The Yoneda Perspective 

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.

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.

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!

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

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!

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

Very well done , best explanation on RU-vid.

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

Good Video; you explained it well.

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

excellent video

Well done

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

That's interesting sir

Platinium end

baza yobanaya

😷 MASK

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