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!
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!
@@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.
@@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
