Kan Academy: Introduction to Limits

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  • Опубликовано: 24 янв 2025

Комментарии • 119

  • @SheafificationOfG
    @SheafificationOfG  5 месяцев назад +14

    Thank you Brilliant for sponsoring!
    To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/GSheaf/ . You’ll also get 20% off an annual premium subscription!

    • @aleksanderadamczyk4191
      @aleksanderadamczyk4191 4 месяца назад

      Hi! The 30 day trial period doesn't seem to work. The discount is applied as expected though

    • @shdvvch1ld
      @shdvvch1ld 11 дней назад

      @@aleksanderadamczyk4191 wdym? you didn't get contents? or system just skips free 30 days?

  • @susanafaciolince7755
    @susanafaciolince7755 5 месяцев назад +391

    this is the first time i see a math video and think “this title is actually morally bad”. and i love how diabolical this is.

  • @danielprovder
    @danielprovder 5 месяцев назад +221

    If you play the video backwards the title changes from ’introduction to limits’ to ‘introduction to colimits’ & the head of every arrow becomes a tail

  • @tildejustin
    @tildejustin 5 месяцев назад +147

    going into calc this year, this video has been a great help

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +96

      With limits done, you're well on your way to learn how to derive funct....ors!

    • @dogedev1337
      @dogedev1337 5 месяцев назад +5

      @@SheafificationOfG pls make a video about deriving functors

    • @francescominnocci
      @francescominnocci 5 месяцев назад +6

      you should *totally* make a video about deriving functors, it would be a wonderful (kan) extension to this one

    • @pyropulseIXXI
      @pyropulseIXXI 3 месяца назад +3

      I seriously doubt that, lmao

    • @char8169
      @char8169 Месяц назад

      hi

  • @rtg_onefourtwoeightfiveseven
    @rtg_onefourtwoeightfiveseven 5 месяцев назад +96

    0:27 "Introduction to That Thing, but only for people who already know it"
    As someone who tried (key word: tried) to read Mac Lane's text on category theory, that hits close to home.

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +44

      Mac Lane's is more a reference than learning material. I've heard much better things about Riehl's Categories in Context if you're still trying to learn it.

    • @d.l.7416
      @d.l.7416 5 месяцев назад +6

      I had the same experience with Mac Lane's book lol. I'd also recommend Category Theory in Context, it's really good.

    • @cybergoth2002
      @cybergoth2002 3 месяца назад

      idk, I tried to read Category theory in context just before my first year of graduate school with absolutely no prior experience, and I thought it was too hard. It was only once I had gotten into differential geometry and started studying homological algebra that any of it started to make sense. I can't imagine the material in that book (especially the first few chapters) can be very enlightening if the reader hasn't already seen any interesting examples where they can apply what they're learning

    • @rtg_onefourtwoeightfiveseven
      @rtg_onefourtwoeightfiveseven 3 месяца назад

      @@cybergoth2002 "The Joy of Abstraction" by Eugenia Cheng is a pretty decent category theory intro for those with less mathematical background, I find. It's not that lightweight - it goes up to the Yoneda lemma with proofs along the way - but it is designed for a less-technical audience.

    • @funktorial
      @funktorial 2 месяца назад +1

      While Mac Lane can be a primary text, it can be more than little difficult (and occasionally outdated) so I don't really recommend it unless you've had a lot of other math prior to it. Riehl's Category Theory in Context is great. I also recommend Awodey's Category Theory, which is basically a "rewrite" of Mac Lane's in more elementary & modern terms. I've heard Leinster's Basic Category Theory is pretty nice too. Finally, if you're interested in the more applied side of things, Fong & Spivak's Seven Sketches in Compositionality is rather different in flavor and totally accessible to beginners.

  • @celivalg
    @celivalg 5 месяцев назад +33

    What I like about your videos is you always start from somewhere simple where I can definitely understand, and you gradually ramp up without loosing me, and by the end you are summoning demons and somehow I understand it.

  • @alexsere3061
    @alexsere3061 5 месяцев назад +21

    As someone with a bachelor in math, I understood about 1/3 of this, but your charm made it very enjoyable

  • @pandavroomvroom
    @pandavroomvroom 3 месяца назад +25

    bro embodies chaotic good

  • @zyansheep
    @zyansheep 5 месяцев назад +24

    I think this is now officially my favorite channel :D

  • @michaelearnest1983
    @michaelearnest1983 5 месяцев назад +8

    This is the best explanation of limits I have ever seen, thank you for making this. You motivated all of the abstract nonsense so well that it doesn't feel like nonsense anymore! The bit at the end about how limits are just sequences of things with conditions was also really illuminating.

  • @CognitiveOffense
    @CognitiveOffense 5 месяцев назад +53

    I'm absolutely delighted to say that I got many of the jokes. A few months ago I wouldn't have been able to tell where the math stopped and the abstract nonsense started.
    Good stuff. Thanks for sharing.

    • @schweinmachtbree1013
      @schweinmachtbree1013 5 месяцев назад +2

      the math never stopped :p

    • @CognitiveOffense
      @CognitiveOffense 5 месяцев назад

      @@schweinmachtbree1013 So you're saying the math is unlimited?! ... ;)

    • @rexrex600
      @rexrex600 5 месяцев назад

      abstract nonsense is maths smh

  • @linuxnoodle8682
    @linuxnoodle8682 5 месяцев назад +43

    I'm not proud to say that I almost thought this was an unironic video until you started bringing out the commutative diagrams

    • @schweinmachtbree1013
      @schweinmachtbree1013 5 месяцев назад +12

      commutative diagrams are needed for every introduction to categorical limits! though I will say I’m impressed that (g+)+ explained the idea of limits without universal properties

  • @francescominnocci
    @francescominnocci 5 месяцев назад +12

    This is a masterpiece and a pleasure to watch, especially as I'm going through Riehl's book and trying to reason more categorically in everyday life
    14:39 i would love a video about \infty-categories, perhaps "What does it mean for two things to be equal" and the thumbnail is an interrogative equal sign between two similar emojis and philosphers faces in the background, or as an alternative something misleading about "higher dimensions" with 4-dimensional geometric clickbait

    • @ffc1a28c7
      @ffc1a28c7 Месяц назад +1

      "everyday life" lol.

  • @danielesantospirito5743
    @danielesantospirito5743 5 месяцев назад +5

    This video was the best so far, in my opinion! Thanks to you I'm begging to appreciate for the first time how category theory works... Great videos, great explanations, thank you again!!

  • @DinoMomPlays
    @DinoMomPlays 5 месяцев назад +43

    0/10 video did not construct the Bolzano notion of a limit by checking that a certain natural transformation is a limit in the category of filters on the real numbers (nice video, I enjoyed it).

  • @recursive0
    @recursive0 5 месяцев назад +10

    Whoops, accidentally starting to understand category theory thanks to your videos. (I was saving it for later)

  • @Vaaaaadim
    @Vaaaaadim 5 месяцев назад +17

    I need to sit down and learn some category theory so that I can get all the jokes.

  • @asdfghyter
    @asdfghyter 3 месяца назад +1

    wow, that opening was brilliant!

  • @Filup
    @Filup 5 месяцев назад +5

    I am at that point of my academic career where once I see math that I have not seen before, I should be genuinely worried. I love category theory lmao

  • @brooksbryant2478
    @brooksbryant2478 3 месяца назад

    This is my new favorite RUclips channel

  • @TH-tp4pn
    @TH-tp4pn 4 месяца назад +2

    The reason I opened this video is to check my understanding on limits after graduating from university. And the reason I finished this video is I understand none of it but enjoy the show

  • @许玄清
    @许玄清 5 месяцев назад +78

    sheafification video when

    • @RealJoux
      @RealJoux 5 месяцев назад +5

      He'll have to make it twice though

  • @Zahrizi
    @Zahrizi 5 месяцев назад +5

    I didn't understand this at all, great video👍. /pos

  • @denzelcurnow8308
    @denzelcurnow8308 5 месяцев назад +8

    I'm also at my limit!!! 💪💪💪

  • @Maxence1402a
    @Maxence1402a 2 месяца назад

    Thanks a lot for this rather easy introduction to limits in category theory! Having no education at all in category theory but a master's degree in functional analysis, and a teaching degree at undergraduate level, I had to rewatch some bits of the video until I got the meaning though.

  • @jennifersmith2743
    @jennifersmith2743 5 месяцев назад +6

    Finally, Acerolla for math.

    • @Alceste_
      @Alceste_ 5 месяцев назад

      There are similarities (for instance, how often the word `projection` comes up), but there's a small difference:
      I easily understand & digest Acerola content. :')

  • @decare696
    @decare696 5 месяцев назад +8

    the puns are strong with this one
    Is there a name for the property that you can calculate limits with a single object? And does it apply to topological or metric spaces? If not, those could've made for a better example than chain complexes (and if it does, I'll be surprised and understand your choice of example a bit better).

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +8

      That the limits are "calculated" using a single object is a bit disingenuous on my part.
      That being said, an object S is usually called a "separator" if mapping out of it can distinguish arrows of your category, which is *kinda* like being able to use a single object to identify the "underlying set" of any object of your category.
      The category of topological spaces does have a separator (the singleton), as does the category of metric spaces (the singleton, again), so the underlying set of the limit of any diagram in either category can be determined by probing with the singleton.
      However, it takes a bit more thinking to recover the rest of the structure. For example, in the category of topological spaces, knowing
      1. the underlying set of the limit, and
      2. that the projection maps are continuous
      is not sufficient to determine the topological structure of the limit, as there may be several spaces satisfying both conditions (e.g., the discrete space on the set will always satisfy these conditions). However, it's pretty much enough: the correct topology is the coarsest topology that makes the projection maps continuous.
      The story is similar for metric spaces (the metric in the limit is the supremum of the metrics of the components), except now it's possible that limits don't exist.
      [As mentioned in the video, the category of chain complexes also has a separator, but this is like choosing the set of natural numbers as the separator for Set.]

    • @decare696
      @decare696 5 месяцев назад

      @@SheafificationOfG it being the coarsest is of course the natural result, since all the other candidates have to map into it, so that could've been an entry point to the general concept, but it also feels somewhat contrived, so not much better than what you settled on imo.
      OTOH, Topology and Contrivances, name a better duo.
      To be slightly clearer on my wording, by "calculate with a single object", I meant "fully determined by probing against a single object, making all the others redundant", just for the record.

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +2

      @@decare696 I think some formalisation of your second paragraph is asking for an object S such that Hom(S, -) is conservative. In this case, I can at least say that any category C monadic over Set has a canonical choice of such an object, since the forgetful functor C -> Set is conservative, and it is corepresentable by the free object generated by the singleton set (i.e., U = Hom(S, -), where S = F(*)).
      This covers Vect_R (giving us R again), as well as other varieties of algebras (groups, rings, etc.).

  • @m3morizes
    @m3morizes 17 дней назад +1

    I might be trippin', but are the compositions at 17:28 reversed? Are you using a different convention? I could also just be sleepy, it's 4:00 AM here.

    • @SheafificationOfG
      @SheafificationOfG  17 дней назад +1

      Nah you're right
      The diagram went through some changes that were not reflected in the compositions

    • @m3morizes
      @m3morizes 17 дней назад +1

      @SheafificationOfG Thanks for the clarification 👍, really good video 👌

  • @dustin20541
    @dustin20541 5 месяцев назад +2

    Hey G, thanks so much for making these videos on category theory. Have you considered making a series similar to what Bartosz Milewski did 7 or so years ago where he worked on a whiteboard to help programmers understand category theory a bit better. I really wish he would have continued this series or at least did a refresh on it because this stuff is super interesting to programmers like myself.

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +1

      I've never actually looked at Milewski's work (though I've heard of it). I'll probably stick to something close to my current style, but you can at least expect more category theory flavoured videos in the future!

  • @duncanw9901
    @duncanw9901 5 месяцев назад

    These are like MLG montages for math nerds who drink loose-leaf tea instead of COD players who drink mountain dew.
    Trying to realize the treasure planet triangle meme at 2.5x speed is the true zoomer experience

  • @funktorial
    @funktorial 2 месяца назад

    that point about the empty set is interesting. on the one hand, of course, it's true that all ZF sets are well-founded. on the other hand, though, you can't really express in ZFC something like "the set of all sets that contain themselves," because obstructing such comprehensions is the reason ZFC exists in the first place. I think if you let yourself talk about classes too, you can prove that the class of sets containing themselves must be a set (of course, the empty set). just to say you need more than ZFC to express that such a set really exists in ZFC, I think

  • @romajimamulo
    @romajimamulo 5 месяцев назад

    I had to step away once I realized this wasn't going to be calculus, you brilliant prankster

  • @denizgoksu9868
    @denizgoksu9868 5 месяцев назад +6

    This was truly our Ran Academy
    Lan Academy when?

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +3

      Watch this one at -1x speed in the meantime!

  • @JW-zs1ik
    @JW-zs1ik 5 месяцев назад +6

    Incredible title

  • @StepanMakarenko
    @StepanMakarenko 20 дней назад

    this is so cool! really motivates to study it properly. but why are they called limits tho?

  • @someguy1507
    @someguy1507 Месяц назад

    I don't know anything about category theory. Is there a book or introduction I ought to refer to? Should I learn group theory first?

    • @SheafificationOfG
      @SheafificationOfG  Месяц назад

      Good category theory can only come from having a solid background in more "real" fields of maths (like in abstract algebra or related fields), so I def agree that you should start with group theory!

    • @wingless_avian
      @wingless_avian 19 дней назад

      And if you want to start learning abstract algebra then I advise you to Take a look at Herstein's "Topics in algebra".

  • @wahidislamlinad
    @wahidislamlinad 4 месяца назад +2

    yours is the only channel that doesn't come to my feed still i intentionally search it up regularly just so i don't miss a video.

    • @SheafificationOfG
      @SheafificationOfG  4 месяца назад +2

      Damn, RUclips out there shadowbanning me or smt?

    • @wahidislamlinad
      @wahidislamlinad 4 месяца назад +1

      @@SheafificationOfG if you're getting ad revenue than most probably not. i know having a low bit rate in a video deem it to be low quality so the "holy algorithm" tends to push less the video, but I'm not sure if audio also has the same effect. Although personally speaking I feel very pleased hearing the fan noise in background. It's quite soothing tbh.

  • @schweinmachtbree1013
    @schweinmachtbree1013 5 месяцев назад

    more category theory or homological algebra pls!

  • @templarsam
    @templarsam 11 часов назад

    Wow! A good video video for freshmen that struggle with calculus-1! Thank you, I am going to never study math again!

  • @dehiscence
    @dehiscence 5 месяцев назад

    thanks

  • @BrimmingCuriosity
    @BrimmingCuriosity 5 месяцев назад +10

    Wife left you? Doesn't matter, Sheafification of G just dropped a video!

  • @Androidonator
    @Androidonator 2 месяца назад

    Where is the gsheaf new mic fund?

  • @NoLongerBreathedIn
    @NoLongerBreathedIn 3 месяца назад

    ...I was really hoping for Kan extensions

  • @thephysicistcuber175
    @thephysicistcuber175 2 месяца назад

    23:41 in fact you need products of zero elements too.

  • @asparagus4040
    @asparagus4040 5 месяцев назад +1

    woah 😳

  • @ethan_martin
    @ethan_martin 4 месяца назад

    limitless puns i’m dying

  • @noThankyou-g5c
    @noThankyou-g5c 5 месяцев назад

    this seems like a really good video but for future reference you lost me at 3:48 ish

  • @notu483
    @notu483 5 месяцев назад

    Please cover filters next

  • @brooksbryant2478
    @brooksbryant2478 3 месяца назад

    Colimit video next?

  • @berlinisvictorious
    @berlinisvictorious 5 месяцев назад

    Ah yes always a good appendix for scheme theory video

  • @badabing3391
    @badabing3391 2 месяца назад

    😊

  • @df-163
    @df-163 4 месяца назад

    Coming next: Kan't Academy
    where we discuss the Spectral theorem for Commutative C* algebras

  • @pyropulseIXXI
    @pyropulseIXXI 3 месяца назад

    You can escape limits; it is called non-standard analysis. Also, limits weren't scary at all; they were super easy, barely an inconvenience

  • @ke9tv
    @ke9tv 5 месяцев назад +3

    Replacing the singleton * with the 1-dimensional vector space over R: Why R, rather than an arbitrary field k (with characteristic other than 2, of course...) of course replacing 1 with the unit of k ?
    Love your spoof of mathematical astronautics (ascending to greater and greater heights of abstraction until you run out of oxygen)!

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +12

      I chose the category of vector spaces over R to keep it real 😉. If vector spaces were allowed to be over arbitrary fields, then idk what reasonable maps between vector spaces of mixed characteristics would be (and then I would probably have to kiss arbitrary products goodbye).
      In the category Vect_k, for *any* field k, I could've replaced the singleton with k. This works even if k has characteristic 2/

  • @vitulus_
    @vitulus_ 5 месяцев назад +15

    I like your videos but please apply basic noise removal to them. You can get Audacity for free and remove noise that way.

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +20

      You sure noise removal won't just cut out my voice?
      (jk, I'll give it a go next video)

    • @pmmeurcatpics
      @pmmeurcatpics 5 месяцев назад +5

      ​@@SheafificationOfGoh I thought that was actually a part of the gig, perhaps even the only fan of yours itself blowing in the background

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +5

      @pmmeurcatpics the fan was definitely part of the gig originally, but it's because my mic kept picking it up haha

    • @brekike
      @brekike 5 месяцев назад

      @@SheafificationOfG Please never remove the fan noise, that's the funniest part of the videos(I don't understand most of the category theory jokes though)

  • @jespervalgreen6461
    @jespervalgreen6461 5 месяцев назад

    That was great. Now do it again, slower and in more detail. Assume that I am ignorant and dim, but persistent.

  • @natepolidoro4565
    @natepolidoro4565 3 месяца назад

    Woah there buddy, thats a bit much. I'm dead.

  • @zakkatz9144
    @zakkatz9144 3 месяца назад

    "will always satisfy a weak terminal object"... WILL ALWAYS BE THERE IN THE END!!

  • @user-du9ch3tn2v
    @user-du9ch3tn2v 5 месяцев назад +3

    Kan academy love it!
    Let's make it big together (with a discord?)

  • @aloysiuskurnia7643
    @aloysiuskurnia7643 5 месяцев назад

    Ah yes, category theory, where the "concrete example" is still two-tier abstract.

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +1

      Sets seem pretty concrete (they're definitionally concrete, in fact!)

  • @nektariosorfanoudakis2270
    @nektariosorfanoudakis2270 Месяц назад

    These stuff are trivial compared to what I want to study, and I'm scared. 😢

  • @MooImABunny
    @MooImABunny 5 месяцев назад +3

    I've never seen a category arrow diagram that was clearer and more concise than saying what you want to convey with words 🤷🏽‍♂️

    • @pmmeurcatpics
      @pmmeurcatpics 5 месяцев назад +3

      Yeah but the picture is at least here to see in its completeness all the time. With words, on the other hand, you end up forgetting how it began by the time you reach the end.
      Or at least that's what I have, my short-term memory is abysmal. Maybe it's just that the video is so good, but I thought that I was really getting it with the help of the diagrams (they were kind of tame in comparison to more complex ones, however - but those would be exponentially more difficult to put into words as well)

    • @SheafificationOfG
      @SheafificationOfG  5 месяцев назад +7

      You become more grateful for diagrams when you start dealing with many more than just 3-4 objects and arrows between them. For instance, a lot of homological algebra would be a huge pain to parse.
      Also, it's much more difficult to be comprehensive about relationships between compositions of arrows using words, but it's all there in a diagram.

  • @Yusuketh443
    @Yusuketh443 4 месяца назад +1

    hi :3 UwU

  • @samuelwaller4924
    @samuelwaller4924 2 месяца назад

    I got confused between nu and v lol. I paused for so long...
    v_ν

  • @redgit9905
    @redgit9905 7 дней назад

    I for one find myself at mine

  • @tomholroyd7519
    @tomholroyd7519 17 дней назад

    cardinality

  • @imtryingmybes7
    @imtryingmybes7 4 месяца назад

    so glad i left pure math

    • @SheafificationOfG
      @SheafificationOfG  4 месяца назад +2

      I'm not!! :(

    • @imtryingmybes7
      @imtryingmybes7 4 месяца назад

      @@SheafificationOfG :(, maybe you'll eventually see things differently

  • @tristenarctician6910
    @tristenarctician6910 3 месяца назад

    Kan't Academy

  • @aziz0x00
    @aziz0x00 3 месяца назад

    😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂

  • @Omeomeom
    @Omeomeom 5 месяцев назад

    better than wodzicki

  • @chrissun9068
    @chrissun9068 5 месяцев назад

    I don’t understand this

  • @chair547
    @chair547 3 месяца назад

    is this real math?

  • @duoasch
    @duoasch 4 месяца назад

    so many puns i dont understand

  • @sis_sos
    @sis_sos 5 месяцев назад

    Have you ever considered that you have a negative impact on mathematics youtube?

  • @MDNQ-ud1ty
    @MDNQ-ud1ty 5 месяцев назад +4

    I'm not a fan of all the meme's. Call me a boomer if you want(I'm not). It's distracting IMO. Maybe one odd one here and there but not a constant barage.