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It's been great teaching this course. While this is the end of the course, there is much more to say about applied category theory, which is quite a large subject. And it's far from done: its best days, I believe, are still ahead. Maybe you can help develop it further!

What should you do next? Well, it makes a lot of sense to finish reading our textbook:

- Brendan Fong and David Spivak,
*Seven Sketches in Compositionality*. (See also the website with videos.)

But it will also be good to read this 50-page 'booklet':

- Tai-Danae Bradley,
*What Is Applied Category Theory?*(Blog article here.)

Tai-Danae wrote this based on Applied Category Theory 2018; she attended both the 'school' and the following 'workshop'. She's great at explaining things, so this short book is a lot of fun to read.

Tai-Danae focuses on two examples of applied category theory.

First, she explains the 'decorated cospan categories' invented by Brendan Fong when he was doing his thesis with me. These are a general way of dealing with categories where the morphisms are *networks*. You can learn more about them in Section 6 of *Seven Sketches*, and even more here:

- Brendan Fong,
*The Algebra of Open and Interconnected Systems*, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)

Brendan used them to study electrical circuits, and we've gone further with that application here:

- John Baez and Brendan Fong, A compositional framework for passive linear networks. (Blog article here.)

But Tai-Danae focuses on another application of decorated cospan categories - namely, to chemical reaction networks! This was developed Blake Pollard, another student of mine:

- Blake Pollard,
*Open Markov Processes and Reaction Networks*, Ph.D. thesis, U. C. Riverside, 2017.

Second, she explains Lambek's approach to linguistics based on 'pregroup grammars'. If you enjoyed how this course focused on posets, I bet you'll really like pregroups. A pregroup is just a monoidal poset that's compact closed!

We started talking about *commutative* monoidal posets all the way back in Lecture 21, but in applications to linguistics we don't want commutativity - because the order of words matters! We discussed compact closed *symmetric* monoidal categories in Lecture 74, but Tae-Danae explains more general compact closed categories that are just monoidal, not symmetric monoidal - again, because the order of words matters.

So, if you understood this course, you only need a tiny bit more to get the idea of a pregroup. The fun part is to see how pregroups are used to study words, phrases and sentences!

Lambek was a very interesting guy - perhaps the first to understand the connection between category theory and the lambda calculus - and his work is fun to read:

- Joachim Lambek, Pregroups and natural language processing,
*The Mathematical Intelligencer***28**(2006), 41–48.

But the current revival of interest in pregroup grammars may have been started by this paper:

- Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark, Mathematical foundations for a compositional distributional model of meaning, in the Lambek Festschrift, special issue of
*Linguistic Analysis*, 2010. (Blog article here.)

This should be enough to keep you going for a while. I hope you have many pleasant adventures... and don't be afraid to email me at baez@math.ucr.edu with questions or comments!

## Comments

Thanks for the offered roadmap..I will try to guide myself with it...

`Thanks for the offered roadmap..I will try to guide myself with it...`

I'm not worried about the end.

We can always ask you questions and leave comments on this forum and over twitter.

`I'm not worried about the end. We can always ask you questions and leave comments on this forum and over twitter.`

Thank you so much for teaching this class, John! Reading and (modestly) participating finally got me to a level of understanding from where I can read other resources about applied category theory. I hope that I can apply what I learned to software verification and programming one day!

At the moment I am busy with other things, but I hope to revisit this class soon and fill up the "No Pain, No Gain"-Threads started by me to make it easier for newcomers to catch up :)

`Thank you so much for teaching this class, John! Reading and (modestly) participating finally got me to a level of understanding from where I can read other resources about applied category theory. I hope that I can apply what I learned to software verification and programming one day! At the moment I am busy with other things, but I hope to revisit this class soon and fill up the "No Pain, No Gain"-Threads started by me to make it easier for newcomers to catch up :)`

Thanks, folks! I may add Daniel's "No Pain, No Gain" threads, and others, up to the "officially announced" section of the discussions to make them easier to find.

There's a lot more pain, and gain, available from the readings I just listed. Tai-Danae Bradley's book should give people the least pain, but still plenty of gain.

`Thanks, folks! I may add Daniel's "No Pain, No Gain" threads, and others, up to the "officially announced" section of the discussions to make them easier to find. There's a lot more pain, and gain, available from the readings I just listed. Tai-Danae Bradley's book should give people the least pain, but still plenty of gain.`

I am not certain how to express my gratitude for this wonderful course. I have gained so much during the past 6 months learning what I have only dreamed of understanding one day. It is quite amazing where guidance from a great teacher can get you. It was a pleasant surprise when I realized that I can actually read articles on nLab and other academic resources; something I could not have done couple months ago. So a sincere thank you to John for your amazing gift to guide and teach students your wealth of knowledge and for your generosity to teach this free course even through your busy schedule!

Also shout out to everyone on this forum. It was an honor just breathing the same air with experts from around the world! Thank you.

`I am not certain how to express my gratitude for this wonderful course. I have gained so much during the past 6 months learning what I have only dreamed of understanding one day. It is quite amazing where guidance from a great teacher can get you. It was a pleasant surprise when I realized that I can actually read articles on nLab and other academic resources; something I could not have done couple months ago. So a sincere thank you to John for your amazing gift to guide and teach students your wealth of knowledge and for your generosity to teach this free course even through your busy schedule! Also shout out to everyone on this forum. It was an honor just breathing the same air with experts from around the world! Thank you.`

I add my thanks to John for this course (and for all the other work that he has been doing for years to explain complicated mathematics in ways that even a non-mathematician like myself can find enlightening). I also want to thank all the other course participants whose questions, answers (and figures!) have helped so much with my understanding of the course material.

`I add my thanks to John for this course (and for all the other work that he has been doing for years to explain complicated mathematics in ways that even a non-mathematician like myself can find enlightening). I also want to thank all the other course participants whose questions, answers (and figures!) have helped so much with my understanding of the course material.`

Oh, I have an idea:

Puzzle KEP:Find literature on category theory that will continue the others learning.`Oh, I have an idea: **Puzzle KEP:** Find literature on category theory that will continue the others learning.`

I'd want also to add the others in giving John a big Thank You for all the effort and guidance, the helping hand of a field expert has done charms and made a difference, and prepared us for better self-learning. I'm going to miss this advice. Can't help but envy John's grad students. On the bright side the course has acted as a pole attracting a clever community, and we should manage to not lose contact and disband in all directions. I'd be open to furhter exchange here and elsewhere and talk, time permitting, about pending book chapters, Tai-Danae booklet and other applied topics. Nel mezzo del cammin, I could be useful to some starters and would welcome dialog with the experienced guys.

`I'd want also to add the others in giving John a big Thank You for all the effort and guidance, the helping hand of a field expert has done charms and made a difference, and prepared us for better self-learning. I'm going to miss this advice. Can't help but envy John's grad students. On the bright side the course has acted as a pole attracting a clever community, and we should manage to not lose contact and disband in all directions. I'd be open to furhter exchange here and elsewhere and talk, time permitting, about pending book chapters, Tai-Danae booklet and other applied topics. Nel mezzo del cammin, I could be useful to some starters and would welcome dialog with the experienced guys.`

Answer KEP:I plan on reading this, maybe others will also find it interesting.`**Answer KEP:** I plan on reading [this](https://www.amazon.com/Axiomatic-Method-Category-Theory-Synthese-ebook/dp/B00FWH57D6/ref=sr_1_1?ie=UTF8&qid=1538412903&sr=8-1&keywords=axiomatic+category+theory&dpID=41q5atBsQFL&preST=_SY445_QL70_&dpSrc=srch), maybe others will also find it interesting.`

Hi, folks! I'm teaching a course on category theory at UCR, and there are some hand-written notes here:

I'm not sure it's possible to follow the course from these notes; there's a lot that I say out loud, and even some that I write on the board, that didn't find its way in here. But they might at least give you some ideas for things to study.

`Hi, folks! I'm teaching a course on category theory at UCR, and there are some hand-written notes here: * [Category theory course](https://johncarlosbaez.wordpress.com/2018/10/13/category-theory-course/). I'm not sure it's possible to follow the course from these notes; there's a lot that I say out loud, and even some that I write on the board, that didn't find its way in here. But they might at least give you some ideas for things to study.`