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](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf)_. (See also the
[website with videos](http://math.mit.edu/~dspivak/teaching/sp18/).)

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

* Tai-Danae Bradley, _[What Is Applied Category Theory?](https://arxiv.org/abs/1809.05923)_ (Blog article here.)

Tai-Danae wrote this based on [Applied Category Theory 2018](https://golem.ph.utexas.edu/category/2017/09/applied_category_theory_1.html); 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, "https://arxiv.org/abs/1504.05625">A compositional framework for
passive linear networks
. (Blog article "https://johncarlosbaez.wordpress.com/2015/04/28/a-compositional-framework-for-passive-linear-networks/">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](https://forum.azimuthproject.org/discussion/2084/lecture-22-chapter-2-symmetric-monoidal-preorders/p1), 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](https://forum.azimuthproject.org/discussion/2338/lecture-74-compact-closed-categories/p1), 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](http://www.math.mcgill.ca/rags/JAC/124/Lambek-Pregroups-s.pdf), _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](https://arxiv.org/abs/1003.4394), 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!





**[To read other lectures go here.](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Chapter_4)**