Category Theory For Beginners: Natural Transformations

In this video we define natural transformations. We also define natural isomorphisms and functor categories. We define cones. We also define vertical and horizontal natural transformations, and describe how the associated categories are related to the category of categories. We also state the interchange law.

[@juliosong9851]

Wish I had seen this a year ago. This is definitely one of the best tutorials I’ve seen on the topic and maybe also the *only* one that doesn’t conceal the complexity behind the horizontal composition of natural transformations or leave the details to students as homework. Thanks! 👍👍

[@defunct1373]

These definitions entail a very intricate story, and you are an amazing storyteller. Thank you for the lesson.

[@i6g7f]

Wonderfully explained.

[@gucker]

Very clean explanation, thank you! These "orthogonal" (vertical and horizontal) categories of functors and their natural transformations are very interesting. I guess there are many beautiful examples where natural transformations, universal properties and other categorical constructs are used.

[@RichardSouthwell]

You know it ! Limits are kinds of universal morphisms that are defined in terms of natural transformations that are vast generalisations of categorical products. And the way limits are defined is very elegant. My video on limits will be released on June 17.

[@gucker]

Thank you, I am looking forward to this video!

[@Phantomz101]

Just curious, is that last equation correct?
I thought you would interchange alpha bar and beta instead of vice versa. Unless the horizontal natural transformation is commutative which is not assumed

[@Silviaa451]

Thank you so much for these videos, they are very useful!

[@msk.nfazal9757]

i'm so thankful to you sir.

[@nomukun1138]

By using a koan to expand the identity, you can gain a deeper understanding of Buddha natural transformations.

[@mochen9282]

what's the name of your toy? I want one!

[@RichardSouthwell]

www.amazon.co.uk/McDou-Magnetic-Construction-Educational-Random/dp/B07CLVT1P7

[@westhamdd84]

The last few videos I'm getting less out of because there are no real examples of the concepts.

[@RichardSouthwell]

Understandable. I suggest you skip forward to my video on graphs and dynamical systems th-cam.com/video/ime-EJbCUe0/w-d-xo.html where you will see many concrete examples of functors and natural transformations

[@roywang1294]

Thank you, Richard, for this video. The last equation in the video, concerning the interchange law, I believe it should be:
(mu * beta) dot (gamma * alpha) = (mu dot gamma) * (beta dot alpha)
Here, I have alpha_bar (from video) is gamma, beta_bar (from video) is mu.
See the text around Definition 22.14 about the interchange law, (pg 348 in the Oct 2022 print version) from Eugenia's book. ISBN 978-1-108-47722-2.