions themselves and their composition rather than worrying about what an element is or how it’s mapped or whatnot.You literally just described abstraction.
>>16928860All of mathematics is an abstraction.Sets are abstraction too.>dude it's just random things in a container>*actually just a bunch of curly brackets and commas*There is no such thing as a "set of socks" or a "set of playing cards" in actual rigorous set theory like ZFC. Everything is a set and sets only contain other sets. This doesn't get trannies' clitties melting, but saying "you can do this algebraic operation in any order you like and there's a thing that acts like 1 in multiplication" apparently does.
It's not that there's anything wrong with Set Theory, it's just that the concept of a Set is too basic. We want a "collection of things", but we want it to have a fundamental distinction between the things-in-themselves and the way they "relate" to each other, i.e., what functions actually define the things-in-themselves, because that's how we can actually do analysis with these collections, instead of having to add that on as desiderata later on.Furthermore, the category, because it represents both the objects/sets and their relations using a commutative algebra, is therefore fundamentally better suited for analysis, and for exploring the homology (and cohomology) of a subject/analysis within a topos of other similar subjects and analysis. It is simply so much more powerful than naive Set Theory once you really grasp how it all "works". Especially Galois Theory. Galois connections are one of the most important concepts in modern mathematics one can know about if they do not already know about them.Again, it is not that Set Theory is "wrong". Indeed, most of basic category theory is taught using set theory and its notation. It's just that categories contain more information than sets do, information which makes doing the sorts of math we do these days in fields like algebraic geometry a lot easier compared to the sorts of math that engineers and physicists are doing trying to figure out what happens when black holes collide. Kind of ironic that. QM is kind of more difficult than pure math in a sense because it's bogged down with a lot of what you might call "classical" set theoretical logic that refuses to play nicely with intuitionist approaches (like fuzzy open sets).
>>16928888Personally, I think in most cases both work fine, as its about what you can physically accomplish more than anything. The point is to work with whatever method is the simplest for the particular problem being tackled, and in many cases set theoretic approaches are the simplest, while in other cases its less of a headache to take more category theoretic paths. Set theory is still popular when its the simplest methodology to use for the problem at hand.I work in Topos Theory so I know how fast category theoretic approaches can become moronic in certain situations, and I have seen it abused too much at times.
>>16928894You keep making this argument that Set Theory is better in general though, like, somehow it's simplicity gives it more power. It doesn't. You're wrong, it's precisely the opposite. That simplicity isn't a power, it's a laziness born out of tradition and nothing else, not precision, not insight, not elegance, not anything that mathematics is supposed to be, and is in the hands of brilliant people like Arthur Grothendieck. It is that which holds back many of the physical sciences, such as quantum mechanics, like I mentioned, for the reasons I mentioned.Furthermore, this kind of "get it done" attitude is something worrying to me that I see everywhere in academia these days. It's this "shut up and calculate" vibe going around that discourages anyone from being creative or utilizing their intuition to come up with novel approaches. The whole "occam's razor" argument is, in my opinion, nonsense in this context. Especially, as I said, given the vital importance that every modern mathematician know about Galois Theory at this point. It is, in my opinion, a goddamn travesty that people actually disagree with what I just said. I regard any such person as some kind of broken autistic robot or something that can do math just fine but has no clue what math even is or meant to be used for because they have no intuition at all. Basically, a humanoid biological LLM that can autocomplete really well.Many such cases.
>>16928872you could include atoms desu
>You keep making this argument that Set Theory is better in general though, like, somehow it's simplicity gives it more power.I am an intuitionist who works in Topos Theory for start, and secondly I never said that; I said to use the right tools for the right job.>Furthermore, this kind of "get it done" attitude is something worrying to me that I see everywhere in academia these days. It's this "shut up and calculate" vibe going around that discourages anyone from being creative or utilizing their intuition to come up with novel approaches. Do not put words in my mouth. To use methods not suited to the problem is complexity for the sake of complexity, and you are not making our field look good by implying such things. Many already accuse us of being faggots who engage in fluff-work.>novel approachesA novel approach is the opposite of spamming category theoretic methods on everything, so you are contradicting yourself with this sentiment. Or, like an "broken autistic robot" as you might say.>"occam's razor"Knowing how to solve a problem with an appropriate toolset is not "occam's razor," its called being a mathematician. One can expound upon the lovely aspects of going for purist approaches to solving a problem without neglecting everything else.
>>16928872lowest IQ post I have ever read on this board
If you're thinking of getting into category theory, I warn you that a lot of modern category theory has mainly devolved into stupid nonsense (infinity category/Lurie stuff), rife with meaningless jargon and results whose "truth" rests on unverifiable claims and in-group refereeing. Basically it's a cult and you should stay far away.As far as classical category theory, it all "stupid abstract nonsense" but all very easy. You just have to get good at chasing diagrams, but everything in Mac Lane is "easy" once you stare at it long enough.t. mathematician
>>16928921Topos Theory keeps me sufficiently grounded as to be safe from the black hole of faggotry.
>>16928818It's weird how young people are drawn to category theory and I've seen many students become so-called "infinity students", where they can talk ad nauseum about abstract nonsense and drop many buzzwords, but then struggle with basic calculations and seem completely averse to doing any mathematics involving numbers. Probably because category theory has a lot of inherent "gatekeeping" mechanisms, including its jargon, which makes it attractive to midwits looking to just learn enough in order to gatekeep and flatter themselves as "smart" by being able to rattle off esoteric results and facts. But put them next to anyone with true problem solving ability and they get BTFO.True, meaningful, mathematical research will involve concrete objects: geometry, number theory, analysis. Category theory is the midwit's opium. >>16928923Topos theory is basically just sheaves and points. It is also prone to devolving into bullshit, even people like Scholze are starting to get stuck in the site/topos quagmire of unverifiable cult-mathematics. (See condensed mathematics = metastasized Lurieism.) Deligne wrote a lot about topoi several decades ago, and his papers are always very handwavey, but anything of note that people outside category theory use in modern topos theory basically comes from him and Gabber (Gabber never writes down shit though).
>>16928924I care more about being benefited as a geometer via my proficiency in Topos Theory. Category theory is only useful insofar as it benefits the other fields, which is what it was actually intended for. The only reason why there has been so much culty bullshit is because people forgot this and their research devolved into mysticism (which category theory was bound to do without the tether of being suboordinate to other fields).But yes my field is only held together by extreme care not to fall down the wrong path.
>>16914575Absolutely idiotic advice. Mac Lane is the gold standard. Anything written on classical category theory after 2000 is pozzed, woke bullshit and niggerdry.
>>16928926>Category theory is only useful insofar as it benefits the other fields, which is what it was actually intended for. Bingo, and as a positive side effect it has also become a filter for babbling cultic idiots which allows me to know whom to filter out when attending seminars/conferences/refereeing. >The only reason why there has been so much culty bullshit is because people forgot this and their research devolved into mysticism (which category theory was bound to do without the tether of being suboordinate to other fields).You're actually touching on a key point. Category theory, as I see it, is just a modern evolution of "kabbalism", i.e. mystical, phariseeic judaism, where scholars/rabbis would debate each other and discuss meaningless nonsense ad nauseum (in yeshiva, etc.), in meta-logical and mystical ways, where the truth wasn't so important as much as mentally stroking each other off and developing "in-groups" and "out-groups" and most importantly a "sanhedrin" of "experts" who had the power to decide what was true or not (independent of objective truth). Which, who woulda guessed it, is exactly what is happening to many areas of modern mathematics. And guess which tribe people promoting this kind of mathematics belong to.Geometry is safe, you can be sure that anything you actually prove in that area will be a meaningful contribution to mathematics. Unfortunately, more algebraic areas (like arithmetic geometry) are becoming infected with post-truth cultic mathematics. The more concrete the area, the more resistant to bullshit. Keep doing actual mathematics and never get intimidated by the infinity cult of idiocy.
>>16928930The funny thing is that all the mysticism doesn't even have anything inherently wrong with it; there are just so many issues as a consequence of frequent failures in demarcation of these mental excursions and grounded mathematics. If they kept to their corner and didn't try to stick their fingers up everyone's asses then there wouldn't have been an issue.
>>16928934There's nothing wrong with it as long as they stay in their Yeshiva and stick to jacking each other off. But as soon as they try to spread their way of "thinking" and corrupt true mathematics with the intention of reducing it to meaningless babble, so that they can control, gatekeep, and then gain the power to decide and decree what is "true" or not, then it's a big problem and should be fought against with every fiber of one's being. Obviously you and I see that this is not what's happening and instead their cancer is spreading (hopefully not so much in your area as in mine). They are absolute scoundrels and should be called out publicly for their bullshit whenever possible, which they of course know which is why they hide in their own conferences and referee each other's papers. It is a big problem. Basically the same thing already happened to physics (at the initiative of the same tribe of phariseeic jews) and now they are trying to "physicify" all of pure mathematics as well. It is the jewish dream to become "God" and wield the power to decide what "truth" is.
ITT: people who have never written a paper or even spoken seriously with a mathematician complain about the state of math research.
>>16928943Speak for yourself.
>>16928939I really do love category theory. I hope it doesn't fall beyond the reach of hope like physics has.
>dude papers lmaoa wild jeet appears itt
>>16928834based 400iq frogposter
>>16928888based cat anon and checked quads of truthdo not reply to these LARPing pseudsI gotchu hommie>lol trust me bro I work in topos theory>I'm a REAL mathematician bro>not like those tranny cat theory faggots that cant even into numbers>buncha jewish mystics I tells ya>*proceeds to degenerate into complete psychosis*>dude bro category theory is basically kabbalahlol lmaoyou do not work in topos theoryyou are a larping pseud shitposting on 4chan looking for attention and validationyou have high functioning autism and/or some weird neurotic inferiority complexsimple asyou probably do not even have a degreeor a single real human friend for that matter/thread
>>16929439nigger nobody cares about your coping olympics. >>>/lgbt/
>>16929441seethe elsewhere bitch>>>/trash/
>>16929444keep coping land roping lmao>>>/lgbt/
>>16929447whoops>land*andSorry I wanted to correct my grammar before you got confused and roped the land instead of yourself.
