A Sensible Introduction to Category Theory

Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the basics of category theory, I attempt to remove it.
27 Unhelpful Facts About Category Theory: • 27 Unhelpful Facts Abo...
MetaMaths on category theory: • Intuitive Introduction...
My dissertation on the equivalence between the category of monoidal categories and the category of representable multicategories: drive.google.com/file/d/1hAkV...
FURTHER READING
Basic Category Theory (Tom Leinster): arxiv.org/pdf/1612.09375.pdf
Categories for the Working Mathematician (Saunders Mac Lane): www.mtm.ufsc.br/~ebatista/2016...
Category Theory for Computing Science (Michael Barr and Charles Wells): www.math.mcgill.ca/triples/Ba...
Category Theory for the Sciences (David Spivak): math.mit.edu/~dspivak/CT4S.pdf
Bartosz Milewski on category theory: • Category Theory 1.1: M...
Emily Riehl on category theory: • What is Category Theor...
MUSIC
Meditation Aquatic
369 (Epidemic Sound)
Nights Full of Overthinking
Lionel Quick (Epidemic Sound)
Oregano
Vendla (Epidemic Sound)
Wash
Timothy Infinite (Epidemic Sound)
Wind
Osoku (Epidemic Sound)

[@Mutual_Information]

My buddy has a PhD in pure mathematics and says.. "Studying category theory is like eating your vegetables." Not sure what that means but it has never left my mind.

[@hasko_not_the_pirate]

It means you’ll live healthy

[@fyggy5480]

probably "you'll hate it but it's good for you"

[@lorefox201]

you don't want to but it's good for you

[@brendawilliams8062]

Noether and algebraic suggestions would be my guess.

[@seneca983]

I guess studying group theory is like eating someone else's vegetables then.

[@tentativegazer]

Sure I'll watch a half-hour video on category theory, a thing about which I know nothing about.

[@dashockpixle4140]

Yeah you got me there

[@inakibolivar664]

Well, it’s an introduction

[@gwh0]

I took a graduate course in algebraic topology in 1980 from which I learned nothing except to avoid category theory like the plague.

[@mastershooter64]

yes...that's usually why people watch these kinds of videos...to yk learn about things that they know nothing about...

[@inakibolivar664]

@@mastershooter64 yeah the most ironic part of the comment is that the video is an introduction, if you don’t know nothing about a topic you watch an introduction, it’s just logical

[@galgrunfeld9954]

For anyone curious, the motivation behind doing category theory is so you don't have to do any actual math.

[@pwmiles56]

Sounds like you got it.

[@sharif1306]

care to elaborate?

[@danielyuan9862]

@@sharif1306 category theory isn't actual math

[@sharif1306]

@@danielyuan9862 what is it then? Alchemy?

[@404errorpagenotfound.6]

Sounds like my kind of maths.

[@kikivoorburg]

Category theory is awesome, it feels like the most 'human' thing ever: "Yeah so we have this thing called 'maths' which we use to simplify the world and make connections between different phenomena by putting everything in neat little boxes. ...so then we started wondering whether or not we could make a system of bigger boxes to help sort our existing boxes and find connections, which we were able to do. ...and then when we tried to sort the bigger boxes we realised they already explained themselves!" "Wow! So you can sort anything then?" "Nope, there are still things we can think of that don't fit into the system." "..."

[@U20E0]

Then let’s create more systems that together encompass everything and can be sorted into boxes

[@Soken50]

Even the best library has an "other" section :)

[@U20E0]

@@Soken50 unrelated, but it sort-of interesting how “Even the worst…has…” and “Even the best…has…” essentially mean the exact same thing

[@xXJ4FARGAMERXx]

@@U20E0 Imma need a minute to process this Even the best phone has a limited battery = all phones have limited battery Even the worst phone has a limited battery = all phones have limited battery Even the best phone has a limited battery = Even the worst phone has a limited battery phone = A a limited battery = x Even the best A has x = Even the worst A has x = All A has x

[@ammyvl1]

@@U20E0 no. "even the worst" is used for positive qualities, whereas "even the best" is used for negative qualities.

[@timseguine2]

The problem I have always had with category theory is the existence of a forgetful functor from the category of versions of myself who understand things about category theory to the category of versions of me who don't

[@leptogenesis3558]

This is okay, because there is a free functor Learn from that category of versions of yourself that do not understand category theory to the category of versions of you that understand category theory, making the diagram commute.

[@jsmdnq]

@@leptogenesis3558 It's not because my adjoint is alway co-operating on my co-self to forget my freeness.

[@timseguine2]

@@jsmdnq That's the one thing I never seem to forget about category theory, and I regularly make internal jokes to myself about categorical dual notions of every day objects and activities. Jokes I can share with precisely nobody because almost nobody would understand them, and most of the people who would understand them would not find them funny.

[@jsmdnq]

@@timseguine2 It's that comonad inside you that is conflicting because you haven't dualized the universal limit.

[@petevenuti7355]

Doesn't that all make you isomorphic between you and yourself?

[@silentobserver3433]

This is actually the most understandable introduction to category theory I've ever seen. True, I'm somewhat familiar to it already, but explanations like that somehow make much more intuitive sense than "monad is a monoid in the category of endofunctors" stuff. Good job!

[@imacds]

Everybody asks "What is a monad?". Nobody asks "How is a monad?". :

[@marcus3d]

Well, it's understandable if you already have a grasp on it, but otherwise it's not.

[@mine_churros]

@@marcus3d I only had classes of linear algebra and it's still somewhat understandable

[@garnet1918]

I'm glad someone else hates that description of monads with a burning passion.

[@notsojharedtroll23]

@@imacds LMAOOOO

[@JakubWaniek]

Imagine having the balls to share your dissertation publicly before knowing that it's right... big respect

[@donaastor]

i dont think the respected property here is his strong self-confidence, but rather his true mathematical curiosity - that he wants to know the amswer no matter what. (i don't know which is the one you referred to in your comment)

[@zwishking6032]

the reality of mathematics is that most papers are read by virtually no one. if someone finds an error in your work, its usually a net positive. firstly, the work usually can mostly be salvaged, next, it means someone actually cared about your work which is always nice.

[@mathboy8188]

Also, a math PhD dissertation ain't like handing in a homework assignment. You can be sure he's poured over every detail of it - many times. We're all human, so mistakes are always possible, but by the time you're sophisticated enough to pass your quals and take all those grad courses, your confidence and capability in checking your own work will be enormous, and justified.

[@JakubWaniek]

@@mathboy8188 A PhD dissertation, sure. But this is a master's dissertation. In fact, I'm just putting the final touches on my own :)

[@gornser]

That's what you do when you write a dissertation. You put it out to be judged.

[@VincentKun]

A teacher in my class of Type Theory in computer science degree, while explaining Category Theory he was doing a lot of little parenthesis about things that seemd unrelated and he said: "What i'm explaining now is like we're going into a journey, and i can't resist to stop and enjoy the landscape and describing how beautiful it is to you."

[@kasugaryuichi9767]

This is why platforms like KZbin are meant to exist. Love your work

[@dr.c2195]

They are meant to exist to serve Google with our information. But his work is great, because video uploaders like him are what draws people to Google's information collection platform.

[@michaelvaller]

After watching just a third of the video I must say I indeed learned a bit, and I am studying category theory for almost a year

[@MetaMaths]

Thanks for the notice ! I now feel obliged to continue my CT series …

[@MetaMaths]

@@PefectPiePlace2 Yes, but I named my channel before knowing this proof assistant. And once I knew, I added an extra "s"

[@pamdemonia]

Yes please!

[@oliviaaaaaah1002]

Congratulations on finishing your master's!

[@jacobneu-videos]

What an excellent introduction to the subject! I wouldn't have believed that in 26 minutes someone could explain (in terms a beginner could perhaps understand) all the way to equivalences & natural transforms, but you sure did. Very jealous of your production values too. Cheers mate, bravo. Your comments about adjunctions & Yoneda have tempted me to want to try and intuitively motivate them. So here's another explanation to add to the pile of unhelpful, vague descriptions you've clearly encountered: - Adjunctions are the category theorist's version of minimization/maximization problems. Of course, the adjunction between free & forgetful functors is the paradigm example: the functor F:Set - > Mon solves a minimization problem -- it gives the least solution, subject to a particular constraint. The "least" bit here is the universal mapping property of the free monoid, which is the definition of adjunction applied to this example: F(X) doesn't have any more structure than it has to. The "constraint" we have to "solve" is that, for a given set X, we want X to embed into U(F(X)). So F(X) is the least monoid whose underlying set contains a copy of X. On the other hand, right adjoints solve maximization problems. Consider how the category of groupoids is a coreflective subcategory of Cat, that is, that the inclusion functor Grpd -> Cat has a right adjoint. This functor takes a category C to its "core groupoid" core(C), which consists of all the objects of C but only isomorphisms. This solves a maximization problem: core(C) is the largest groupoid which is a subcategory of C. Other category-theoretic concepts which also perform a minimization/maximization function (e.g. equalizers give the largest subobject equalizing two morphisms) are all instances of adjunctions. - I like to think of the Yoneda Lemma as a category-theoretic version of Euler's Identity (that e^it = cos t + i sin t). Doing algebraic manipulations with cosines directly sucks because cos is not an algebraic object. Hence all the annoying "trig identities" that precalculus students have to memorize and later forget. This is a serious issue in fields like electrical engineering, where they have to do stuff like cos(2t) and stuff all the time (e.g. doubling the frequency of a wave function). So what do they do? Use complex numbers! Instead of working with cos(t), they'll work with e^it, which is super nicely behaved algebraically (because exponentiation _is_ algebraic). Then, at the end of the day, they take the real part of the answer, and it was like they were working with the cosines all along. A pretty neat trick -- indeed it's (basically) the reason complex numbers were invented in the first place -- and it all works because of Euler's identity. The Yoneda Lemma plays the same role in category theory: often a category you're working with will be kinda crappy (e.g. not having (co)limits). But the category Prsh(C) of presheaves on C is a very nice category indeed: it has all limits, colimits, exponential object, subobject classifier, etc. The Yoneda embedding allows you to embed C into Prsh(C) in a nice way, and do all your algebra in Prsh(C). If the solution you get (e.g. of taking a limit) is "real" (representable), then a standard Yoneda Lemma argument says that the solution (e.g. limit) existed in C all along. So Yoneda basically allows you to have imaginary (co)limits/exponentials/whatever, which might turn out to be real all along. Very nifty, and very useful. Hope that helps, and hope you make more videos about category theory!

[@kasugaryuichi9767]

Thanks

[@freddyfozzyfilms2688]

my hero

[@Pietro-qz5tm]

What a wonderful insight. I'm a math graduate student tackling with an exam on category theory right now, I'll try to keep in mind those points of view studying the subject. Thanks!

[@travisnell6849]

Commenting before viewing, but hopefully the Yoneda explanation finally works for me. Yet to find it explained to my learning style, and I legit have a Phd in math.

[@bb010g]

This is a very friendly introduction to the Yoneda lemma! Thanks. The adjoint introduction reminded me of what I've read on nLab, and also works well.

[@Sebastian-xb5hj]

My favourite description of equivalence of categories comes from Awodey who, in his book on the subject, says: 'One can think of equivalence of categories as “isomorphism up to isomorphism”.'

[@chanlaoshi8634]

And isomorphic means equal up to an isomorphism :)

[@nullmeasure6155]

I love that book! really helped me grasp the subject as well as I do model theory finally

[@cherryblossom000]

Thank you, I finally understand the difference between isomorphism and equivalence. I’m a hobbyist Haskell programmer so the nitty gritty details of category theory aren’t usually required, so all I really knew was (endo)functors, natural transformations, and handwaving of isomorphisms as ‘these types can be converted into each other without loss of information’. Still trying to wrap my head around as adjunctions and representable functors though.

[@iteo7349]

Imo, you can't really grasp them without knowing some very important examples of them. If you learn moduli spaces, you will 200% understand representable functors. Adjoints are a bit less clear to me too -- sometimes my mental picture is "something with the flavour of free generation" (e.g. forgetful functor GROUP->SET and "take the freely generated group" SET-> GROUP functor are adjoints), sometimes it's something a bit less abstract which unfortunately idk how to get across (the main example for me is pushforward and pullback of sheaves, but I assume that's too specific).

[@kaidenschmidt157]

This was the most fun I’ve experienced from a KZbin video in a while! You’ve done something casual and very fun, and I’m looking forward to looking through the links you’ve included to learn more

[@Zhantyzgz]

I'm making a presentation at uni next week about this exact topic, so this is incredibly good timing I've wanted a gentle undergrad introduction to category theory for years now, so it's pretty funny that I find it now that I've taken matters into my own hands and started learning it formally lol

[@superstarben37]

This is a fantastic video, wow! Props to you dude, took a subject that is generally INCREDIBLY boring and unintuitive, and turned it into an engaging introduction where I finally felt I could follow along. Clearly a lot of work went into this, I'm looking forward to seeing what else you come up with in the future!

[@AutumnWynds]

Your video comes at the perfect time. I haven't been reading category theory per se, but instead a translation of Gödel's work on undecidable propositions. Even though this is unrelated, I came out from watching this with a greater sense of understanding. Thanks!

[@carvas18]

adjunctions are actually quite simple, if you have two functors L : C D : R then you essentially have a representation of C in D (via L) and of D in C (via R) the adjunction then just tells you that for every c in C and d in D you have a natural isomorphism D(Lc, d) ~ C(c, Rd) which essentially means that whatever relations you find between objects in the image of L and the category D at large are "mirrored" as relations between objects in the category C at large and those in the image of R. The quintessential example is the Tensor - Hom adjunction in abelian groups where it tells you that (a b, c) ~ (a, Hom(b, c)) which essentially tells you that providing a bilinear map from the product abelian group (a X b) to c is the same as for each a providing a linear map from b to c which is not that suprising after all that once you "fix" one of the variables in a bilinear map you get a linear map on the second variable.

[@augustinecelmina6765]

"cathegory theory is not a religion" *dramatic music* "HERE WE BEGIN TO GLIMPSE THE TRUE POWER OF CATHEGORY THEORY"

[@RazgrizDuTTA]

This is gold! Thanks! I am doing a PhD in engineering simulation and I am trying to formalize the wizardry I do to my sets of eigenvectors before building my reduced order models. I am not from a pure math background so videos like this one are invaluable to kickstart my understanding of these math branches :)

[@schow176]

I had just finished my a levels a few weeks ago and will be studying maths as my major. This is my first time learning something about category theory. The video is wonderful and quite comprehensible to me. Thank you for such a wonderful video and I hope I’ll be able to explore more through this channel👍

[@oliverfalco7060]

That was some pretty cool dense mind blowing shit I feel I've barely understood but still I've enjoyed. It's always nice to have new videos from you Oliver

[@dunemeister5718]

Don’t know if you’ll see this, but I actually go to Bath university as well, and after watching your 27 facts video, asked one of my lecturers about category theory as I knew that he studied it at PhD. That lecturer happened to be the very Thomas Cottrell that supervised you! Funny how the world works

[@dcterr1]

Wow, great video! I've heard some very difficult math lectures on category theory, so I never really understood it very well. But this video is excellent because you give several useful concrete examples, which greatly helped me to understand what's going on.

[@pamdemonia]

Congrats on your thesis being submitted and a very clear video on category theory to boot!

[@maximchurilov1889]

Thank you so much for this! Hope to see more of such content on this channel

[@dakotaschuck]

Hey, statistician here who's been really loving learning about CT, and your funny video was actually quite helpful. Thanks for this one which is perhaps more helpful... perhaps. Thanks for the delightful and informative content ✨

[@krumpy8259]

I can‘t resist to ask for more. Very instructive video. I think I can say I came a little bit closer to the spirit of category theory. I watched a lot about this topic and seeing a lot of explanations. The main issue for me regarding CT is that it is almost always everywhere explained the same way regardless of the known problems that one faces with it. But you go in the right direction.

[@b1odegradable559]

I studied pure maths and we only quite briefly touched this topic as preparation of module and ring theory. I am still amazed how you can use the general definitions and see them play out in specific ones. Very interesting introduction, thanks a lot for the effort. :)

[@Anthsytar]

Physicist with a little passion for abstract math here. Well done! I was very confused about why I would be seeing them when learning about differentiable manifolds when I thought Categories would just be ultra-abstract things for algebraists. Subscribed!

[@darkarchon2841]

I gotta say, I learned more from this video about category theory than from my (failed) course on category theory in uni. Though I was a lot less diligent studen then than I am now. Great video, thanks for doing it!

[@exsurgemechprints2671]

this video gave me a bird's eye view of my lifetime math adventure. From 1st grade maths until I graduated college and any other math I encountered. Different math rules, new kinds of numbers but all feel somewhat the same. That sameness was explained by this video.

[@jotaro6390]

I have always been thinking that category is similar to groups but have different rules. But in the video said the most epic thing that I've ever heard in math: "let's take a special case of category - sets"

[@vdo7571]

Probably the best introduction video to category theory out there. Please elaborate on the 'downsides' of category theory that you alluded to in part 2 video

[@lachlanperrier2851]

Absolutely sick video. Please make more!

[@therealjordiano]

thanks so much for the clear explanation dude, very nicely put together

[@Dr.Cassio_Esteves]

This is by far the best introduction to category theory I have seen (and I have seen quite a bit of those). Brilliant! Sorry any mistakes, my english is a work in progress.

[@gornser]

This was a nice refresher on things I studied 20 years ago

[@Axman6]

Best intro ever, thank you so much for the karma I got on the Haskell subreddit for your last video. I hope you post this one there before I do.

[@pmmeurcatpics]

The moment where you introduced the category of categories (21:00) was such a sublime experience, it might well be the happiest math-related moment in my life so far. Thank you for this wonderful video:)

[@bobtheblob728]

I love the way you think about this!! So many interesting analogies. I've taken classes on category theory, read countless Wikipedia articles, and lots of KZbin, and this video helped me get the point the best way I've ever seen

[@Lumeone]

The best! Getting to the essence quickly is the sign of mastery. Thank you.

[@Encysted]

Hey, glad you got to do this!

[@Vegedow]

Thankyou thankyou thankyou! Thanks to the coconut video I became able to understand the maths in an advanced physics video course, it's really nice to know more details of it.

[@EnordAreven]

3/10 didn't do a Morbius meme during the morphism section.

[@maximofernandez196]

Man, I'm at the first year of my career, and you made me think in everything in such an abstract level. I fucking love it

[@NoNTr1v1aL]

Finally, the long-awaited sequel is here!

[@kruksog]

I subbed to you on that fateful day. Glad you made this follow up.

[@lennyuniverse]

As a math major and enthusiast, greatly enjoyed the approach here

[@michaelklaczynski3650]

This is very helpful! I'm reading some abstract AI papers right now, and just knowing what all the symbols mean makes it much more comprehensible!

[@FeliciaSopokovitch]

Thanks a lot! I was wondering how much priority it should take in my self-study of maths, and understanding that categories are a certain simple mathematical structure that happens to be useful to encode some aspects of maths, but doesn't easily deal with *everything*, is super helpful.

[@treetheoak8313]

Holy crap you actually did it you madlad! Also congrats on your channel starting to pick up steam!

[@alanv7251]

congrats on finishing the dissertation!

[@mozarteanchaos]

i absorbed very little of this but i think if i was more awake or had a better grasp on maths this would've made a lot of sense. maybe. i think your video is good

[@RaphaelAndrieux]

Thank you for the great content !

[@nerdsgalore5223]

I've been trying to get into more abstract mathematics like group theory, and this is a great and intuitive explanation of category theory!

[@irrelevant_noob]

You still should start with group theory tho... That's like, maybe not kindergarten stuff, but high-school-ish. Can be understood even with relatively little prowess in the abstract. Category on the other hand, is post-grad stuff, you better be prepared for a war of attrition with it. ^^

[@utof]

YES!!!! IVE been waiting for this since that joke video! Thank you!!

[@IustinThe_Human]

i like that you chose to reprezent composition of 2 morphism as fade animation to the new object instead of drawing another line

[@geradoko]

Thank you very much for that excellent explanation! Several times I had to stop or repeat parts of the video to read thoroughly the symbols or to think over what I had heard. But just this is the advantage of videos compared to lessons. I studied math over forty years ago (in Germany) but we never had this topic. Yet I know isomorphisms from group theory ... It's easier to understand category theory if you are familiar with some examples.

[@hugospinat451]

I watched the other video like 3 days ago so happy to get this one now x)

[@ArvedRockt]

Awesome video! First time that I kinda got what Category Theory is actually about.

[@trannusaran6164]

Sick, that was a pretty clear explanation! Thx :3

[@officebatman9411]

This is such a good video! Can someone give an example of difference between equivalence and isomorphism?

[@4doorsmoorhoors542]

A coffee cup and a donut are equivalent in shape- (they both have a hole). We would have to define the functions of the two objects and define more categories before we can be considered if they are isomorphic. This video and 9th grade algebra class are all I am going off of to answer this question. Hope this helps! 😂

[@tantzer6113]

Congratulations on the MS thesis. No comment was on the math, but the writing seems clear, simple, and engaging.

[@brennanlawson6108]

Categories remind me of classes in programming. Good video!

[@fakechuck7659]

The Terence Howard joke took it over the top. Well done.

[@unsightedmath7040]

Thankyou so much for this video.

[@bigman3274]

i've only ever taken a discrete math course and this seems pretty dope

[@josemanuelramirezgomez6206]

This is a wonderfull piece of math information. Thanks!!!

[@kanewilliams1653]

Part two please! I am waiting!!!!

[@farty555]

Ohhh that's why coffee and donuts go so well together! 😮

[@johnrickert5572]

At 19:58, I certainly understand why you omit going from Sets to free groups, but for the interested viewer, it's really quite simple, beautiful, and for me at least, fascinating. As always, thanks for your consideration.

[@josvanderspek1403]

Great indtroduction! Next video: natural transformations, limits and colimits, and the Yoneda lemma? Perhaps Adjuncionts? (Which would probably would require a video of its own?)\ Great job, looking forward for more!

[@jaraddemarco7664]

Since I started school for electrical engineering, I felt like the math classes were leaving something out. There’s something I’m looking for, like some kind of explanation of math that my teachers haven’t given me. In my mind I could barely explain to myself what I was looking for. I knew it had to do with boxes of logic. If you start with AB CD for example, there would be two different boxes of explaining this backwards CD AB and DC BA. See how the two different things can be seen as two different ideas based on how big the chunks are? Those are like boxes to me. I want all of math organized in boxes. Keep in mind I’m not even sure if I’m expressing the real thing on my mind. But I’m looking. I didn’t completely understand the video, but I’m drawn to it, and every once and a while in the video my curiosity was satisfied. If you could make a video, in MUCH MUCH more detail, I’d appreciate it. It seems like every five seconds of this video could be expanded into another 2 hours. Thanks for the video, as is though.

[@georgekomarov4140]

I'm impressed with you having enough balls to stop yourself from talking about elements of object without authomatically saying something like "given that \mathcal{C} is concrete", or talking about "category of categories" without mentioning "small"

[@alexanderwu]

How do you make a video about such a dense and obscure topic so entertaining to watch? I envy your supernatural powers

[@oportbis]

I'm doing a masters degree in september in Algebra, geometry and number thoery and I have to know the basics of category theory and your video helped me know that I guessed right the fact that you could have a "category" category (not in those terms but the idea is equivalent). That makes me feel like I got what it takes to success in getting my PhD

[@oportbis]

BTW the subscribtion was fast. Damn I love the way you express yourself

[@carly09et]

Very nice ... it explains some errors in my learning. This helps with the problem of equivalence.

[@harrisonbennett7122]

Awesome Video dude!

[@leg10n68]

Would've been cool if this was in SoME2

[@OliverLugg]

I considered it, but I do have another idea or two I could try for that. Won't make any promises though.

[@michalhoransky1214]

​@@OliverLugg SoME 4 coming up?

[@JoeShmowYo]

wow that rock paper scissors example really clicked for me. now i get how important associativity is for keeping structure simple and consistent. if i invented rock paper scissors i’d be pretty upset about it not generalizing to larger groups of people

[@Zxv975]

I'm struggling to understand it. Could you explain it?

[@wehpudicabok6598]

A+ use of the "they're the same picture" meme there

[@draziraphale]

A very good intro! Thanks

[@davidpatawaran925]

Thank God for this man’s mind

[@NitrogenDev]

This has been a nice follow-up to the notorious nut video, haha. If categories alone are still somewhat understandable, categories of categories completely stump my intuition.

[@xrhsthsuserxrhsths]

Adjoint functors make sense if you think of them this way: The category C (dom of left adjoint and codomain of right adjoint) is the category of interest The category D is a category where you can manipulate stuff and use it to "predict outcomes" The left adjoint is like encoding a vague object (or morphism due to functoriality) of C The right adjoint is like decoding a well understood object (or morphism due to functoriality) of D The unit shows where you land if you immediately decode something after encoding it The co-unit shows where encoding something you have decoded lands after immediately encoding it again The triangle laws show how functoriality and naturality work in this framework and the homset isomorphism shows that there is a bijective correspondence between manipulations of encoded objects to yield certain outcomes/predictions and processes from "unencoded" objects to decoded ones. So in a sense, adjuntions are like a version of substituting morphisms of C with an easily encodable domain, with processes of D with an easily decodable co-domain. It is somehow like predicting the weather using theories instead of waiting for the phenomena to happen.

[@popularmisconception1]

I was thinking exactly the same thing... in totally different words

[@xrhsthsuserxrhsths]

@@popularmisconception1 So there is intuition behind adjunctions after all!!

[@jjaggers7918]

Please do more videos, Id love to learn more.

[@dcterr1]

I understand that Eugenia Cheng is an expert in category theory and has managed to apply it to many useful aspects of society, which has recently sparked a great deal of interest for me in the subject.

[@thmstbst]

This is the first video I've watched of yours, and I'm already in tears laughing :P

[@benzflynn]

Nicely paced intro to categories. As usual the term has a different meaning to its everyday sense, i.e. sets whose elements have a shared characteristic. Yet the divergence in meaning isn't as bad as say synchronous & asynchronous operations in software. Kinda weird how mathematicians are always trying to generalize on existing concepts in order to determine allowances and limits for all concepts. I'm not surprised that this "new" branch of mathematics has already found wide application as covers the process of shifting perspective on a math construct, something that we often do to make its analysis easier.

[@ivanjermakov]

As an amateur Haskell developer, this introduction makes a lot of sense

[@jeffreygreen7860]

I always enjoyed (the out of print book) Arrow, Structures, & Functors by Arbib and Manes.

[@EngMorvan]

What an interesting subject! Now I love Cat 😻 theory.

[@MarcDonis]

I remember when the internet was all about stupid cat videos (and pr0n). This gory cat video isn't what I signed up for! My head hurts. Serious now, that was heavy stuff. You may think this is trivial, Oliver, since this your thesis is faaaaaaaaaar more obscure, but to those of us casually interested in math, it was really hard. Anyway, you have a great voice. Keep doing what you do, please!

[@MsCane666]

a categorad is a categoroid in the category of endocategors

[@dessertion]

The only case where adjunction really obviously makes sense to me is the fact that currying/uncurrying form an adjunction (since (X * Y) -> Z and X -> (Y -> Z) are basically the same thing). In all other cases my eyes start glazing over as the nLab page functs me into the (Category of Pain)^op and my brain becomes naturally isomorphic to strawberry icecream mmmmmmmmmmmmmmmmmmmmm

[@anonymoose3423]

Take a look at forgetful-free adjunctions. Constructions like free groups, free monoids, and free modules all arise from such adjunctions.

[@MatthijsvanDuin]

nLab is a dangerous place... you can go there to look up a concept you know and understand, and after reading their exposition of it realise you understand it no longer.

[@jsmdnq]

Category theory is a unification of all of mathematics because it abstracts the structure of mathematics in to a common language. It is like a universal language. It would be analogous to spoken languages. If you learned a "master language" it would allow you to speak and understand all other spoken languages. It's not actually this because no such master language could exist as it would essentially have to be all languages and category theory doesn't actually let you understand other mathematics. What it does though is provide a common abstract interface so all those different mathematical areas can be translated in to a common language and then one can see how the various different areas relate to each other and see the deeper structural aspects. It turns out that almost all mathematics(maybe all) uses basic common ideas and ways of thinking which one wouldn't know otherwise. Category theory is extremely powerful because it lets one see the underlying machinery. It, say, is analogous to how understanding that all matter is made of atoms. From the atomic perspective it unifies all things and things that might have seemed different is just one variation on the atomic theme. Category theory is difficult at first because one doesn't realize the concepts it presents are "universal"(unless they have already learned a large amount of mathematics). Like anything one just has to learn it to see it's use.

[@TGMResearch]

The Functor sounds a bit like an invariance in physics. In order to ascertain an invariance in or among mathematical structures, you first need a way to describe them on a meta level so to speak, and that is a Category. Don't know how I got here, all I knew about Category Theory was that it exists. Thanks Oliver!