The Yoneda Perspective

Credits:
I use a (modified version of) 3Blue1Brown's animation library "manim".
The main inspiration was this series of blogs on math3ma (an AMAZING website)
www.math3ma.com/blog/the-yone...
www.math3ma.com/blog/the-yone...
www.math3ma.com/blog/the-yone...

[@Jaylooker]

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.

[@sadface7457]

This is quite profound. This lemma is perhaps the foundation for the knowledge graph which stores representation of objects and entities and hyperlinks between them.

[@aron8999]

I think you're just thinking of a graph.

[@sumitmittal88]

I think me meant knowledge graphs ie multi layered graph HNSW

[@JakubWaniek]

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!

[@gabrielrhodes9943]

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

[@Karolkoks6]

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!

[@gabibensimon9]

Very well done , best explanation on CZcams.

[@sadface7457]

There are so few channels exploring what would be considered advance mathmatical topics. Its incredible. Hats off.

[@nathanryan12]

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

[@AlessandroZir]

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

[@arghyachakraborty1151]

excellent video

[@ignaciogimon6894]

Good Video; you explained it well.

[@JosiahWarren]

Well done

[@IshanBanerjee]

That's interesting sir

[@AlessandroZir]

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

[@sdfdsf4162]

baza yobanaya

[@bragzanma725]

Platinium end

[@dianadsouza7380]

😷 MASK

[@annaclarafenyo8185]

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

[@paulpetricevic6949]

boy what

[@imperfect_analysis]

Or that he's not studying math in the UK or the US
Girl/boy what's your damn problem? Not all mathematicians are English language masters

[@annaclarafenyo8185]

@@imperfect_analysis I know, just providing context so his video can improve. I had the same problems starting out, you know "Yoo-ler", "Ho-mo-to-py", etc. The comment is not mean spirited in intent.

[@imperfect_analysis]

@@annaclarafenyo8185 alright:) sorry if I sounded mean but you're right

[@greenland8376]

@@annaclarafenyo8185you cant just claim afterwards that it wasnt mean spirited when it clearly is. claiming that he „never spoke to a mathematician“ is dumb and hurtful. try better