Category Theory III 7.2, Coends

[@siyuanchen659]

I have completed all three series, from completely lacking knowledge of category theory to now being able to read many category theory literatures on the Internet. There is no mysterious jargon anymore. This is really the best course for programmers to learn category theory. The next thing to do is to read books, blogs, papers and waiting for the 4th part. Thanks again, Dr. Milewski!

[@DrBartosz]

Congratulations! Very few people complete the whole course.

[@Bratjuuc]

That was some wild journey. Deep thanks to you, Bartosz! This journey is one of it's kind and the tuition was unprecedentally cool and intuitive. We, the viewers, are so lucky the lectures like this even exist, I can't even express how lucky we are. This gift is just priceless.

[@quant-trader-010]

Finally completed all three series. What a journey! Thank you so much Dr. Milewski!

[@christianfrohlich3835]

I love your lectures! Do you consider making a series on Homotopy Type Theory? :)

[@tunairaiol]

Thank you so much for all your lectures. I hope you are doing well and can't wait to see more.

Thank you Dr. Milewski, you are an awesome teacher. I didn't study mathematics at university and I was still able to (mostly) follow through the whole course!

[@aDwarfNamedUrist]

This leg of the journey has come to a close - and what a journey it was! Bartosz, you are an excellent teacher of category theory, and this course greatly aided in my aspirations of learning it. Much thanks

[@IsaacAndersonMedia]

Miss your videos

[@JanKowalski-oq6ie]

Awesome series. Do You plan on doing 4th part?

[@DrBartosz]

Maybe

[@AussieDnB]

Please, please do. This series is spectacular, and we only managed to cover about half the topics listed in the first episode. I was really looking forward to Kan extensions, topoi, enriched categories, n-categories, and HoTT...

[@MyPs123456]

4th part + 1. Finished all three parts + the book. The books is also pretty well written but your lectures really made them much easier to understand.

[@clarejang9088]

1. Can I think the result of Ninja Yoneda lemma (in 3:04) as "Free functor of F is isomorphic to F if F is already a functor"? 2. Does continuous functor preserve end too?

[@DrBartosz]

1. What do you mean by free functor? The one obtained by the left Kan extension? 2. Strictly speaking, a continuous functor is defined to preserve all small limits. So it depends on your category.

[@clarejang9088]

I mean the one obtained by right Kan extension.

[@adrianmiranda5531]

Are you talking about the right kan extension of F along itself? I know that as the (underlying functor of the) codensity monad of F.

[@srghma]

At 31:39 you started defining ninja yoneda dual with "integral from infinity to x of C(x, a) ...", but in book (page 441) it's "integral from infinity to z of C(a, z) ..."

[@srghma]

Or maybe we have "ninja yoneda", "ninja yoneda dual", "ninja coyoneda" and "ninja coyoneda dual" and on video you made a proof of "ninja coyoneda dual", but in book you made a proof of "ninja yoneda dual"?

[@DrBartosz]

It all depends on whether the functor is covariant or contravariant.

[@DrBartosz]

In fact, somebody made a similar comment in my blog, and I have modified it since to be consistent. bartoszmilewski.com/2017/03/29/ends-and-coends

[@srghma]

tnx, I wrote what is what in my notes github.com/srghma/category-teory-bartosz-milewski-lecture-notes/blob/master/part%203/7%3A%20Natural%20transformations%20as%20ends%2C%20Coends/2.jpg github.com/srghma/category-teory-bartosz-milewski-lecture-notes/blob/master/part%203/7%3A%20Natural%20transformations%20as%20ends%2C%20Coends/5.jpg

[@clarejang9088]

It looks like direction of h in 9:18 is reversed.

[@DrBartosz]

Oops, you're right.

[@ivanli9468]

Haskell needs RankNTypes version of forall for End but ExistentialQuantification version of forall for Coend. So Haskell actually has got direct and precise sense of Exists for Coend. Is it right?

[@ShimshonDI]

By "infinite product" or "infinite sum" you mean potentially infinite, right? As in having the same number of inputs as objects in your category. Are ends/coends just a more common or useful notion in categories with infinite numbers of objects, or is this more a programmer bias (e.g. using the category Hask) than a math bias in general? Also, what's 8.1 going to be? Kan extensions? O.o When is it coming?

[@BartoszMilewski]

You can call it a programmer's bias.

[@timh.6872]

I personally like to think of it less as "infinite" and more as "indefinite". For a "finite" coproduct, all the possible objects that could have been injected into it are listed right there in the type, you just pick one to inject, or handle all of them to project. For an indefinite coproduct, you can inject whatever you want from the diagonal of the domain of the profunctor, and you have to handle every single one of those possibilities mapping out of it (oh, hey those feel like natural transformations...)

[@hujason4944]

my problem with your series is they are getting shorter and shorter :) just 7 lectures for the season

[@capitaldissimulation8877]

hairstylish cocats