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Thanks for your comment! You are very welcome ☺

Thank you for the feedback, I am glad that you enjoyed the topic ☺

Agreed 😁 I hope you enjoy AG ☺

When I look at the big RUclips channels and their questionable quality, then I rather stay small 🤣 Haha, just kidding. I am glad that you like the channel, you feedback is really appreciated ☺

Yes, that should be the difference between the multiplicative group and the additive group. But they are not isomorphic (equivalent) - not sure what you mean with that 🤔

@@VisualMath Sorry, I think I was wrongly conflating the multiplicative group with the multiplicative group law and the additive group with the additive group law.

@@Jaylooker Ah, no worries. I get confused all the time. The way I remember that they are not the same is via the coordinate rings (polynomials versus Laurent polynomials) 😀

@@VisualMath Good point. That clarifies things. I think the same maps appear again with Laurent polynomials having rings R[t, t^-1] and polynomials having rings R[t]. These are still related. The localization of a commutative ring S away from an element s ∈ S is a universal way to invert s. One example is localization of polynomial ring Z[t] which is the Laurent polynomial ring Z[t,t^-1] which provides one map. The other map is the identity of a polynomial ring. Localization also apply to categories and this localization of categories is what I had in mind when inverting the Lefschetz motive.

Oh yes, infinite loop spaces are tricky 😅 Nothing planned right now, but I will see whether they fit naturally somewhere.

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 😀

Welcome 😀

Ah, thanks, that is right. I will put a warning in the description 😀

@@VisualMath I am waiting to see the video about scheme🙂. It is hard for me to understand the concept and its uses and differences between algebraic variety and scheme.

@@tim-701cca Yes, that is a tough one. We will see how that goes when we get there 😅

I am glad that the radical now makes sense 😀 I can feel you: sometimes it takes me years to understand something. That is why talking with people is so important 🙂

Haha. Who can claim that they understand sheaves? I don't, I just make videos about them 😂 But they somehow work to well to ignore them...😀

@@VisualMath I certainly don't understand them 😂 I get the examples, but that's a very different thing

I am glad that you liked the video, and I hope it will be useful. Enjoy our AT journey ☺ P.S.: I go by they/them, so “sir” could be improved.

The example to keep in mind is the alternating group A4: it is solvable but not supersolvable as the Klein four group Z/2Z x Z/2Z appears (that one is not cyclic). That the alternating group A4 (or rather the symmetric group S4) is solvable is the reason why there is a formula for the roots of polynomials of degree 4. Thus, the notion supersolvable is not enough for polynomials and that is why we need the generalization solvable. I hope that makes some sense 😀

@@VisualMath Really helpful! It reminds my mistake. For a supersolvable group, it must have additionally Gi is nomal subgroup of G ( which I have carelessly ignored ). And moreover, it also reminds me that when a normal series is refined to a composition series, the factors must be prime cyclic. It is equivalent and goes well! Sry for my mistake and tks for your help!😄

@@M0n1carK Excellent ☺ I hope you will enjoy algebra!

Thanks for checking in, its always good to have you here ☺ Sphere packing is certainly fun. Last time I checked not that much was known (for the nonregular or lattice case), but you are correct that the fields is growing very fast. I will have another look. I enjoy doing the AG series - thanks for the suggestion!

Thanks, that is a nice analogy. Its a good companion to the “sheaf on a graph” picture that I like a lot 😀

Hah, another misguided field. Just kidding 😂Maybe what I should have said is "in nonscientific context" 🤔

@VisualMath the misguidedness is peaking 😞 BTW, I think the sort of finite state automaton you use for marrow chains is called a Diagram 😭

@@mrl9418 Haha, “Diagram”, what is not a diagram 🤣

@@VisualMath Now that question is on my mind, only unironically. 🤔😭

Thanks for watching 😀

Thanks for watching 😀 I guess you are studying some form of algebraic geometry? At the moment I am not planing anything on the Weyl algebra, but we will see what the future holds.

Thanks for watching 😀 Indeed, it should “not equal zero”; sorry for the typo.

Hmm, that is an interesting question. In the usual categorical setting, I have never seen the notion of “equivalence of functors”. However, when you go to higher categories, then there are many more notions of “equal”, so you should get the infinite hierarchy if you go to higher categories. Maybe these two links help? mathoverflow.net/questions/402558/does-there-exist-a-definition-of-equivalence-of-functors mathoverflow.net/questions/7666/lax-functors-and-equivalence-of-bicategories?rq=1

Thanks for the feedback, I am glad that you like the series. I enjoy doing it and your feedback is very much appreciated ☺

It depends where to put the emphasis 😂 My take is that the objects do not matter. Not in the sense that you do not need them, but rather that you should not care about them 😀

Thanks for watching, you are welcome ☺

Welcome, I am glad that you liked it ☺

Well, nobody is perfect 😅 and every subscription model (free or paid or in between like RUclips 😁) has advantages. Even Python has some advantages 🤣 Anyway, thanks for the additional references, those might indeed be useful for someone.

Haha, I am glad that you liked the long ramble 🤣 Thanks for watching ☺

Yes, its one of my favorite analogies in representation theory. Thanks for watching ☺

Hmm, sorry to hear that. Let us try to improve: can you help me and be specific?

I do not know the book. Sounds fantastic, I will have a look. Thanks 😄

That is tempting, but I don’t know how to tie this together. I know that the Cantor staircase and set as in the video come from SL(2,\bar{F}_p) which is not very far away from SL(2,Z) or even GL(2,Z). Maybe one could try some lifting theory? Hmm...🤔

@@VisualMath There is the Hasse-Minkowski theorem for quadratic forms which is an if and only if statement. Quadratic forms can be described by SL(2, Z) matrices.

Yes, it is. And everything comes from SL2 ☺ Anyway, I am glad that you enjoyed it 😀

The question is not silly at all! It assigns the whole vector space, not just elements, to vertices and edges. In general, a sheaf tries to associate “rich” data to open subsets: having a vector spaces is much better than just having a vector! Even better, with a vector space at hand, we can talk about maps, and they are the key players in all of this. Your Möbius strip example sounds like you have the following in mind (correct me if I am wrong): it is a line bundle over the circle that looks locally like a cylinder, but is not a cylinder globally. That is exactly the type of situation sheaves like. In AG a line bundle is often called “invertible sheaf”, and one can indeed use sheaf theory to prove the statement you mention. (Essentially an invertible sheaf of degree 0 has no non-zero sections unless it is the trivial sheaf.) So, yes, its exactly like you describe it. I would however say that using sheaf theory to prove the nonexistence of such a cut is a bit of an overkill 😅 Anyway, thank you for watching ☺

That is one of my favorite pictures 🤣

Haha, never underestimate graphs 🤣

Categories categorify sets, so you are wondering what categorifies the notion of “set cardinality”? Hmm, seems a natural question, but I am not sure I have ever anyone seen talking about this.

@@VisualMath Yes, because I think the cardinality of a collection of mathematical objects is related to whether it is not a set or a proper class. As in, possibly at its limit as a cardinal is it a proper class. After looking a bit, this refers back to the Russell-Frege definition of cardinals as proper classes. Maybe a similar statement could be made between small and large categories using something that describes their size? This would be reflected in whether the collections of objects in a given category is equivalent to set or equivalent to a proper class.

@@Jaylooker Hmm, I have indeed never seen anything in that direction. The reason why might just be something silly, like most category theorists do not like set theory 🤣

@@VisualMath Approaches to smallness and largeness between categories in relation to cardinality and sets are discussed in “Set theory for category theory” (2008) by Shulman. In section 8 The largeness of categories in relation to inaccessible cardinals (ie uncountable infinites) is discussed. Important to note is that by definition inaccessible cardinals are excluded from being built from bigger sets using Zermelo-Fraenkel set theory with the axiom of choice (ZFC). Example 8.1 describes the presheaf functor for large categories I thought of in my first comment but excluded. In 9 the largeness of categories is relative to Grothendieck universes. Grothendieck universes model ZFC without the axiom of infinity. From this it is apparent that inaccessible cardinals are proper classes and gives in what sense they are larger than sets. The Vopénka principle and other large cardinal axioms affect sets and the categories derived from them. One proposal mentioned to categorify large categories is to follow the elementary theory of the category of sets (ETCS) foundation and define the category set as an elementary topos with a natural number object and axiom of choice. This proposal has small categories equivalent to internal categories of sets and large categories equivalent to indexed categories relative to an elementary topos. Another approach is by algebraic set theory with a category of classes which contains large categories. At the end he also brings up the use of 2-categories to describe large categories. I think there is some merit to it. I think an issue that needs resolving with it that any sheaf over a Grothendieck site is a stack (2-sheaf). Maybe conditions imposed by sheafifying a presheaf allows the transfer from small categories (presheaf) to large categories (sheaf)? A convergent limit of a functor is given by the Taylor tower approximation in Goodwille calculus which may imply a large category. Sheafification approximations the sheaf of a presheaf. Note example 7.7 which considers localization of a large category similar to Bousfield localization. Grothendieck sites behave like open sets of topological spaces. The definition of an open sets of a topological space sounds similar to a Grothendieck universe. Maybe the sheaf is a large category with respect to its Grothendieck site?

Welcome, I hope you enjoyed the video ☺

Haha, welcome 😀

I am glad to hear that, thanks for the feedback ☺

That is right, but I haven't been in Germany for a while 😅 The situation seems to be pretty bad, so let us hope it will not get worse.

I like to think of graphs as the easiest nontrivial structure where sheaves make sense. So sheaves on graphs are for me always the "baby example" I like to keep in mind. The categorical perspective then, as usual, works well if you already know what a sheaf is from examples in the wild 😀

@@VisualMath Good point. It is important to have examples and applications of a mathematical object. Your example of a cellular sheaf is interesting and natural to consider after working with matroids.

Yes, the theorem is excellent, I am glad that you like it ☺

Haha, exactly 🤣 Complex analysis is full of scary theorems 🙃

Welcome ☺

Me too 🤣 Anyway, I am glad that you like the theorem!

Haha, great. I hope you enjoy AT ☺

@@VisualMath thanks :)

Thanks for the feedback, I hope you will enjoy linear algebra ☺

Thanks, I am glad that you liked the video ☺

You did a great job figuring that out 🤣

Ah, I didn't knew those examples. Thanks for sharing ☺