Kan Academy: Introduction to Limits

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  • Опубліковано 7 лис 2024

КОМЕНТАРІ • 99

  • @SheafificationOfG
    @SheafificationOfG  2 місяці тому +9

    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 Місяць тому

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

  • @susanafaciolince7755
    @susanafaciolince7755 2 місяці тому +304

    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 2 місяці тому +161

    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 2 місяці тому +94

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

    • @SheafificationOfG
      @SheafificationOfG  2 місяці тому +65

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

    • @dogedev1337
      @dogedev1337 2 місяці тому +3

      @@SheafificationOfG pls make a video about deriving functors

    • @francescominnocci
      @francescominnocci 2 місяці тому +5

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

    • @pyropulseIXXI
      @pyropulseIXXI 21 день тому +1

      I seriously doubt that, lmao

  • @rtg_onefourtwoeightfiveseven
    @rtg_onefourtwoeightfiveseven 2 місяці тому +64

    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  2 місяці тому +29

      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 2 місяці тому +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 27 днів тому

      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 25 днів тому

      @@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.

  • @celivalg
    @celivalg 2 місяці тому +27

    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 2 місяці тому +14

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

  • @DinoMomPlays
    @DinoMomPlays 2 місяці тому +33

    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).

  • @pandavroomvroom
    @pandavroomvroom 16 днів тому +7

    bro embodies chaotic good

  • @linuxnoodle8682
    @linuxnoodle8682 2 місяці тому +38

    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 2 місяці тому +11

      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

  • @CognitiveOffense
    @CognitiveOffense 2 місяці тому +51

    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 2 місяці тому +2

      the math never stopped :p

    • @CognitiveOffense
      @CognitiveOffense 2 місяці тому

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

    • @rexrex600
      @rexrex600 2 місяці тому

      abstract nonsense is maths smh

  • @francescominnocci
    @francescominnocci 2 місяці тому +10

    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

  • @zyansheep
    @zyansheep 2 місяці тому +21

    I think this is now officially my favorite channel :D

  • @michaelearnest1983
    @michaelearnest1983 2 місяці тому +6

    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.

  • @许玄清
    @许玄清 2 місяці тому +69

    sheafification video when

    • @RealJoux
      @RealJoux 2 місяці тому +2

      He'll have to make it twice though

  • @danielesantospirito5743
    @danielesantospirito5743 2 місяці тому +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!!

  • @recursive0
    @recursive0 2 місяці тому +10

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

  • @asdfghyter
    @asdfghyter Місяць тому +1

    wow, that opening was brilliant!

  • @brooksbryant2478
    @brooksbryant2478 14 днів тому

    This is my new favorite UA-cam channel

  • @Maxence1402a
    @Maxence1402a 2 дні тому

    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.

  • @Filup
    @Filup 2 місяці тому +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

  • @jennifersmith2743
    @jennifersmith2743 2 місяці тому +5

    Finally, Acerolla for math.

    • @Alceste_
      @Alceste_ 2 місяці тому

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

  • @denzelcurnow8308
    @denzelcurnow8308 2 місяці тому +7

    I'm also at my limit!!! 💪💪💪

  • @dustin20541
    @dustin20541 2 місяці тому +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  2 місяці тому +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!

  • @decare696
    @decare696 2 місяці тому +7

    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  2 місяці тому +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 2 місяці тому

      @@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  2 місяці тому +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.).

  • @Vaaaaadim
    @Vaaaaadim 2 місяці тому +15

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

  • @TH-tp4pn
    @TH-tp4pn Місяць тому

    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

  • @JW-zs1ik
    @JW-zs1ik 2 місяці тому +6

    Incredible title

  • @Zahrizi
    @Zahrizi 2 місяці тому +1

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

  • @denizgoksu9868
    @denizgoksu9868 2 місяці тому +4

    This was truly our Ran Academy
    Lan Academy when?

  • @Androidonator
    @Androidonator 2 дні тому

    Where is the gsheaf new mic fund?

  • @wahidislamlinad
    @wahidislamlinad Місяць тому +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  Місяць тому +1

      Damn, UA-cam out there shadowbanning me or smt?

    • @wahidislamlinad
      @wahidislamlinad Місяць тому +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 2 місяці тому

    more category theory or homological algebra pls!

  • @NoLongerBreathedIn
    @NoLongerBreathedIn 25 днів тому

    ...I was really hoping for Kan extensions

  • @zakkatz9144
    @zakkatz9144 15 днів тому

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

  • @notu483
    @notu483 2 місяці тому

    Please cover filters next

  • @brooksbryant2478
    @brooksbryant2478 14 днів тому

    Colimit video next?

  • @pyropulseIXXI
    @pyropulseIXXI 21 день тому

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

  • @romajimamulo
    @romajimamulo 2 місяці тому

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

  • @df-163
    @df-163 Місяць тому

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

  • @berlinisvictorious
    @berlinisvictorious 2 місяці тому

    Ah yes always a good appendix for scheme theory video

  • @jespervalgreen6461
    @jespervalgreen6461 2 місяці тому

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

  • @esphix
    @esphix 2 місяці тому +1

    someday i will understand these videos

  • @dehiscence
    @dehiscence 2 місяці тому

    thanks

  • @ke9tv
    @ke9tv 2 місяці тому +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  2 місяці тому +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/

  • @duncanw9901
    @duncanw9901 2 місяці тому

    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

  • @MooImABunny
    @MooImABunny 2 місяці тому +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 2 місяці тому +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  2 місяці тому +6

      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.

  • @vitulus_
    @vitulus_ 2 місяці тому +13

    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  2 місяці тому +19

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

    • @pmmeurcatpics
      @pmmeurcatpics 2 місяці тому +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  2 місяці тому +5

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

    • @brekike
      @brekike 2 місяці тому

      @@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)

  • @natepolidoro4565
    @natepolidoro4565 25 днів тому

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

  • @asparagus4040
    @asparagus4040 2 місяці тому +1

    woah 😳

  • @user-du9ch3tn2v
    @user-du9ch3tn2v 2 місяці тому +3

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

  • @noThankyou-g5c
    @noThankyou-g5c 2 місяці тому

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

  • @ethanmartin2781
    @ethanmartin2781 Місяць тому

    limitless puns i’m dying

  • @BrimmingCuriosity
    @BrimmingCuriosity 2 місяці тому +8

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

  • @Omeomeom
    @Omeomeom 2 місяці тому

    better than wodzicki

  • @tristenarctician6910
    @tristenarctician6910 24 дні тому

    Kan't Academy

  • @imtryingmybes7
    @imtryingmybes7 Місяць тому

    so glad i left pure math

  • @Yusuketh443
    @Yusuketh443 2 місяці тому +1

    hi :3 UwU

  • @chair547
    @chair547 25 днів тому

    is this real math?

  • @aloysiuskurnia7643
    @aloysiuskurnia7643 2 місяці тому

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

    • @SheafificationOfG
      @SheafificationOfG  2 місяці тому

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

  • @chrissun9068
    @chrissun9068 2 місяці тому

    I don’t understand this

  • @aziz0x00
    @aziz0x00 23 дні тому

    😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂😂

  • @duoasch
    @duoasch Місяць тому

    so many puns i dont understand

  • @sis_sos
    @sis_sos 2 місяці тому

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

  • @MDNQ-ud1ty
    @MDNQ-ud1ty 2 місяці тому +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.