The Yoneda Embedding Expresses Whether, What, How, Why

[@chineduecheruo8872]

This is AMAZING!!!!!! Thank you!!!!!

[@jacquesfaba55]

As a zen koan goes: What is the sound of one homomorphism clapping?

[@nunoalexandre6408]

Love it!!!!!!!!!!!!!!!!

[@math4wisdom]

Nuno, thank you for your encouragement! I hope to make more videos that you like. You and all are welcome to join our Math 4 Wisdom discussion group www.freelists.org/list/math4wisdom

[@numb2023]

It was PERFECT.

[@math4wisdom]

THANK YOU! and I am so curious to learn more about you and your interests!

[@chineduecheruo8872]

Wonderful framing! Thank you. So deep.

[@chineduecheruo8872]

This is great. Please share more of you knowledge!

[@rafaelfreire3792]

This video was so helpful. Thank you!

[@math4wisdom]

Thank you, Rafael! I am curious, why are you learning about the Yoneda lemma? What do you find interesting about it?

[@rafaelfreire3792]

@@math4wisdom I'm studying tensors. The author in the source I'm using defines a tensor as a vector space of multilinear maps between V* and V, in which V* = Hom(V; R). I'm in my second year of graduation in computer science and is the first time I'm seeing abstract algebra so I'm having a hard time with it and a bit lost. Nevertheless your video helped a lot trying to understand homomorphisms.

[@math4wisdom]

@@rafaelfreire3792 Wow! Thank you for explaining. I am glad to know this video helped. I do have lots more to learn about tensors myself. And we can think about what the Yoneda Lemma is saying in this case.

[@BillsonTheRoad]

Good work!

[@math4wisdom]

Thank you, Bill! This is a key example of "Math 4 Wisdom". I look forward to talking with you about it.

[@redpepper74]

This was very thought-provoking, I liked the idea that these objects in category theory can take on the essence of the fundamental building blocks of questions in language. A lot of it did go over my head though. I like more abstract higher math, but as an undergraduate, I’ve been stuck doing calculus and linear algebra for a while. Are there any category theory resources you could recommend for beginners like me? I would particularly like something that includes exercises because playing with mathematical objects is how I do my best learning :)

[@math4wisdom]

Wonderful! Thank you for your comment. Eugenia Cheng has a readable book, "The Joy of Abstraction", which concludes with the Yoneda Lemma books.google.lt/books/about/The_Joy_of_Abstraction.html?id=N_GCEAAAQBAJ&redir_esc=y Lawvere and Schanuel, "Conceputal Mathematics: A first introduction to categories", available online, has lots of exercises and Lawvere is a leading thinker but it doesn't cover the most fruitful ideas, which are adjunctions and the Yoneda Lemma. Tai Danae Bradley's blog posts are very intuitive www.math3ma.com/categories/category-theory and she has a booklet "What Is Applied Category Theory?" arxiv.org/pdf/1809.05923.pdf The great educator John Baez has a tutorial "Lectures on Applied Category Theory" math.ucr.edu/home/baez/act_course/ which is based on Fong and Spivak's book "Seven Sketches in Compositionality: An Invitation to Applied Category Theory" arxiv.org/pdf/1803.05316.pdf More advanced books that I recommend are Emily Riehl, "Category Theory in Context", www.math.jhu.edu/~eriehl/context.pdf Steve Awodey "Lecture Notes" and "Category Theory Book" www.andrew.cmu.edu/user/awodey/SummerSchool/ Tom Leinster "Basic Category Theory" arxiv.org/pdf/1612.09375.pdf

[@math4wisdom]

Also, you and all are welcome to join our Math 4 Wisdom discussion group www.freelists.org/list/math4wisdom which is the hub of our community, including our study groups.

[@math4wisdom]

Steve Awodey's video series is excellent kzbin.info/www/bejne/eHeZnHt6Zql0m7c

[@redpepper74]

@@math4wisdom Wow that’s a lot of stuff! Thanks for taking he time to reply, I appreciate it :)

[@sumdumbmick]

the Wikipedia article says all of this, and more. just look there. how much of a pathetic victim are you?

[@wenaolong]

Wow. The Aristotelian correspondences to the four W-words are about as counterintuitive as they can be. But that makes sense. If one is confused between any element or object in a category, it seems likely that one will be confused about all of them.

[@math4wisdom]

Thank you for your remark. I am curious to know more about your understanding of Aristotle's four causes.

[@fbkintanar]

2:10 "taking us to the world of sets". My impression is that it is a bit more specialized, in that it (the Yoneda embedding functor) is taking us to the world of "sets of arrows/morphisms". An arrow has a bit more structure than a mere element, it has a source and target for example. If we were to think set theoretically, an arrow might correspond to a set of ordered pairs of elements. But category theory doesn't go there, it wants to avoid reductionist thinking of arrows into sets of ordered pairs of elements, because it doesn't want to talk about elements; that would tie us to reasoning at the level of sets and elements and that would not be guaranteed to generalize to categorical reasoning that transfers correctly to other categories. We aren't interested in the global set of arrows, we are interested in the set of local sets of arrows, local to a particular source and target. So the Y emb. functor, when it takes us to the category of sets, we are thinking of the target of the functor as a **family** of local sets of arrows.

[@ἀπόστολος14]

Are you effing kidding me! I kept on getting videos for the Yoneda Lemma just popping up everywhere. It was like God wanted me to know this and kept on putting it off, I finally gave in. I believe it the Yin and Yang switching with Christ has to do with the Yoneda Lemma.

[@the_allucinator]

You resemble a hierophant