Why is every introduction to homotopy theory so categorical? Is category theory necessary to know for calculating homotopy groups?

Great content thank you

Lovely video, too much noise.Thanks for the attempt though

Surface(cos(u/2)cos(v),cos(u/2)sin(v),sin(u)/2),u,0,2pi,v,0,2pi

around 5:50 the conjugation g(\phi) should actually be g*\phi*g^(-1) in order for the later calculations to be correct, right?

Very beautiful and helpful

Your voice ♡♡♡♡♡

Sleepy ahh vibe 😂

Is \gamma really from M to TM? I'm not sure that makes sense.

Yoneda’s lemma and its dual allow any small category C to be described from morphisms into and out of it also known as its hom-set. Given an object X ∈ C there exists a functor Hom(-,X): C^op -> Set describing morphisms into C. For morphisms out of C there are also exists a functor Hom(X,-): C -> Set. This provides a concrete way to implement Grothendieck’s relative point of view of considering morphisms of a category instead of objects of that category in order to understand a category. It is important to note that the functor Hom(-,X): C^op -> Set is a presheaf of the category C. The presheaves are the probing questions or morphisms into C as the maps f: - -> C you mentioned in your examples of a deck of cards and topological space.

Can this twisted form be thought of as analogous/related to a sesquilinear form?

loved it!

never ever play music while talking like this.

2:02 3B1B character?

why only one video :)) loved the explanation

Can't watch because of music. Pls remove.

So far i've watched 3-4 videos on homology/cohomology and they all had creepy music playing in the back and just an eerie vibe. From now on it will forever be creepy math for me

Nice! But why the background music?

Too much music, too little information.

Why are you doing everything to put one to sleep?! Is this some weird ASMR math channel?! Also, why steal from other creators?

Wow

Beautiful video.

The concept of Derivations, and the generalization of the derivative is really interesting to me.

Very cool! I have been very curious about the Yoneda lemma, and this was illuminating.

Why do you say that the tangent plane image is not a good way to think of the tangent space?

Dude turn up the mic.

"a section to the composition functor" what a testament to nLabs tendency to overcomplicate some things.... As if higher math isn't already complicated enough.. No wonder nLab is unusable for many...so unecessary

Please turn the volume up during editing

I would argue that a "nicer" example of a category enriched over itself is the category of k-vector spaces (or more generally, R-modules). Indeed, the set of linear maps between two vector spaces is a vector space, and composition is bilinear (so induces a canonical linear map from the tensor product of hom-spaces). Great video!

good explanation but the music is distracting.

I hate when people record videos forgetting about audio quality

First time that I ever managed to comprehend anything related to this topic. Wikipedia and ncatlab are very good websites, and yet they are bad for introduction to this topic. This is the first video that I watched from this channel about this topic, since I believe in general case first, special case later if ever, and still managed to comprehend the definitions, even if not the motivations or the reasons for the names.

baza yobanaya

It's pronounced "tah-pology" not "tope-ology". It's a small error, but it reveals you've never spoken to a mathematician.

Fantastic explanation! I have not seen Yoneda's Lemma introduced so delicately before and it has been much needed.

Do you have a discord server?

very good explanation! thank you very much; ❤❤❤

6:49 I think colimits are called direct limits while limits are inverse limits. Since it's confusing, most people just don't call them that anymore

Very well done , best explanation on KZbin.

For the (y, 0) vector field (3:14) it was unclear exactly what was being integrated and with respect to what? I assume we are integrating "y" with respect to theta along the circle? OK, later (7:19) I noticed that you are integrating y dx + 0 dy which is the line integral of the tangential component of the vector field with respect to the arc length in the language of vector calculus. I don't know if it is the same thing in the language of differential forms---I am still confused about whether "dx" in differential forms refers to an infinitesimal or a basis of a co-vector. Thanks.

Thank you for sharing your thoughts about such advanced and profound math problems. I think your videos are quite illuminating and I would like to express my respect for your worthwhile work!

Excellent cursory introductory look at a complex subject. It's all starting to take shape in my mind.

Man, you need to improve your diction.

CLARIFICATIONS: 1. At around 0:40 I say something to the effect of "if V_K is isomorphic to W_K as K-vector spaces, it may not be the case that V is isomorphic to W as F-vector spaces". I'm wrong there, that will always be true, since dim_K(V_K)=dim_F(V)=dim_K(W_K)=dim_F(W). I should have brought this point up later on when I had introduced taking G-invariants. The point I should be making is "if X and Y are isomorphic as K-vector spaces, it may not be true that X^G and Y^G are isomorphic as F-vector spaces". The intuition is that, in this latter example, the G-actions on X and Y may not be related, but the G-actions on V_K and W_K are. 2. The clarification (1) was necessary because dimension completely characterizes finite-dimensional vector spaces up to isomorphism. But one can show that the whole theory discussed in the video applies to vector spaces equipped with "(p,q)-tensor structure"s. The structure of an algebra is an example of such a structure. Dimension does not completely characterize algebras up to isomorphism, so it is possible for V_K and W_K to be isomorphic as K-algebras but V and W are not isomophic as F-algebras

Stumbled upon this when looking for a more intuitive explanation of what tangent spaces are. Pretty good video, would be even nicer if you spoke louder tho ;)

So succinct and great amk

Hole at infinity?

thanks! I liked the part of the cards; ❤❤🦊

excellent video

Platinium end