I would say it’s the heart of geometry, rather than saying topology is distinct from geometry. It loses its meaning when it’s distinction from geometry is emphasized: Important though this first order concept may be for the logician/computer scientist. The question of what the essence of geometry is , is not answered by geometry itself! There is true maturity that develops when one uses the ‘system’ to transcend itself: Into the field of group actions on topological structures. To emphasize this vocabulary’s distinction from geometry: removes its life. Though one has the opportunity to indeed Emphasize the distinction. @1:30 As someone who does construct characteristic classes in the name of this deeper form of geometry: Topology: and in particular the study of topological invariants, It is backwards … or, to me, strange to say it’s done ‘for category theory’: Unless maybe youre a computer scientist with a different sense of the value this concept has - The beginning of categroy theory, more naturally, start’s with grothendieck’s Concept of there being classes of objects: which reinterprets the euler characteristic in many different ways: e.g. (1) the edges of a 2-manifold’s subdivision are thought of as a vector space…. Although, my appreciation is probably shallow in comparison to a computer scientist: So maybe eventually in my understanding it’s not truly backwards, But as a geometer, yes it actually is backwards. I apologize for accusing you of emphasizing a distinction: I see that’s what the rest of the video is for. @3:33 One is bringing up noether: one is interested in the property of an abstract ring - Of every ideal : for Z, nZ, is finitely generated. This is simply an observation for this particular ring, but is a coveted property for an abstract ring. There is an implication of there being a readily available construction. I actually work for a literal construction worker in exploration of this concept in mathematics: ‘Construction’ Obviously, the conclusion here is that you have a computer scientists understanding of topology. These chains in homology, Or ‘diagrams’ - which is a degrading term and not representative of the deeper meaning that the diagrams encode - is an excellent technology in understanding the primary structures, … but is only one high-way system on a fertile earfh which is richer a priori: This is the sheaf-theoretical topology, perhaps one could say in the sense of chebyshev: of there truly being Data of epsilons; Instead of using the noncommutativr algebra to hide from analysis. The fertile earth is richer a priori: And the diagrams is a culture built on a very fine fabric of one facet of wealth held in the kernel a prior.
You guys don't counter question each. Instead asks questions which will lead you guys to the next segment of the video. Which fails the method of Socrates.
@dailymemigzugxoyditsi3273 yes, you are right. Sometimes we do it. I’ll improve it for the next videos. Please, tell me us how to do it better in your opinion 😉
@@addisonkendal9867 yes we had an introductory video on the channel proposing videos using the Socratic method. But we are experimenting with different formats, so we took that video down. Next video will have a little different format. Let us know what you think about it 😎
Amazing video for such complex topic, really a good middleground for those who is not a professional mathematician but still has basic understanding. Actually love how you managed to explain functor in 1 minute
One of the many triumphs of this channel is how rigorously it treats "the roadmap." Obviously, any one aspect must necessarily lose a lot of resolution... but anyone new to these regions of math were going to forget all of that anyways. Rather, your depth is the width of the entire territory itself... meaning that I will waste SIGNIFICANTLY less time once I begin the journey of rigor through any one specific aspect. I really wish more channels would not only take this approach, but build on it. Thank you!
@@96capitainek thanks!! Sofia and I want to take all the video suggestions we receive, create simple explanations and post videos about them. Please, let us know something you’d like to have a simple explanation about 😎
My professor was never a fan of the closure requirement and I agree with his reasoning. An operation is a function. A function is a triple (G, X, Y) where X is the domain and Y is the target. If you say that a group has a set and an operation "on that set", you've already stated the requirement that the domain and target of the function of the operation must be that set (or products of it). There is nothing left to check or require as far as closure is concerned.
Yeah, and in the end you have a group operation with a closure property: So that’s an interesting psychological bent your professor has, With no new contribution to the group concept, Because the operation has this closure property. That’s its nature.
Closure is necessary. For example define the set {-1, 0, 1} with the addition operation. This satisfies all the axioms except there is no closure. 1 + 1 = 2 which is out of the set. So it's not a group.
@@smolboi9659 So that’s not a group. That operation doesn’t have the closure property: So I don’t know what your professor is contributing - but no matter how one can logically twist things together, the nature of this concept is that the transformation will have a closure property. That’s correct.
@alessiogarzia6632 Grazie Alessio. (I’ll answer in English because the channel is in English) We are planing to start a channel in Italian as well, but not in the next few months since we need to focus on growing this channel here first. Im glad you like our content 😎
Right, I'm convinced that this video is made with AI. To quote: "The group is the set of keys, and the operation is trying to fit the key into the keyhole" First of all, wrong terminology, second of all, this is obviously a language model misinterpreting what "operation" means in this context, as it is absurd. From that it is also clear that you guys do not know what you are talking about in favour of pumping out as many videos as you can. Also, the description reeks of generative AI, it's genuinely disgusting to read. Please, do better, fact check the stuff that is in these videos, for fuck's sake.
You’re totally right about the description, which we’ve now deleted. This video was published 7 months ago, and we’ve learned a lot since then, (like not using AI and AI generated images). So I’d say that our more recent videos are an improvement with content and fact checking! We try to be better and better every video, and we’ll improve even more in the future :)
@@error.418 well the point was to simplify its definition using an analogy. But let us know what exactly you did not like and what analogy/illustration would better describe it in your opinion please
That’s very rude: This formal concept could mean a true diversity of things. If you feel this way, you should specify that you actually have a specific relationship to it: and then there’s no poor description here. There’s maybe a specific understanding. That’s about it. Perhaps the vocabulary of group actions on topological spaces is more lively than the computer scientific group: But there’s nothing poor about this explanation.
@@dibeos I don't understand what the lock and key analogy was trying to say, and I've studied group theory. That said, the later explanation of groups was fine.
I would say it’s the heart of geometry, rather than saying topology is distinct from geometry.
It loses its meaning when it’s distinction from geometry is emphasized:
Important though this first order concept may be for the logician/computer scientist.
The question of what the essence of geometry is , is not answered by geometry itself!
There is true maturity that develops when one uses the ‘system’ to transcend itself:
Into the field of group actions on topological structures.
To emphasize this vocabulary’s distinction from geometry: removes its life.
Though one has the opportunity to indeed
Emphasize the distinction.
@1:30
As someone who does construct characteristic classes in the name of this deeper form of geometry:
Topology: and in particular the study of topological invariants,
It is backwards … or, to me, strange to say it’s done ‘for category theory’:
Unless maybe youre a computer scientist with a different sense of the value this concept has -
The beginning of categroy theory, more naturally, start’s with grothendieck’s
Concept of there being classes of objects: which reinterprets the euler characteristic in many different ways:
e.g. (1) the edges of a 2-manifold’s subdivision are thought of as a vector space….
Although, my appreciation is probably shallow in comparison to a computer scientist:
So maybe eventually in my understanding it’s not truly backwards,
But as a geometer, yes it actually is backwards.
I apologize for accusing you of emphasizing a distinction: I see that’s what the rest of the video is for.
@3:33
One is bringing up noether: one is interested in the property of an abstract ring -
Of every ideal : for Z, nZ, is finitely generated.
This is simply an observation for this particular ring, but is a coveted property for an abstract ring.
There is an implication of there being a readily available construction.
I actually work for a literal construction worker in exploration of this concept in mathematics:
‘Construction’
Obviously, the conclusion here is that you have a computer scientists understanding of topology.
These chains in homology,
Or ‘diagrams’ - which is a degrading term and not representative of the deeper meaning that the diagrams encode - is an excellent technology in understanding the primary structures, … but is only one high-way system on a fertile earfh which is richer a priori:
This is the sheaf-theoretical topology, perhaps one could say in the sense of chebyshev: of there truly being Data of epsilons;
Instead of using the noncommutativr algebra to hide from analysis.
The fertile earth is richer a priori:
And the diagrams is a culture built on a very fine fabric of one facet of wealth held in the kernel a prior.
You guys don't counter question each. Instead asks questions which will lead you guys to the next segment of the video. Which fails the method of Socrates.
@dailymemigzugxoyditsi3273 yes, you are right. Sometimes we do it. I’ll improve it for the next videos. Please, tell me us how to do it better in your opinion 😉
Why did you say that? Are they supposed to use the Socratic method?
@@addisonkendal9867 yes we had an introductory video on the channel proposing videos using the Socratic method. But we are experimenting with different formats, so we took that video down. Next video will have a little different format. Let us know what you think about it 😎
@@addisonkendal9867 they had a channel video intro in which they mentioned they used this method.
Amazing video for such complex topic, really a good middleground for those who is not a professional mathematician but still has basic understanding. Actually love how you managed to explain functor in 1 minute
@vnshngpnt we are glad you liked it, Anton!! Let us know what other topics you’d like to see explained simply 😎
1.Calculator
2.Pencil case
3.Pen
4.Math notebook (preferably with grids)
5.Formula sheet
6.Textbook
Category theory at last! Amazing!
Thank you dear Di Beo's
One of the many triumphs of this channel is how rigorously it treats "the roadmap."
Obviously, any one aspect must necessarily lose a lot of resolution... but anyone new to these regions of math were going to forget all of that anyways.
Rather, your depth is the width of the entire territory itself... meaning that I will waste SIGNIFICANTLY less time once I begin the journey of rigor through any one specific aspect.
I really wish more channels would not only take this approach, but build on it. Thank you!
Thanks so much for your feedback, we’re really glad to hear that (and completely agree)! Let us know what area you’d like “roadmapped” next ;)
Wow, your work in explaining such an abstract topic as category theory is incredible! I love the video!
@@96capitainek thanks!! Sofia and I want to take all the video suggestions we receive, create simple explanations and post videos about them. Please, let us know something you’d like to have a simple explanation about 😎
@@dibeos If you can go through spinors or other complex manifold stuff, it would be awesome!
@@96capitainek thanks, you’re not the first one who tells me that… I will add to the list of next videos in the channel right now!
HE DID IT 🙌
@@alextrebek5237 oh yeah!! 😎🤙🏻
what are the six things...?
1. Group theory 0:29
2. Abstract algebra 2:51
3. Algebraic topology 5:12
4. functorial semantics 6:41
5. Topos theory 9:21
6. Higher category theory 10:36
Hope that helps!
My professor was never a fan of the closure requirement and I agree with his reasoning. An operation is a function. A function is a triple (G, X, Y) where X is the domain and Y is the target. If you say that a group has a set and an operation "on that set", you've already stated the requirement that the domain and target of the function of the operation must be that set (or products of it). There is nothing left to check or require as far as closure is concerned.
@@neildutoit5177interesting, I’ve never thought about it this way
Yeah, and in the end you have a group operation with a closure property:
So that’s an interesting psychological bent your professor has,
With no new contribution to the group concept,
Because the operation has this closure property.
That’s its nature.
Closure is necessary. For example define the set {-1, 0, 1} with the addition operation. This satisfies all the axioms except there is no closure. 1 + 1 = 2 which is out of the set. So it's not a group.
@@smolboi9659 I think his point is that in this case addition cannot be considered a valid operation for this set
@@smolboi9659
So that’s not a group.
That operation doesn’t have the closure property:
So I don’t know what your professor is contributing - but no matter how one can logically twist things together, the nature of this concept is that the transformation will have a closure property.
That’s correct.
Very informative accessible video. Thank you :)
@logosecho8530 glad you liked it!! Let us know what you are interested in please. This way we can post videos on that.
My lovely show(I mean an rubric) come back
As always :❤
@@SobTim-eu3xu thanks!!! 😎
are you Italian?
@@Topological_Space hi Alessio, I am born in Brazil but we do live in Italy (Udine)
@@dibeosnon ci credo adoro i vostri video
🤌🤌🤌🤌
@alessiogarzia6632 Grazie Alessio. (I’ll answer in English because the channel is in English) We are planing to start a channel in Italian as well, but not in the next few months since we need to focus on growing this channel here first. Im glad you like our content 😎
@@dibeos que? Brasil??? Ólhó.
Right, I'm convinced that this video is made with AI.
To quote: "The group is the set of keys, and the operation is trying to fit the key into the keyhole"
First of all, wrong terminology, second of all, this is obviously a language model misinterpreting what "operation" means in this context, as it is absurd. From that it is also clear that you guys do not know what you are talking about in favour of pumping out as many videos as you can. Also, the description reeks of generative AI, it's genuinely disgusting to read.
Please, do better, fact check the stuff that is in these videos, for fuck's sake.
You’re totally right about the description, which we’ve now deleted. This video was published 7 months ago, and we’ve learned a lot since then, (like not using AI and AI generated images). So I’d say that our more recent videos are an improvement with content and fact checking!
We try to be better and better every video, and we’ll improve even more in the future :)
That's the worst explanation of what a group is I've ever heard
I disagree with you here, I think it was explained quite well considering the sole Concept of a "group" is plain simple.
@@error.418 well the point was to simplify its definition using an analogy. But let us know what exactly you did not like and what analogy/illustration would better describe it in your opinion please
@@sisyphs thank you. Yes, it is simple. The goal of the key and locks analogy was just to illustrate its usefulness and purpose
That’s very rude:
This formal concept could mean a true diversity of things.
If you feel this way, you should specify that you actually have a specific relationship to it: and then there’s no poor description here.
There’s maybe a specific understanding.
That’s about it.
Perhaps the vocabulary of group actions on topological spaces is more lively than the computer scientific group:
But there’s nothing poor about this explanation.
@@dibeos I don't understand what the lock and key analogy was trying to say, and I've studied group theory. That said, the later explanation of groups was fine.
cool
@@ZeubZebi-f6i 😎