I like to think of Hom(X, -) as a device that allows you to take x-ray images of X from any angle you want. This is fairly precise, because morphisms usually correspond to projections. e.g. you can picture the surjection from (Z, +) to (Z/12Z, +) as: arrange the number line into a helix, then x-ray the helix down onto a clock face.
Great, I am glad that the explanation was helpful! For me personally the Yoneda embedding is relatively natural, while the Yoneda lemma itself is quite mysterious. Its good to know that this way of approaching the Yoneda lemma worked for you as well!
I love this perspective on the Yoneda perspective! I am wondering, is there any way to make the gold foil analogy more literal? That is, to construct an appropriate category where one of the objects could represent a gold atom, such that the Yoneda lemma for this category implies that one could discover the structure of the gold atom by firing electrons at it? If so, which category would this work for I wonder?
Thanks, I love the analogy as well; its lovely ☺ Hmm, I am not sure how to answer your question. Let me give it a shot, but please complain id this is not what you are looking for. For me the gold foil experiment can be categorical summarized by the idea to replace objects by the functors of “taking hom into them” (like in the gold foil experiment) or, somewhat dually, “taking hom out of them”. So the corect categories to start with should be some form of a functor category where the gold objects is replaced by a functor. The electrons are then the objects that you can input into this “gold functor”.
Fascinating. But what I'm puzzled about is that: As a covariant functor, Hom (-, X): Cop to Set should preserve the composition of Cop (because it reverses the composition of C) . So, precisely, should the Cop on the left side of the slides be C?
Hmm, excellent question. A typical “sign” error? I have no idea 🤣 Let me still try: First, a contravariant functor F from C to D is a functor from C^(op) to D. Ok, this way we can get rid of the “contra” and focus on usual functors. Next, it seems then its en.wikipedia.org/wiki/Yoneda_lemma#Contravariant_version Maybe what is confusing is that I tried not to mention contravariant functors?
@@VisualMath Hmm, I may get it. By studing functors via functor category, as objects in this functor cat Fop: Cop to Dop must be the same as F: C to D in some sense. So if define F: Cop to D, then is it just the same as Fop: C to Dop? Since arrows are more significant than objects, covariant functors masy just provide a "reference" for contravariant functors. And which one is co- and the other is contra- makes nosense though. Well, I also agree with the idea of not to mention contra-. From the learner view, maybe describing both C and Cop simutaneously is better?(since the usage different symbols for C and Cop in the previous video) Then every contra- functor may just constructed wih aid of the functor C to Cop. I think this may be helpful.
@@M0n1carK Yes, exactly. At one point we have to face a choice whether we prefer, say for groups, f(ab)=f(a)f(b) over f(ab)=f(b)f(a). I feel the first is nicer 😅 Whatever is then studied in CT should then be an extension of "familiar" constructions, hence I like to ignore contravariant functors 😀
Is Cayley's lemma in group theory a special case of Yoneda lemma? The lemma basically says you can consider each group element as a group action on the some set, which is same to this philosophy. Now I am getting the what is a philosophy of category theory. Category theory wants to investigate the interactions between objects, not the internal structures of the objects.
Thank you for the reminder: I already forgot about this lovely example. The answer is YES! Well, it is rather the Yoneda embedding than the lemma itself: Every group is a subgroup of Aut(X) for some X, that is Cayley. Every category is a subcategory of End(C), that is Yoneda. The connection goes via taking C=G, the “group category” with one object and arrows being the group elements with the induced composition. For a precise and detailed discussion see math.stackexchange.com/questions/1701/yoneda-lemma-as-generalization-of-cayleys-theorem And YES! again: you summarize very nicely what category theory wants to study. I personally like to replace “interactions” with “relations”, but then we are on the same page ;-)
Let me try to cook up a one-to-one correspondence. Gold foil, action of shooting, particles, observations, interpretations do they corresponds to category C, functors F:C->SET, an object in C, functors F:C->SET, SET, SET?
I like to think of Hom(X, -) as a device that allows you to take x-ray images of X from any angle you want. This is fairly precise, because morphisms usually correspond to projections.
e.g. you can picture the surjection from (Z, +) to (Z/12Z, +) as: arrange the number line into a helix, then x-ray the helix down onto a clock face.
I like that perspective a lot as well 🙂
I found the gold foil experiment analogy to be a very fruitful way to think about this. Nice video!
Same here - I am glad that you liked the analogy 😀
Thank you very much, your presentation and interpretation of the Yoneda Embedding really made things much clear to me. Great Video!
Great, I am glad that the explanation was helpful! For me personally the Yoneda embedding is relatively natural, while the Yoneda lemma itself is quite mysterious. Its good to know that this way of approaching the Yoneda lemma worked for you as well!
Incredible video! Thank you!!
Thank you so much. The Yoneda lemma is wonderful, I am glad that you liked it!
Amazing video thank you very much for all the work you put into this
Thank you so much for the feedback - I am glad that you enjoyed the video. I hope you will enjoy category theory in general - I am a big fan ;-)
I love this perspective on the Yoneda perspective! I am wondering, is there any way to make the gold foil analogy more literal? That is, to construct an appropriate category where one of the objects could represent a gold atom, such that the Yoneda lemma for this category implies that one could discover the structure of the gold atom by firing electrons at it? If so, which category would this work for I wonder?
Thanks, I love the analogy as well; its lovely ☺
Hmm, I am not sure how to answer your question. Let me give it a shot, but please complain id this is not what you are looking for.
For me the gold foil experiment can be categorical summarized by the idea to replace objects by the functors of “taking hom into them” (like in the gold foil experiment) or, somewhat dually, “taking hom out of them”. So the corect categories to start with should be some form of a functor category where the gold objects is replaced by a functor. The electrons are then the objects that you can input into this “gold functor”.
Fascinating.
But what I'm puzzled about is that:
As a covariant functor, Hom (-, X): Cop to Set should preserve the composition of Cop (because it reverses the composition of C) .
So, precisely, should the Cop on the left side of the slides be C?
Hmm, excellent question. A typical “sign” error? I have no idea 🤣
Let me still try: First, a contravariant functor F from C to D is a functor from C^(op) to D. Ok, this way we can get rid of the “contra” and focus on usual functors. Next, it seems then its en.wikipedia.org/wiki/Yoneda_lemma#Contravariant_version
Maybe what is confusing is that I tried not to mention contravariant functors?
@@VisualMath Hmm, I may get it.
By studing functors via functor category, as objects in this functor cat Fop: Cop to Dop must be the same as F: C to D in some sense.
So if define F: Cop to D, then is it just the same as Fop: C to Dop? Since arrows are more significant than objects, covariant functors masy just provide a "reference" for contravariant functors. And which one is co- and the other is contra- makes nosense though.
Well, I also agree with the idea of not to mention contra-. From the learner view, maybe describing both C and Cop simutaneously is better?(since the usage different symbols for C and Cop in the previous video) Then every contra- functor may just constructed wih aid of the functor C to Cop. I think this may be helpful.
@@M0n1carK Yes, exactly. At one point we have to face a choice whether we prefer, say for groups, f(ab)=f(a)f(b) over f(ab)=f(b)f(a). I feel the first is nicer 😅 Whatever is then studied in CT should then be an extension of "familiar" constructions, hence I like to ignore contravariant functors 😀
Is Cayley's lemma in group theory a special case of Yoneda lemma? The lemma basically says you can consider each group element as a group action on the some set, which is same to this philosophy. Now I am getting the what is a philosophy of category theory. Category theory wants to investigate the interactions between objects, not the internal structures of the objects.
Thank you for the reminder: I already forgot about this lovely example. The answer is YES!
Well, it is rather the Yoneda embedding than the lemma itself: Every group is a subgroup of Aut(X) for some X, that is Cayley. Every category is a subcategory of End(C), that is Yoneda. The connection goes via taking C=G, the “group category” with one object and arrows being the group elements with the induced composition. For a precise and detailed discussion see math.stackexchange.com/questions/1701/yoneda-lemma-as-generalization-of-cayleys-theorem
And YES! again: you summarize very nicely what category theory wants to study. I personally like to replace “interactions” with “relations”, but then we are on the same page ;-)
Let me try to cook up a one-to-one correspondence. Gold foil, action of shooting, particles, observations, interpretations do they corresponds to category C, functors F:C->SET, an object in C, functors F:C->SET, SET, SET?
Sorry I am not following 😰 What is it you are trying to do?
I'm trying to find a one-to-one correspondence between the mathematical objects in the theorem and the objects in the physical experiment@@VisualMath