Guys, don't give up if you feel like you hit a brick wall with understanding or appreciating the Yoneda Lemma. This is the first example that really shows the broad power of category theory to generalize results from many areas of math. I was about to admit defeat in learning category theory for now. Then I collected myself and carefully reviewed all the definitions involved. To summarize, that includes: -the axioms and definitions of the basic building blocks of category theory: categories, functors, and natural transformations. In particular, make sure you can explain to someone else exactly how functors and natural transformations preserve structure. Naturality, for example, is analogous to linearity for linear transformations in linear algebra. -the definition of a hom functor, which is based on the definition of a hom set. Remember from the previous video that a hom functor encapsulates the totality of morphisms emanating from an object, or alternatively impinging upon it. -the definition of a representable functor: a functor which is naturally isomorphic to a hom functor. Do everything you need to do to solidify your understanding of these definitions, because the Yoneda Lemma depends on all of them. Open a tab to review videos from Milewski's first course. Open another tab to google definitions and examples of whatever you're less than 100% clear about. It is worth it to get a taste of the beauty and power that category theory offers through results like the Yoneda Lemma. Google the Yoneda Lemma to see some examples of what it generalizes. Even if most of these examples are from areas of math you haven't studied, it's still amazing just to see how many there are. Someone once wrote a poem about the Yoneda Lemma in relation to the theorems it generalizes: One Ring to rule them all, One Ring to find them, One Ring to bring them all and in the darkness bind them
I think one thing that Bartosz Milewski failed to mention is what I heard from the Yeneda lemma lecture of the TheCatsters(ua-cam.com/video/TLMxHB19khE/v-deo.html): What the situation that yoneda lemma can apply to is that when you think there's a lot of things goning on but in fact you have very little choice of what't going on and every thing that's going on everywhere is really controlled by one little small thing
Correct me if I'm wrong, but there should be an intrinsic variant of the yoneda lemma for any category that has exponentials and a terminal object with arrows to all non-initial objects. Maybe that's a really strong condition, but I really distrust SET and much prefer intrinsic constructions independent of any specific category. Proof "sketch": We consider some category C meeting the above requirements. The generalization of the yoneda lemma to C is that every natural transformation α from the covariant exponential functor anchored at some a to any endofunctor F are fixed by some arrow from the terminal object (they're all uniquely isomorphic) to F a. First we can pick out the identity "member" of a->a by selecting the arrow i from T to a->a that when combined with any arrow x from T to a to make an arrow [x, i] from T to a × a->a such that x = eval . [x, i]. We know that such an arrow must exist by specializing the adjunction diagram for exponentials and products on T, a, and a. Since we now have i, we compose it with α_a. This must commute with some arrow from T to F a, call it p. Then we just compose with F f (where f is some arrow from a to some object y), and we end up at F y. Now we take the other route in the naturality square: (f . _) after i gives us an arrow from T to a->y, but because of the identity nature of i, that arrow is precicely the one that "picks" f. Then we compose that arrow with α_y and end up with another arrow from T to F y. By naturality these paths commute, but that means the composition of picking an f with α_y is entriely determined by what α_a after i composes to. QED. It still feels cheap and cheesy, and I want to make a more rigorous argument as to why the exponential functor acting on f after i is equivalent to the "picking" morphisim that grabs f, but I suspect that's just diagram chasing through a few adjunctions. Maybe all categories that meet the requirements are faithfully representable in SET and my inclination towards intrinsic constructions is for naught, but I think it's worth considering for something like CAT (or maybe CAT's larger variants).
So there are exactly as many different apple cakes F(a) as there are orange cake recipes F(x) that are based on a way to swap apples with oranges a->x ? And that should even true for all other ingredients x ?
I just realized C x [C, Set] is a category, so the Yoneda Lemma applies to it We’ll call this C1 The Yoneda Lemma then applies to C2 = C1 x [C1, Set], C3 = C2 x [C2, Set], etc.
question. Fa is a set: how to define two distinct elements of Fa, say p and q. If p = alpha_a(ida), then what is q? The identity ida is fixed I assumed, so is q = alpha'_a(ida)? If so what is alpha'_a? I'm guessing alpha'_a is another natural transformation from C(a, -) to Fa, but I'm not sure. Thank you
Guys, don't give up if you feel like you hit a brick wall with understanding or appreciating the Yoneda Lemma. This is the first example that really shows the broad power of category theory to generalize results from many areas of math. I was about to admit defeat in learning category theory for now. Then I collected myself and carefully reviewed all the definitions involved. To summarize, that includes:
-the axioms and definitions of the basic building blocks of category theory: categories, functors, and natural transformations. In particular, make sure you can explain to someone else exactly how functors and natural transformations preserve structure. Naturality, for example, is analogous to linearity for linear transformations in linear algebra.
-the definition of a hom functor, which is based on the definition of a hom set. Remember from the previous video that a hom functor encapsulates the totality of morphisms emanating from an object, or alternatively impinging upon it.
-the definition of a representable functor: a functor which is naturally isomorphic to a hom functor.
Do everything you need to do to solidify your understanding of these definitions, because the Yoneda Lemma depends on all of them. Open a tab to review videos from Milewski's first course. Open another tab to google definitions and examples of whatever you're less than 100% clear about. It is worth it to get a taste of the beauty and power that category theory offers through results like the Yoneda Lemma. Google the Yoneda Lemma to see some examples of what it generalizes. Even if most of these examples are from areas of math you haven't studied, it's still amazing just to see how many there are. Someone once wrote a poem about the Yoneda Lemma in relation to the theorems it generalizes:
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them
I think one thing that Bartosz Milewski
failed to mention is what I heard from the Yeneda lemma lecture of the
TheCatsters(ua-cam.com/video/TLMxHB19khE/v-deo.html): What the situation that yoneda lemma can apply to is that when you think there's a lot of things goning on but in fact you have very little choice of what't going on and every thing that's going on everywhere is really controlled by one little small thing
Wdym naturality is analogous to linearity
Thank you. I am making progress and your comment was very helpful.
Thank you, very inspritational!
I’m in junior year highschool I don’t know why I’m even watching this
Because you, young man, have excellent taste. Good on yer, Mate! Keep it up.
When he said 'is this a contravariant functor? you might think so.... but it's not!', I wish someone had done a dramatic 'dam dam daaam'
Exactly two thirds into the lecture I am gone.
Nice work :)
if we have w::a->x in C(a,x), C(a,x) is covariant in "a" and is indeed sitting in the left position.
Correct me if I'm wrong, but there should be an intrinsic variant of the yoneda lemma for any category that has exponentials and a terminal object with arrows to all non-initial objects. Maybe that's a really strong condition, but I really distrust SET and much prefer intrinsic constructions independent of any specific category.
Proof "sketch":
We consider some category C meeting the above requirements. The generalization of the yoneda lemma to C is that every natural transformation α from the covariant exponential functor anchored at some a to any endofunctor F are fixed by some arrow from the terminal object (they're all uniquely isomorphic) to F a.
First we can pick out the identity "member" of a->a by selecting the arrow i from T to a->a that when combined with any arrow x from T to a to make an arrow [x, i] from T to a × a->a such that x = eval . [x, i]. We know that such an arrow must exist by specializing the adjunction diagram for exponentials and products on T, a, and a.
Since we now have i, we compose it with α_a. This must commute with some arrow from T to F a, call it p. Then we just compose with F f (where f is some arrow from a to some object y), and we end up at F y. Now we take the other route in the naturality square: (f . _) after i gives us an arrow from T to a->y, but because of the identity nature of i, that arrow is precicely the one that "picks" f. Then we compose that arrow with α_y and end up with another arrow from T to F y. By naturality these paths commute, but that means the composition of picking an f with α_y is entriely determined by what α_a after i composes to. QED.
It still feels cheap and cheesy, and I want to make a more rigorous argument as to why the exponential functor acting on f after i is equivalent to the "picking" morphisim that grabs f, but I suspect that's just diagram chasing through a few adjunctions. Maybe all categories that meet the requirements are faithfully representable in SET and my inclination towards intrinsic constructions is for naught, but I think it's worth considering for something like CAT (or maybe CAT's larger variants).
There is a generalization of the Yoneda lemma for enriched categories that doesn't use Set.
So there are exactly as many different apple cakes F(a) as there are orange cake recipes F(x) that are based on a way to swap apples with oranges a->x ? And that should even true for all other ingredients x ?
Thank you.
I just realized
C x [C, Set] is a category, so the Yoneda Lemma applies to it
We’ll call this C1
The Yoneda Lemma then applies to C2 = C1 x [C1, Set], C3 = C2 x [C2, Set], etc.
33:58, he looks at the class, everyone is open mouthed and mind blown
simple
question. Fa is a set: how to define two distinct elements of Fa, say p and q. If p = alpha_a(ida), then what is q? The identity ida is fixed I assumed, so is q = alpha'_a(ida)? If so what is alpha'_a? I'm guessing alpha'_a is another natural transformation from C(a, -) to Fa, but I'm not sure. Thank you
27:59 we went from Fa -> Gb by Fa->Ga->Gb, we could also go Fa->Fb->Gb right? can i do something similar or the second proof?
I just had to stop and thank you for the negative * negative part
The red diagram at 27:30 doesn't yet prove that we have a functor [C,Set]×C→Set, right? We would need to check that it preserves composition?
Yes, this and the identity law. But they just follow from regular function composition and identity. It's a simple exercise.
@@BartoszMilewski There's that word 'simple' again. 🙂
Amazing!
I have no idea whats happening
Hahaha ☹
I had the same feeling after reading this chapter in the book, but after this video I've at last was able to see all the beauty.
I find Tom Leinster's book "Basic Category Theory" very helpful for understanding Yoneda.
A-ha!
i almost get it
Thanks Gaddafi!
>for pOrgammers