Really good video, you really helped me with this topic.

Thank you 🙏

That is a really great explanation. Thanks

Wonderful explanation

Looks like a super pullback

This was awsome

Amazing!

These are great videos on category theory. Very well done.

H...how did you know I was eating chocolate cake???

"It's a naturality square! Stop the clock!!" hehe

jolly well...haha. Good set of videos!

Does anyone else get the distinct impression that this is what it's like to be taught Category Theory by a Timelord? Awesome videos, Simon!

I appreciate you!!!

Just to say a much belated (by 6 years) thanks for the clear explanation between a collection and a set. Must look further into the basics of this intriguing subject though as, from my naive understanding, Russell's paradox seemed to show that the set of all sets is an abstract entity that cannot exist (as you point out). It seems a little odd that its non-existence could be altered by changing its name from set to class (so I'm still clearly missing something!) - maybe there is an axiom of set theory that the demonstration of Russell's paradox relies on, that does not apply to these bigger 'classes' (so side-stepping the paradox showing non-existence)? Hopefully will make bit more progress that this in the next 6 years! :)

Like

5:00 These non-examples were really helpful, thanks.

Thanks, I watched all of your videos. very useful likeeeeeeeeeeeeeeeeeeeeeeee

Thank you so much. Finally I understand this stuff

Beautiful.

Could you please link the follow up videos proving the Yoneda Lemma and that Yoneda Embedding is an embedding?

2:00 Can't see the board.

Thank you

keep going and you'll discover knot theory.

Wonderful videos. Great insight into Category Theory!

The "AxB versus BxA" example of "unique up to unique isomorphism" around 7:40 was very enlightening.

I am a huge fan of your videos, and I believe that the videos are fine as they are right now. There are so many books and lecture notes on categories to obtain tons of examples, and I don't think audience is supposed to grasp the concepts just by watching these videos but by getting their hands dirty somewhere else. I hope you guys keep making these videos no matter what criticism you get (though it may be just my selfish wish).

Smile happens while watching your lectures. I like your style very much. Thanks:)

Love these videos.

I don't understand your question. When in the video are you referring to?

Generally speaking in mathematics, if you have a map f:X->Y then sometimes you can 'pullback' something from Y to X. For instance, categorical pullbacks seem to have their roots in pulling back a bundle E->Y to one over X, sometimes written f*E->X. In integration you have a function g:M->R you wish to integrate on some space M. when changing coordinates you will have some function f:U->M, from some nice space U: you pull back the function to U and then integrate on U. Very brief, sorry!

Oh, sweet, a presentation of this definition that actually goes into detail on the natural isomorphism in question! I know you even allude to lack of detail in some other sources, but I've gotten stuck on this issue so many times with so many different pieces of CT lit it's not even funny..

I'm new to category theory and got a bit lost in this lecture. Are there lecture notes or supplementary readings? I followed the limits lecture series up to here, but got quite lost when talking about the existence of a natural iso between the morphisms C(V,U) with the totality of cones with Vertex V over D. I see how this implies U is a limit, but not how a limit implies the existence of the natural iso. Once the yoneda lemma hit, I was completely lost.

At 7:28 I think the other triangle identity comes from 3 not 4, by setting g=ey, x=GY and Y'=Y.

This is a great introduction. Thanks for sharing!

is there any relation between the limit in calculus and the limit in category theory. can anyone explain it?

Much more explicative than the MacLane-Eilenberg book, at least up to this point. They explain this in the third chapter, and they use an awfull amount of technology. In fact, this goes right after the Yoneda lemma.

truly inspirational

Many thanks for these instructions!

The product x is the cartesian product of sets. The identity map id: G -> G is the function which sends and element to itself. On the other hand, e is the unit element in the group G, but we can think of it as the corresponding function * -> G which sends the the one element of * to e.

For any set S you have a canonical isomorphism S ~ S x *.

1:33 "The thing that you have in sets: the obvious isomorphism" What isomorphism?

On the associativity (6:21) The cross product is referring to the group product? Or Cartesian product (and how would that work)? Or ... ? Cannot really read the diagram (too small) at 6:48. Nor the unit law at 7:18. I'm also not clear on e vs id.

cool thanks!

ohhhhh i get it now this so exciting

I wish more instructors had this kind of enthusiasm! Amaaaaaaazzzing!

You all are awesome! Your doing a great thing for us aspiring mathematicians that don't get much for lectures, (or not so well done ones). Thank you.

"it jolly well ought to" :D I am an American so I am entertained by such things.

Hello & thank you for your videos! I could not find good introductory material for category theory earlier on when I needed it; I don't know how I missed your videos searching youtube for Category Theory but anyway I found them now searching for monoids and semigroups.

2:20 listening to people trying to explain how something can be vacuously true is my favorite past time=p=p it makes sense, and yet it always sounds like it doesnt

Very impressive. To my mind, the term string diagram is not correct. It's interesting - who inventing such a term and why?(Im not well-verse in category theory)