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As you may have noticed, my production of lectures has been slowing down. There are a few reasons:

I'm trying to finish a bunch of papers. I usually get started writing around noon or 1 pm, and when I get into it it's hard for me switch gears and write a lecture, especially since I've been trying to go to the gym almost every day at 6.

It's getting harder to write the lectures as the book proceeds and the 'sketches' get more sketchy, leaving me to fill in more details.

I have the feeling that many students have fallen behind the rather quick pace of the lectures, leaving only a small band of energetic followers.

My energy is slowly running out.

As for 2, I don't know if I *should* be filling in so many details. Maybe people would be happier if I gave more of an overview. This will be even more of an issue soon. Fong and Spivak give just a rough definition of 'monoidal category' in Section 4.4.3. The definition is a bit complicated, but it's a fundamental concept in category theory. Should I spend time to fill in the details or not? This is just one example of the decisions we face.

As for 3, it would be great to hear from people who *aren't* in the small band of energetic students who are leaving lots of comments on the lectures and solving lots of puzzles.

As for 4, that's mainly my problem, but I should warn you: I'm considering finishing at the end of chapter 4, after a good explanation of monoidal categories, compact closed categories, and PERT charts as another application of \(\mathcal{V}\)-enriched profunctors. I've put a lot of energy into this course and hate the idea of quitting before its done, but it's also tough to wake up each morning and know I need to spend an hour or two writing lectures along with papers. This will get a bit tougher on Wednesday when I go to Singapore.

## Comments

on #3. I am in that other group that had fallen off. I could not keep up, unfortunately. Although I check the lectures once in a while. And I think material is great. I am very very happy are doing this.

It gives such strong foundational background to understanding measurement models, contract management, programming language & database transformation models, etc.

For me, I got stuck somewhere on between lectures 6 and and 11 (in May), and could not become fluent in calculating left and right adjoins for all the puzzles you and other students posted. So everything was a struggle. Also I felt, I had to parse out every time, carefully -- differences in notations reflecting inverses/images etc. And some other things folks were mentioning (eg lattices/etc). Although, in the process, I think, I understood how to check my own solutions (which is a prerequisite for a good self-study). And yours and others feedback comments on what I had posted --was invaluable in that.

I felt, to keep up with the pace, and filling in my own gaps, in the process -- I needed to be putting continuously 3-5 hours a day.

Ultimately the efforts for my other projects -- was also too much, cognitively, and I felt they were higher priority for me at this time. So I had to deprioritize the course.

My next thought was to print out various cheat-sheets/pictorial overviews -- from your course, and plaster it all over my office space -- so that when my eyes wonder of my work screen -- I can see the category theory :-).

`on #3. I am in that other group that had fallen off. I could not keep up, unfortunately. Although I check the lectures once in a while. And I think material is great. I am very very happy are doing this. It gives such strong foundational background to understanding measurement models, contract management, programming language & database transformation models, etc. For me, I got stuck somewhere on between lectures 6 and and 11 (in May), and could not become fluent in calculating left and right adjoins for all the puzzles you and other students posted. So everything was a struggle. Also I felt, I had to parse out every time, carefully -- differences in notations reflecting inverses/images etc. And some other things folks were mentioning (eg lattices/etc). Although, in the process, I think, I understood how to check my own solutions (which is a prerequisite for a good self-study). And yours and others feedback comments on what I had posted --was invaluable in that. I felt, to keep up with the pace, and filling in my own gaps, in the process -- I needed to be putting continuously 3-5 hours a day. Ultimately the efforts for my other projects -- was also too much, cognitively, and I felt they were higher priority for me at this time. So I had to deprioritize the course. My next thought was to print out various cheat-sheets/pictorial overviews -- from your course, and plaster it all over my office space -- so that when my eyes wonder of my work screen -- I can see the category theory :-).`

John, here's my experience with the course.

Filling in the details is the greatest feature of this course actually, which comes with a price - much more time is needed because of numerous rabbit holes encountered by the way. The touch on logic was really fascinating, many things clicked for me at once. Another cool example is the tropical algebra, which we encountered during Chapter 2.

The concept of adjoints didn't click for me until I started my own investigation and discovered that functors, natural transformations and adjoints have weak analogies in topology, namely with continuous maps, homotopies and (weak) homotopy equivalences respectively. After refreshing my scarce knowledge in topology (probably a lecture during undergraduate calculus a decade ago), using Hatcher's Chapter 0, it became clear that all physical/visual intuitions humans have concerning shapes, their deformations and retractions into each other, were usurped by the modern definition of topology, which deals with undirected, abstract continuous spaces. So, as I see it at the moment, it is no wonder that Mac Lane and Eilenberg have to come up with a more general concept, which allowed to work with directed, discrete spaces, while conceptualizing the same natural notions of continuity, shape and deformation, and called it category theory. So in my view it is the biggest omission of the book and the course not to mention such correspondences in the Chapter 1, because they naturally appeal to human physical intuitions. It is clear that most (all?) of mathematicians use some sort of "visualizations" for concepts they use, but since they try to avoid to be imprecise and too wordy in their papers, such intuitions are rarely appear there. But in an introductory book and a course one has such a freedom :)

Also, it seems to be a mistake not to provide more time for Chapter 3, or make a pause for at least a few weeks after it (the concept is called spaced learning). This is the cornerstone chapter - either you understand it, and everything collapses at once, or there is no reason to actually proceed. And even if you got this "closure", you need some time for everything to settle down, and proceed refreshed. So I slowed my pace, filled some gaps, looked on things from different angles, and almost completed it, just need to reread 3.4 + limits with a gained perspective.

My advice is not to give up John, but in order not to burn yourself down, postpone new lectures until the October/November. Also don't give up on your style and the style of the book - what we are actually learning is not just category theory, but the skills of generalization, engineering of arbitrary theories and algebras - on a number of nice mathematical and applied examples.

And thank you for everything you are doing, that's really cool!

`John, here's my experience with the course. 1. Filling in the details is the greatest feature of this course actually, which comes with a price - much more time is needed because of numerous rabbit holes encountered by the way. The touch on logic was really fascinating, many things clicked for me at once. Another cool example is the tropical algebra, which we encountered during Chapter 2. 2. The concept of adjoints didn't click for me until I started my own investigation and discovered that functors, natural transformations and adjoints have weak analogies in topology, namely with continuous maps, homotopies and (weak) homotopy equivalences respectively. After refreshing my scarce knowledge in topology (probably a lecture during undergraduate calculus a decade ago), using Hatcher's Chapter 0, it became clear that all physical/visual intuitions humans have concerning shapes, their deformations and retractions into each other, were usurped by the modern definition of topology, which deals with undirected, abstract continuous spaces. So, as I see it at the moment, it is no wonder that Mac Lane and Eilenberg have to come up with a more general concept, which allowed to work with directed, discrete spaces, while conceptualizing the same natural notions of continuity, shape and deformation, and called it category theory. So in my view it is the biggest omission of the book and the course not to mention such correspondences in the Chapter 1, because they naturally appeal to human physical intuitions. It is clear that most (all?) of mathematicians use some sort of "visualizations" for concepts they use, but since they try to avoid to be imprecise and too wordy in their papers, such intuitions are rarely appear there. But in an introductory book and a course one has such a freedom :) 3. Also, it seems to be a mistake not to provide more time for Chapter 3, or make a pause for at least a few weeks after it (the concept is called spaced learning). This is the cornerstone chapter - either you understand it, and everything collapses at once, or there is no reason to actually proceed. And even if you got this "closure", you need some time for everything to settle down, and proceed refreshed. So I slowed my pace, filled some gaps, looked on things from different angles, and almost completed it, just need to reread 3.4 + limits with a gained perspective. My advice is not to give up John, but in order not to burn yourself down, postpone new lectures until the October/November. Also don't give up on your style and the style of the book - what we are actually learning is not just category theory, but the skills of generalization, engineering of arbitrary theories and algebras - on a number of nice mathematical and applied examples. And thank you for everything you are doing, that's really cool!`

Theorem 1.This course is awesome.Proof:John, thanks very much for your idea of teaching this course to all of us. Having your Lectures to supplement the book has been tremendously useful; I don't think I would have understood the material if I had just read the book on my own.I'm still finishing my digestion of Chapter 3. I haven't posted much, but I'm definitely still here! It's been surprisingly difficult to establish an intuition for adjunctions, but I'm glad you've given us enough detail to make the effort, because I can tell it's important material and struggling through it makes me feel like I'm really learning something and not just being a tourist.

I look forward to starting on Chapter 4 and reading all the stuff you've written for us. If you decide to make this a 4-month (free!) course instead of a 6-month course, I'm sure we'll all understand and still appreciate your efforts just as much.

I wonder if it would be helpful and fun if you ended with a few "greatest hits" lectures to give us a taste of the later material without trying to teach it thoroughly. If you gave us some idea of what we're missing, and why it's exciting, that might help us stick with the subject and continue our self studies. I'd like to know other things, like adjunctions and monoidal categories for example, that you consider especially important if one wants to get a true sense of how category theory is put together.

One bit of feedback if you decide to do another course next summer. I appreciate the give and take in the comments as you let us students work through the puzzles ourselves. However, all the confusions and false starts make the comment thread less useful as a reference when I go back to a lecture for repeat study. When you give us puzzles to solidify our understanding, it would be helpful if the discussion could be distilled to a correct set of answers "blessed" by you. Perhaps after a couple of weeks you could pick correct student answers and copy them into an answers thread or some sort of hidden spoiler tag? That would make the Lectures a more useful long-term reference, if it's not too much work.

The forum software makes it hard to sustain conversations in old threads, especially without a working notification system, or a place for "front page" announcements. It can feel lonely once someone has fallen behind. I wonder if something like periodic email surveys could reveal whether there's a substantial number of students still working through earlier sections. If so, perhaps it would be good to have a "rest week" now and then for everyone to go back and try some new puzzles together to help those students catch up. But I guess realistically, once someone falls way behind they're more likely to stop attending the course altogether.

Perhaps you could reduce the effort somewhat by spacing out the Lectures more? For example, what if you followed each Lecture with a day or two where all you do is add new puzzles for that topic. This might also give students less speedy with the material more chances to get first crack at a puzzle, rather than the "usual suspects".

Thanks again for the course! It has been awesome. \( \blacksquare \)

`**Theorem 1.** This course is awesome. *Proof:* John, thanks very much for your idea of teaching this course to all of us. Having your Lectures to supplement the book has been tremendously useful; I don't think I would have understood the material if I had just read the book on my own. I'm still finishing my digestion of Chapter 3. I haven't posted much, but I'm definitely still here! It's been surprisingly difficult to establish an intuition for adjunctions, but I'm glad you've given us enough detail to make the effort, because I can tell it's important material and struggling through it makes me feel like I'm really learning something and not just being a tourist. I look forward to starting on Chapter 4 and reading all the stuff you've written for us. If you decide to make this a 4-month (free!) course instead of a 6-month course, I'm sure we'll all understand and still appreciate your efforts just as much. I wonder if it would be helpful and fun if you ended with a few "greatest hits" lectures to give us a taste of the later material without trying to teach it thoroughly. If you gave us some idea of what we're missing, and why it's exciting, that might help us stick with the subject and continue our self studies. I'd like to know other things, like adjunctions and monoidal categories for example, that you consider especially important if one wants to get a true sense of how category theory is put together. One bit of feedback if you decide to do another course next summer. I appreciate the give and take in the comments as you let us students work through the puzzles ourselves. However, all the confusions and false starts make the comment thread less useful as a reference when I go back to a lecture for repeat study. When you give us puzzles to solidify our understanding, it would be helpful if the discussion could be distilled to a correct set of answers "blessed" by you. Perhaps after a couple of weeks you could pick correct student answers and copy them into an answers thread or some sort of hidden spoiler tag? That would make the Lectures a more useful long-term reference, if it's not too much work. The forum software makes it hard to sustain conversations in old threads, especially without a working notification system, or a place for "front page" announcements. It can feel lonely once someone has fallen behind. I wonder if something like periodic email surveys could reveal whether there's a substantial number of students still working through earlier sections. If so, perhaps it would be good to have a "rest week" now and then for everyone to go back and try some new puzzles together to help those students catch up. But I guess realistically, once someone falls way behind they're more likely to stop attending the course altogether. Perhaps you could reduce the effort somewhat by spacing out the Lectures more? For example, what if you followed each Lecture with a day or two where all you do is add new puzzles for that topic. This might also give students less speedy with the material more chances to get first crack at a puzzle, rather than the "usual suspects". Thanks again for the course! It has been awesome. \\( \blacksquare \\)`

The course is amazing! (And the proof has been given in the comment by Pete :) ) I'm so sorry for my delay -- after several travels within the last months and new/old papers, I fell behind but I'm trying to catch up. I'm so happy to go through the course material and reading the lectures, very thought-provoking. There is a depth of thinking that goes way beyond a beginning course stuff, and I've the feeling that I should learn this thinking style too. It's a bit out of topic, but I hope you can enjoy a rhythm I derived from the numerical sequence on the 'Just For Fun 1.' Many new ideas arise! :) Thank you for all of this!

`The course is amazing! (And the proof has been given in the comment by Pete :) ) I'm so sorry for my delay -- after several travels within the last months and new/old papers, I fell behind but I'm trying to catch up. I'm so happy to go through the course material and reading the lectures, very thought-provoking. There is a depth of thinking that goes way beyond a beginning course stuff, and I've the feeling that I should learn this thinking style too. It's a bit out of topic, but I hope you can enjoy a rhythm I derived from the numerical sequence on the 'Just For Fun 1.' Many new ideas arise! :) Thank you for all of this!`

Igor wrote:

Actually the first half of Chapter 4 - the half I've covered so far - uses almost nothing from Chapter 3. It picks up where Chapter 2 left off: it continues the study of preorders, and \(\mathcal{V}\)-enriched categories where \(\mathcal{V}\) is a preorder. So don't let any difficulties with Chapter 3 stop you from starting Chapter 4.

But yes, Chapter 3 touches on a huge amount of material: it's the first place one meets full-fledged

category theoryinstead of preorder theory! I could easily have spent twice as much time on it. The main reason I didn't is that I knew it would condemn me to not finishing the course by the time I have to start teaching (not for fun - for pay!) in late September.I was relieved that Chapter 4 returned to preorder theory, which is much easier to explain since one never needs to discuss equations between morphisms.

However, the second half of Chapter 4 is about monoidal categories, so here one needs to understand full-fledged categories again. And I would like to cover monoidal categories.

The amount of time it would take to fully explain the chapters keeps increasing after that.

I'm going to be busy teaching actual classes from the end of September until April... including an advanced course on category theory, but also undergraduate courses like calculus, and graduate courses like real analysis. I'll have three students finishing their PhD theses next spring, so I'll need to spend a lot of time reading and editing their theses. And then I have three other grad students, and I'm working part-time for Pyrofex and Metron. So, the first time I could manage to continue this course would be April 2019... and frankly, I'm getting tired just thinking about it!

`Igor wrote: > Also, it seems to be a mistake not to provide more time for Chapter 3, or make a pause for at least a few weeks after it (the concept is called spaced learning). This is the cornerstone chapter - either you understand it, and everything collapses at once, or there is no reason to actually proceed. Actually the first half of Chapter 4 - the half I've covered so far - uses almost nothing from Chapter 3. It picks up where Chapter 2 left off: it continues the study of preorders, and \\(\mathcal{V}\\)-enriched categories where \\(\mathcal{V}\\) is a preorder. So don't let any difficulties with Chapter 3 stop you from starting Chapter 4. But yes, Chapter 3 touches on a huge amount of material: it's the first place one meets full-fledged _category theory_ instead of preorder theory! I could easily have spent twice as much time on it. The main reason I didn't is that I knew it would condemn me to not finishing the course by the time I have to start teaching (not for fun - for pay!) in late September. I was relieved that Chapter 4 returned to preorder theory, which is much easier to explain since one never needs to discuss equations between morphisms. However, the second half of Chapter 4 is about monoidal categories, so here one needs to understand full-fledged categories again. And I would like to cover monoidal categories. The amount of time it would take to fully explain the chapters keeps increasing after that. > My advice is not to give up John, but in order not to burn yourself down, postpone new lectures until the October/November. I'm going to be busy teaching actual classes from the end of September until April... including an advanced course on category theory, but also undergraduate courses like calculus, and graduate courses like real analysis. I'll have three students finishing their PhD theses next spring, so I'll need to spend a lot of time reading and editing their theses. And then I have three other grad students, and I'm working part-time for Pyrofex and Metron. So, the first time I could manage to continue this course would be April 2019... and frankly, I'm getting tired just thinking about it!`

Pete - I'm glad you're liking this course.

I could definitely do that.

This would take a lot of work, which I don't have time for now... but I understand why it would be good. I had naively hoped that confused students (e.g., wondering which answers were right) would ask questions. But I agree that's nowhere near as good as having all the right answers to the puzzles nicely organized in one place.

I'm kinda planning to convert my lectures into a 'book', or at least a PDF file. If I do this, I hope to include good answers to the puzzles.

`Pete - I'm glad you're liking this course. > I wonder if it would be helpful and fun if you ended with a few "greatest hits" lectures to give us a taste of the later material without trying to teach it thoroughly. I could definitely do that. > When you give us puzzles to solidify our understanding, it would be helpful if the discussion could be distilled to a correct set of answers "blessed" by you. This would take a lot of work, which I don't have time for now... but I understand why it would be good. I had naively hoped that confused students (e.g., wondering which answers were right) would ask questions. But I agree that's nowhere near as good as having all the right answers to the puzzles nicely organized in one place. I'm kinda planning to convert my lectures into a 'book', or at least a PDF file. If I do this, I hope to include good answers to the puzzles.`

... and maybe, if I'm extra energetic, the official "exercises" and the answers to those, since we've got those here too!

`... and maybe, if I'm extra energetic, the official "exercises" and the answers to those, since we've got those here too!`

Despite the fact that I haven't kept up with exercises or participated recently (particularly with the last chapter - paper deadlines have been getting in the way), this course has been really incredible and I've read all of your lectures and learned a lot from them. I want to echo what other people have said, that the lectures very concretely add to the book, by showing how to think mathematically. Some other thoughts:

It seems to me that these lectures, and some of the discussion of the puzzles should be refined into re-usable teaching resources. I'm certainly going to go back and use the lectures as reference material, but if you're running out of energy (highly understandable - I don't even know how you've been keeping it up so far), I feel like the most utility is in taking everything so far and smoothing out all the wrinkles. I think courses like this take more than one iteration to get totally right.

The main things I got from the course were: (i) finally having some grasp on adjunctions and their relation to quantifiers, (ii) the trick of looking at preorders to understand hard concepts in the full categorical setting, (iii) the concept of naturality - although I still need to work on this (and feel a lecture or two more on it wouldn't have been amiss) and (iv) the concept of closure vis a vis self-enrichment. I'm excited to catch up on the diagrammatic "logic" of compact closed categories - I don't yet understand how the feedback examples work.

`Despite the fact that I haven't kept up with exercises or participated recently (particularly with the last chapter - paper deadlines have been getting in the way), this course has been really incredible and I've read all of your lectures and learned a lot from them. I want to echo what other people have said, that the lectures very concretely add to the book, by showing how to think mathematically. Some other thoughts: 1. It seems to me that these lectures, and some of the discussion of the puzzles should be refined into re-usable teaching resources. I'm certainly going to go back and use the lectures as reference material, but if you're running out of energy (highly understandable - I don't even know how you've been keeping it up so far), I feel like the most utility is in taking everything so far and smoothing out all the wrinkles. I think courses like this take more than one iteration to get totally right. 2. The main things I got from the course were: (i) finally having some grasp on adjunctions and their relation to quantifiers, (ii) the trick of looking at preorders to understand hard concepts in the full categorical setting, (iii) the concept of naturality - although I still need to work on this (and feel a lecture or two more on it wouldn't have been amiss) and (iv) the concept of closure vis a vis self-enrichment. I'm excited to catch up on the diagrammatic "logic" of compact closed categories - I don't yet understand how the feedback examples work.`

I would also like to echo the others and voice a huge thank you for all the great lectures you've put together. Without your exposition I surely wouldn't have managed to get this far through the book. I like the topic selection from the book, but I find at times their exposition a bit too abrupt; so it's been great working through the topics at a slower pace.

Regarding the third point, there were times when I lagged behind, but I've tried to get myself back on track; I usually try to visit the forum daily and read materials (although sometimes at a more superficial level).

Pete Morcos suggested:

I like this idea as well. Also, it would be great if you could provide a list of further topics or resources that you would have covered if time permitted.

Looking forward for the exposition on monoidal categories :)

`I would also like to echo the others and voice a huge thank you for all the great lectures you've put together. Without your exposition I surely wouldn't have managed to get this far through the book. I like the topic selection from the book, but I find at times their exposition a bit too abrupt; so it's been great working through the topics at a slower pace. Regarding the third point, there were times when I lagged behind, but I've tried to get myself back on track; I usually try to visit the forum daily and read materials (although sometimes at a more superficial level). Pete Morcos suggested: > I wonder if it would be helpful and fun if you ended with a few "greatest hits" lectures to give us a taste of the later material without trying to teach it thoroughly. I like this idea as well. Also, it would be great if you could provide a list of further topics or resources that you would have covered if time permitted. Looking forward for the exposition on monoidal categories :)`

For me the course has already been greatly beneficial and I'm finding excruciating that the obligations of a 'day job' don't let me engage and commit fully to this opportunity to squeeze out all the goodness. In my case I cannot afford to fight the puzzles but in weekends I sit down and advance the book and the lectures religiously and do choosen problems. Ch. 3 is a difficult passage and I invested a couple of weekends convincing myself I was confortable with it, writing down in full a left Kan extension. After that I got tired and had to delay ch. 4 to summer holydays that I started enjoying, so I'm going to get ahead of schedule again. I reaped sweet benefits from the course. The concept of Kan extension is written in the books, but It wasn't until being told here from mouth to ear what it was, that I understood it. The political analogies of liberal and convervative adjoints are perfect also and pure oral tradition, not in textbooks. Now we are visiting enrichment that is exactly what I need to access certain writings of Willerton. I was amazed by the energy displayed and would understand whatever has to happen, for all the effort already freely given, thanks.

`For me the course has already been greatly beneficial and I'm finding excruciating that the obligations of a 'day job' don't let me engage and commit fully to this opportunity to squeeze out all the goodness. In my case I cannot afford to fight the puzzles but in weekends I sit down and advance the book and the lectures religiously and do choosen problems. Ch. 3 is a difficult passage and I invested a couple of weekends convincing myself I was confortable with it, writing down in full a left Kan extension. After that I got tired and had to delay ch. 4 to summer holydays that I started enjoying, so I'm going to get ahead of schedule again. I reaped sweet benefits from the course. The concept of Kan extension is written in the books, but It wasn't until being told here from mouth to ear what it was, that I understood it. The political analogies of liberal and convervative adjoints are perfect also and pure oral tradition, not in textbooks. Now we are visiting enrichment that is exactly what I need to access certain writings of Willerton. I was amazed by the energy displayed and would understand whatever has to happen, for all the effort already freely given, thanks.`

Another semi-lapsed student here: I keep reading lectures and comments, but due to work and family commitments I stopped doing exercises and puzzles a few weeks ago - around the time of Kan extensions - with the natural consequences for my understanding of the material.

I join in the thanks to John for all his work - and thanks also to the more active students in the forum!

I think the course is great and I plan to go over the material more carefully when I have more time (btw, yes, having it all collated in a single PDF file, if not also a properly edited printed book, would be extremely useful!)

`Another semi-lapsed student here: I keep reading lectures and comments, but due to work and family commitments I stopped doing exercises and puzzles a few weeks ago - around the time of Kan extensions - with the natural consequences for my understanding of the material. I join in the thanks to John for all his work - and thanks also to the more active students in the forum! I think the course is great and I plan to go over the material more carefully when I have more time (btw, yes, having it all collated in a single PDF file, if not also a properly edited printed book, would be extremely useful!)`

John, we have been moving through the course at a pretty glacial pace in our study group....but we are still moving! :) This course has been tremendous. I fully realized this just the other day glancing again through a book I had picked up in the library a few months back. Just months ago it was completely incomprehensible to me, yet now I found myself nodding, recognizing, and even understanding the basics of things like symmetric monoidal categories. This was incredibly motivating, and furthered my resolve to keep working through the rest of the material.

I realize it's difficult sometimes to see the effect of all this incredible work you have put into writing and answering questions.....but it is there, and will continue to help those like myself for quite a while to come as I continue through the course.

`John, we have been moving through the course at a pretty glacial pace in our study group....but we are still moving! :) This course has been tremendous. I fully realized this just the other day glancing again through a book I had picked up in the library a few months back. Just months ago it was completely incomprehensible to me, yet now I found myself nodding, recognizing, and even understanding the basics of things like symmetric monoidal categories. This was incredibly motivating, and furthered my resolve to keep working through the rest of the material. I realize it's difficult sometimes to see the effect of all this incredible work you have put into writing and answering questions.....but it is there, and will continue to help those like myself for quite a while to come as I continue through the course.`

Thanks, everyone! I urge you all to continue reading beyond Chapter 4.

Chapter 5 is about signal flow graphs in control theory, and "props", which are certain nice monoidal categories. My students Jason and Brandon wrote their theses on these topics:

Jason Erbele,

Categories in Control: Applied PROPs, Ph.D. thesis, U. C. Riverside, 2016.Brandon Coya,

Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2018. (Blog article here.and Brendan has worked on these too, and so have I:

Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids,

Theory and Applications of Categories32(2017), 380–395. (Blog article here.)John Baez, Brandon Coya and Franciscus Rebro, Props in network theory,

Theory and Applications of Categories33(2018), 727–783. (Blog article here.)so this material is almost painfully familiar to me.

Chapter 6 is on circuit diagrams, and both Brandon and Brendan did their theses with me on this topic. Brandon Coya's thesis is above, and Brendan's is here:

The Algebra of Open and Interconnected Systems, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)based in part on this paper, which is still fighting its way through the referees:

Chapter 7 is on topos theory, which is very different, but also fascinating.

`Thanks, everyone! I urge you all to continue reading beyond Chapter 4. Chapter 5 is about signal flow graphs in control theory, and "props", which are certain nice monoidal categories. My students Jason and Brandon wrote their theses on these topics: * Jason Erbele, <i><a href = "https://arxiv.org/abs/1611.07591">Categories in Control: Applied PROPs</a></i>, Ph.D. thesis, U. C. Riverside, 2016. * Brandon Coya, <i><a href = "https://arxiv.org/abs/1805.08290">Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective</a></i>, Ph.D. thesis, U. C. Riverside, 2018. (Blog article <a href = "https://johncarlosbaez.wordpress.com/2018/05/19/circuits-bond-graphs-and-signal-flow-diagrams/">here</a>. and Brendan has worked on these too, and so have I: * Brandon Coya and Brendan Fong, <a href = "https://arxiv.org/abs/1601.02307">Corelations are the prop for extraspecial commutative Frobenius monoids</a>, <i>Theory and Applications of Categories</i> <b>32</b> (2017), 380–395. (Blog article <a href = "https://johncarlosbaez.wordpress.com/2016/02/02/corelations-in-network-theory/">here</a>.) * John Baez, Brandon Coya and Franciscus Rebro, <a href = "https://arxiv.org/abs/1707.08321">Props in network theory</a>, <i>Theory and Applications of Categories</i> <b>33</b> (2018), 727–783. (Blog article <a href = "https://johncarlosbaez.wordpress.com/2018/04/27/props-in-network-theory/">here</a>.) so this material is almost painfully familiar to me. Chapter 6 is on circuit diagrams, and both Brandon and Brendan did their theses with me on this topic. Brandon Coya's thesis is above, and Brendan's is here: * Brendan Fong, <i><a href = "https://arxiv.org/abs/1609.05382">The Algebra of Open and Interconnected Systems</a></i>, Ph.D. thesis, University of Oxford, 2016. (Blog article <a href = "https://johncarlosbaez.wordpress.com/2016/10/23/open-and-interconnected-systems/">here</a>.) based in part on this paper, which is still fighting its way through the referees: * John Baez and Brendan Fong, <a href = "https://arxiv.org/abs/1504.05625">A compositional framework for passive linear networks</a>. (Blog article <a href = "https://johncarlosbaez.wordpress.com/2015/04/28/a-compositional-framework-for-passive-linear-networks/">here</a>.) Chapter 7 is on topos theory, which is very different, but also fascinating.`

Maria wrote:

I'd like to hear it!

I've been thinking of buying Wotja, a program for composing 'generative music'. By the way, do you know any experts on the software and hardware needed to record electronic music and a bit of live music? More precisely: any experts who'd be willing to talk to me?

(Maybe you're an expert on this - that would be best. But if not, you may know a friendly one.)

`Maria wrote: > It's a bit out of topic, but I hope you can enjoy a rhythm I derived from the numerical sequence on the 'Just For Fun 1.' I'd like to hear it! I've been thinking of buying [Wotja](https://intermorphic.com/wotja/), a program for composing 'generative music'. By the way, do you know any experts on the software and hardware needed to record electronic music and a bit of live music? More precisely: any experts who'd be willing to talk to me? (Maybe you're an expert on this - that would be best. But if not, you may know a friendly one.)`

Of course! Thank you so much! The information and the links about the mathematical rhythm are here: https://forum.azimuthproject.org/discussion/comment/18314/#Comment_18314 https://soundcloud.com/maria-mannone/mathematical-rhythm

I am not an expert on this, but thanks for asking! I think that NIME folks would be excited to talk with you! In particular, I would recommend Pamela Z, classical soprano and electronic music composer, who works with electronic and live music. Also, I will send you privately the email contact of an Italian scholar.

`Of course! Thank you so much! The information and the links about the mathematical rhythm are here: https://forum.azimuthproject.org/discussion/comment/18314/#Comment_18314 https://soundcloud.com/maria-mannone/mathematical-rhythm I am not an expert on this, but thanks for asking! I think that NIME folks would be excited to talk with you! In particular, I would recommend [Pamela Z](http://www.pamelaz.com/pzbios.html), classical soprano and electronic music composer, who works with electronic and live music. Also, I will send you privately the email contact of an Italian scholar.`

Maria - thanks very much! I'll check out those links.

I know a lot of math, and I like to think I have some musical taste, but I'm really ignorant about the best way to, say, improvise on a keyboard, create a MIDI file, then edit this, then play it on some electronic instrument and record it, then do sound processing like adding reverb and other effects. I'd also love to be able to whistle or sing and have that converted to MIDI. I'd like a system that's easy for beginners but could keep being extended as I get better. I'm a bit scared to buy Ableton or Cubase. So, it would be good to talk to people who would enjoy trading these music skills for mathematics.

`Maria - thanks very much! I'll check out those links. I know a lot of math, and I like to think I have some musical taste, but I'm really ignorant about the best way to, say, improvise on a keyboard, create a MIDI file, then edit this, then play it on some electronic instrument and record it, then do sound processing like adding reverb and other effects. I'd also love to be able to whistle or sing and have that converted to MIDI. I'd like a system that's easy for beginners but could keep being extended as I get better. I'm a bit scared to buy [Ableton](https://www.ableton.com/en/) or [Cubase](https://www.steinberg.net/en/products/cubase/start.html). So, it would be good to talk to people who would enjoy trading these music skills for mathematics.`

The course is going fine. I personally was having trouble with understanding

Bool-profunctors lately, so I decided to take a break to clear my head.I think my problem was that I was treating

Bool-profunctors as regular functors.`The course is going fine. I personally was having trouble with understanding **Bool**-profunctors lately, so I decided to take a break to clear my head. I think my problem was that I was treating **Bool**-profunctors as regular functors.`

@John I use (the sadly late) Paul Hudak's Euterpea. Paul's original paper is worth reading. Outputs include Lilliput scores and midi devices. I don't know what soft synth windoze folks use. I use Fluidsynth which is linux/mac.

Trying to link to the Euterpea github account I came across Sonic Pi which looks interesting.

`@John I use (the sadly late) Paul Hudak's [Euterpea](http://www.euterpea.com/). Paul's [original paper](http://haskell.cs.yale.edu/wp-content/uploads/2015/03/HSoM.pdf) is worth reading. Outputs include [Lilliput](https://musescore.com/user/27052214/scores/4874469) scores and midi devices. I don't know what soft synth windoze folks use. I use Fluidsynth which is linux/mac. Trying to link to the Euterpea github account I came across [Sonic Pi](https://sonic-pi.net/) which looks interesting.`

I got up to chapter 3 of the book. Since then I've just tried to work through the daily forum posts as I doubted I'd ever catch up otherwise. I've also bookmarked all the papers people have linked to and am slowly reading them. Thanks to @Matthew I'm also still trying to get round to posting a decent diagram to go with Kleisli's Every standard construction is induced by a pair of adjoint functors (super-short and super-important IMO) instead of some attempt at writing a YAMT (yet another monad tutorial).

`I got up to chapter 3 of the book. Since then I've just tried to work through the daily forum posts as I doubted I'd ever catch up otherwise. I've also bookmarked all the papers people have linked to and am slowly reading them. Thanks to @Matthew I'm also still trying to get round to posting a decent diagram to go with Kleisli's [Every standard construction is induced by a pair of adjoint functors](http://www.ams.org/journals/proc/1965-016-03/S0002-9939-1965-0177024-4/S0002-9939-1965-0177024-4.pdf) (super-short and super-important IMO) instead of some attempt at writing a YAMT (yet another monad tutorial).`

I would also like to express my gratitude along with amazement at the pace you've managed to keep up. I got caught up with finishing a lab project and now with exam preparation. However, I'll be done with exams on Friday and have a month before my next project starts, time which I was planning to fill with maths. I've currently worked the book up to the end of chapter 4, but still need to start digesting your posts on the subject.

Thinking about this material daily and reworking basic category theory has illuminated and solidified many of the concepts. Your perspective has really helped in this regard. I've finally got a good grasp on adjoints and all of the maths I'm encountering seems an entire step or two easier in the uptake. I'm also starting to get a feeling of how one could effectively use monoidal categories. When browsing some of your work before this course, I had trouble understanding exactly what was going on when tensoring and couldn't divorce the idea from categorical (co)products. The lead-up via monoidal preorders has changed my perspective on what these structures are.

I would really appreciate your thoughts (particularly big picture stuff) on the networks from chapters 5 and 6. Additionally, a compiled list of references for further study in some actionable order at the end of the course would be really helpful. The lists on your website have proved invaluable for my studies.

Regarding music recording: I'm no expert, but I've messed around with home recording and some DJing from an early age. A short list of what you'll need based on your request for electronic and live music:

1) You'll want some sort of USB audio interface. This is essentially an external sound card that includes the requisite input/output channels and allows you to record latency free (the sound card on your computer is probably to slow for this). These come in a variety of configurations and price ranges. I don't have any concrete recommendations, but you will want something that has a MIDI input as well as one or two audio jack/XLR (for microphone) inputs along with headphone outputs. You should be able to find an all-around solution for around 200$.

2) Midi Controller: If you don't already own a keyboard with MIDI output, I would highly recommend getting such a thing. Even if you don't play piano, you should have some device that's more tactile than a computer keyboard and mouse to tap out rhythms and melodies ect. You can get bare bones MIDI keyboards that only cover 2 or 3 octaves (they can sit on your desk) fairly cheaply (under 100$). If you want weighted keys it gets more expensive. Alternatively, if you are more into rhythmic expression, there are a plethora of pad-based MIDI controllers (imagine an array of square tappable rubber pads) out there, also available in basic options for around 100$.

3) Microphone: Most instruments like digital keyboards, guitars, basses ect., will have audio jack outputs that you can plug directly into your USB interface or if you have a guitar amplifier you can output that sound into your computer. If you want to record voice, ambient sound or acoustic instruments you will need a microphone. It's worth getting a decent one. I have the Rode NT1-A which comes in a practical starter kit including pop-shield, cradle, and cable. This would set you back 150$

4) Cables and adaptors. Rule of thumb: There will always be one essential piece missing, no matter how much you've considered things before ordering. Maybe apply some category theory.

Here things get a little complicated. As far as I'm informed most people doing recording work either in Logic Pro if they're on Mac or Cubase if on Windows. Imagine the software platform as a framework that you bolt plug-ins on as desired. Most of the specialized functionality professionals use comes from these. The base software will, however, cover most of your needs and there are many free plug-ins online. I've personally used Cubase. It's a little intimidating to get set up (audio drivers), but after that, it's not too difficult. I'm sure there are tutorials on youtube. I recommend you get full-fledged software that's in standard use.

The way these software platforms work is that you have a timeline where you can edit your tracks (eg. cut, paste, ect.). There is a strong division between audio tracks (e.g from a microphone) which are handled in waveform and midi tracks, which are just a pitch plus an intensity and duration. Audio will sound like whatever you piped in. You can layer effects on top of this (e.g reverb, delay, compression, (co)equalizer :). The software comes with a basic suite of standard effects and tools. For more specialized/refined ones (e.g good guitar amp emulations) there are plug-ins. MIDI tracks, on the other hand, has to be 'instrumentized' in order to sound. The software uses so-called VST instruments for this. These can either be synthesizers (compute sound based on algorithmic wave-form generation) or sample-based instruments (take recordings of an instrument's sound and adapt it to your MIDI track). Generally, the VST instruments included in the base software aren't satisfactory. Good synthesizer plugins can be found for free or cheap online (only requires programming to produce) while good sampled instruments (e.g symphonic instruments) get really pricey since people have to go out and record thousands of samples for each instrument. The standard plug-in used by composers ect. is called Native Instruments and costs 500+, but can produce orchestral tracks that are for the most part indistinguishable from an actual recording.

But again, a standard software will get you started and once you mess around and find your work-flow you will know what you need additionally. If you want to perform you're composed electronic music creatively in a live setting, that's a separate can of worms.

That concludes my mini-lecture. Hope this gives you some idea of what you're in for. All in all, I would say getting into the software aspect is on the level of learning basic photoshop or equivalent. You will only make use of a small subset of the features. And there will be tons of resources out there to help get you started with basic work flows. Most of the fun comes with messing around. Feel free to ask any further questions. I should get back to studying.

`I would also like to express my gratitude along with amazement at the pace you've managed to keep up. I got caught up with finishing a lab project and now with exam preparation. However, I'll be done with exams on Friday and have a month before my next project starts, time which I was planning to fill with maths. I've currently worked the book up to the end of chapter 4, but still need to start digesting your posts on the subject. Thinking about this material daily and reworking basic category theory has illuminated and solidified many of the concepts. Your perspective has really helped in this regard. I've finally got a good grasp on adjoints and all of the maths I'm encountering seems an entire step or two easier in the uptake. I'm also starting to get a feeling of how one could effectively use monoidal categories. When browsing some of your work before this course, I had trouble understanding exactly what was going on when tensoring and couldn't divorce the idea from categorical (co)products. The lead-up via monoidal preorders has changed my perspective on what these structures are. I would really appreciate your thoughts (particularly big picture stuff) on the networks from chapters 5 and 6. Additionally, a compiled list of references for further study in some actionable order at the end of the course would be really helpful. The lists on your website have proved invaluable for my studies. Regarding music recording: I'm no expert, but I've messed around with home recording and some DJing from an early age. A short list of what you'll need based on your request for electronic and live music: - Hardware: 1) You'll want some sort of USB audio interface. This is essentially an external sound card that includes the requisite input/output channels and allows you to record latency free (the sound card on your computer is probably to slow for this). These come in a variety of configurations and price ranges. I don't have any concrete recommendations, but you will want something that has a MIDI input as well as one or two audio jack/XLR (for microphone) inputs along with headphone outputs. You should be able to find an all-around solution for around 200$. 2) Midi Controller: If you don't already own a keyboard with MIDI output, I would highly recommend getting such a thing. Even if you don't play piano, you should have some device that's more tactile than a computer keyboard and mouse to tap out rhythms and melodies ect. You can get bare bones MIDI keyboards that only cover 2 or 3 octaves (they can sit on your desk) fairly cheaply (under 100$). If you want weighted keys it gets more expensive. Alternatively, if you are more into rhythmic expression, there are a plethora of pad-based MIDI controllers (imagine an array of square tappable rubber pads) out there, also available in basic options for around 100$. 3) Microphone: Most instruments like digital keyboards, guitars, basses ect., will have audio jack outputs that you can plug directly into your USB interface or if you have a guitar amplifier you can output that sound into your computer. If you want to record voice, ambient sound or acoustic instruments you will need a microphone. It's worth getting a decent one. I have the Rode NT1-A which comes in a practical starter kit including pop-shield, cradle, and cable. This would set you back 150$ 4) Cables and adaptors. Rule of thumb: There will always be one essential piece missing, no matter how much you've considered things before ordering. Maybe apply some category theory. - Software: Here things get a little complicated. As far as I'm informed most people doing recording work either in Logic Pro if they're on Mac or Cubase if on Windows. Imagine the software platform as a framework that you bolt plug-ins on as desired. Most of the specialized functionality professionals use comes from these. The base software will, however, cover most of your needs and there are many free plug-ins online. I've personally used Cubase. It's a little intimidating to get set up (audio drivers), but after that, it's not too difficult. I'm sure there are tutorials on youtube. I recommend you get full-fledged software that's in standard use. The way these software platforms work is that you have a timeline where you can edit your tracks (eg. cut, paste, ect.). There is a strong division between audio tracks (e.g from a microphone) which are handled in waveform and midi tracks, which are just a pitch plus an intensity and duration. Audio will sound like whatever you piped in. You can layer effects on top of this (e.g reverb, delay, compression, (co)equalizer :). The software comes with a basic suite of standard effects and tools. For more specialized/refined ones (e.g good guitar amp emulations) there are plug-ins. MIDI tracks, on the other hand, has to be 'instrumentized' in order to sound. The software uses so-called VST instruments for this. These can either be synthesizers (compute sound based on algorithmic wave-form generation) or sample-based instruments (take recordings of an instrument's sound and adapt it to your MIDI track). Generally, the VST instruments included in the base software aren't satisfactory. Good synthesizer plugins can be found for free or cheap online (only requires programming to produce) while good sampled instruments (e.g symphonic instruments) get really pricey since people have to go out and record thousands of samples for each instrument. The standard plug-in used by composers ect. is called Native Instruments and costs 500+, but can produce orchestral tracks that are for the most part indistinguishable from an actual recording. But again, a standard software will get you started and once you mess around and find your work-flow you will know what you need additionally. If you want to perform you're composed electronic music creatively in a live setting, that's a separate can of worms. That concludes my mini-lecture. Hope this gives you some idea of what you're in for. All in all, I would say getting into the software aspect is on the level of learning basic photoshop or equivalent. You will only make use of a small subset of the features. And there will be tons of resources out there to help get you started with basic work flows. Most of the fun comes with messing around. Feel free to ask any further questions. I should get back to studying.`

Well, since we're talking music and math, every sequence on the OEIS can be played and listened to.

For instance, here is Recamán's sequence: https://oeis.org/play?seq=A005132

`Well, since we're talking music and math, every sequence on the OEIS can be played and listened to. For instance, here is Recamán's sequence: https://oeis.org/play?seq=A005132`

Thanks for the detailed advice, Marius! I improvise on the piano, and I have a Yamaha Clavinova, which does MIDI, but it's old - some of the keys are going bad, and it has a floppy disk drive... but more importantly, I've never learned how to create MIDI files by improvising and then edit them to create better compositions.

I've created electronic pieces of music that I edit and process (adding reverb and other effect, cutting and pasting, etc.) using Audacity. This is lots of fun, but I don't have the tight control over rhythm that I'm hoping I'd get using MIDI - I'm just editing sound files directly.

I've got Windows so it sounds like I should get Cubase. I like electronic sounds as long as I have enough control over them so I should start with those and try Native Instruments after I get serious! That sounds exciting but I need to work my way into this.

Exciting!

I will answer your math questions separately when I'm feeling more awake. I took a second dose of a shingles vaccine and it's really knocked me out all day. (My wife too, otherwise I wouldn't believe it's the vaccine!)

`Thanks for the detailed advice, Marius! I improvise on the piano, and I have a Yamaha Clavinova, which does MIDI, but it's old - some of the keys are going bad, and it has a floppy disk drive... but more importantly, I've never learned how to create MIDI files by improvising and then edit them to create better compositions. I've created electronic pieces of music that I edit and process (adding reverb and other effect, cutting and pasting, etc.) using Audacity. This is lots of fun, but I don't have the tight control over rhythm that I'm hoping I'd get using MIDI - I'm just editing sound files directly. I've got Windows so it sounds like I should get Cubase. I like electronic sounds as long as I have enough control over them so I should start with those and try Native Instruments after I get serious! That sounds exciting but I need to work my way into this. Exciting! I will answer your math questions separately when I'm feeling more awake. I took a second dose of a shingles vaccine and it's really knocked me out all day. (My wife too, otherwise I wouldn't believe it's the vaccine!)`

I'm feeling better now. But I guess Marius' math question wasn't too technical! He wrote:

Sure! I've decided that I should do at least something about these chapters, since they're so heavily based on work I've done with students.

`I'm feeling better now. But I guess Marius' math question wasn't too technical! He wrote: > I would really appreciate your thoughts (particularly big picture stuff) on the networks from chapters 5 and 6. Sure! I've decided that I should do at least something about these chapters, since they're so heavily based on work I've done with students.`

I'm late to this thread, but I'm enjoying the course!

I'm still on Chapter 3 :) -- but supplementing my reading with David Spivak's papers, and their AQL IDE which they have kindly made available for download.

Personally I have gained from the exercises, esp the easier/counting ones since I'm not used to thinking about categories in this way... (I'm normally working in a situation where the compiler does it all for me, if at-all!).

With regard to following the book: it's very helpful that they chose a simplified presentation for their material, since we're not all category theory experts. But your lectures are a better road map to better understanding the necessary category theory than say following the references, which being honest I often don't do.

So, thanks! I'm grateful (and impressed, as always) for the energy you've put into this, and I'm sure you'll find a graceful way to wrap up (somehow!)

`I'm late to this thread, but I'm enjoying the course! I'm still on Chapter 3 :) -- but supplementing my reading with David Spivak's papers, and their AQL IDE which they have kindly made available for download. Personally I have gained from the exercises, esp the easier/counting ones since I'm not used to thinking about categories in this way... (I'm normally working in a situation where the compiler does it all for me, if at-all!). With regard to following the book: it's very helpful that they chose a simplified presentation for their material, since we're not all category theory experts. But your lectures are a better road map to better understanding the necessary category theory than say following the references, which being honest I often don't do. So, thanks! I'm grateful (and impressed, as always) for the energy you've put into this, and I'm sure you'll find a graceful way to wrap up (somehow!)`

First of all, thank you for creating your lectures and providing much needed extra examples and perspectives! To use an videogame analogy, this is like starting out at level 1 in a big scary world, but already having a slightly overpowered sword and potion in the backpack.

As for the class: Although I tried to follow the class during semester, I often missed the energy to do more than one exercise or even one. I skimmed here and there, but not in a linear and consistent fashion. Now however there is finally summer break for me! I will spend the next 2 weeks at my grandparents place with a laptop, paper, no internet and a offline copy of this forum - I hope I can do some catching up! :) (My plan is to skim chapter 1 for unkown concepts then work through chapter 2 3 4). I will see how I can maximize my productivity during the day - much more than 4+ hours of intensive math a day wont fly. I guess I will make many walks and playing with dogs in between. I am really looking forward for this time - and I hope I can catch up, I want to contribute! :)

Independently of my own success: You have done something marvelous with this class. Many have benefited and many more will benefit from your work, I am sure. Thank you so much!

`First of all, thank you for creating your lectures and providing much needed extra examples and perspectives! To use an videogame analogy, this is like starting out at level 1 in a big scary world, but already having a slightly overpowered sword and potion in the backpack. As for the class: Although I tried to follow the class during semester, I often missed the energy to do more than one exercise or even one. I skimmed here and there, but not in a linear and consistent fashion. Now however there is finally summer break for me! I will spend the next 2 weeks at my grandparents place with a laptop, paper, no internet and a offline copy of this forum - I hope I can do some catching up! :) (My plan is to skim chapter 1 for unkown concepts then work through chapter 2 3 4). I will see how I can maximize my productivity during the day - much more than 4+ hours of intensive math a day wont fly. I guess I will make many walks and playing with dogs in between. I am really looking forward for this time - and I hope I can catch up, I want to contribute! :) Independently of my own success: You have done something marvelous with this class. Many have benefited and many more will benefit from your work, I am sure. Thank you so much!`

Thank you very much for this course, John, reading a lecture or doing an exercise on my train to/from work makes my day (that's ~5h/week dedication).

Since the beginning, I was unable to keep up with your speed, but it did not matter at all. The lectures (and the solved exercises by our fellas!) are a great permanent resource for self-paced learning. Your explanations are a perfect complement to the sketches. And the fact that you brought together so many people spotting mistakes/typos/suggestions/puzzles to improve Spivak and Fong's draft should be an example to the mathematical community.

I prefer to go slow and do every proof myself than follow your rythm. This is the magic of this non-traditional course. And, personally, it also helps to try to sit every two weeks with a coworker doing the same, to confront our solutions in the non-virtual world. If we didn't have the 'true' solutions to the book exercises, we probably wouldn't be doing the course as seriously we are.

My recommendation, therefore, is that you do not give up but do not get fed up either. This should be pleasant for everyone, especially you. Know that your effort is valued, that it makes a huge impact, and follow your own rythm. Even one lecture per week would be fine. If you have to pause for a while, do so. We won't blame you, we thank you for everything you already gave.

And good luck with the gym! Mens sana in corpore sano.

`Thank you very much for this course, John, reading a lecture or doing an exercise on my train to/from work makes my day (that's ~5h/week dedication). Since the beginning, I was unable to keep up with your speed, but it did not matter at all. The lectures (and the solved exercises by our fellas!) are a great permanent resource for self-paced learning. Your explanations are a perfect complement to the sketches. And the fact that you brought together so many people spotting mistakes/typos/suggestions/puzzles to improve Spivak and Fong's draft should be an example to the mathematical community. I prefer to go slow and do every proof myself than follow your rythm. This is the magic of this non-traditional course. And, personally, it also helps to try to sit every two weeks with a coworker doing the same, to confront our solutions in the non-virtual world. If we didn't have the 'true' solutions to the book exercises, we probably wouldn't be doing the course as seriously we are. My recommendation, therefore, is that you do not give up but do not get fed up either. This should be pleasant for everyone, especially you. Know that your effort is valued, that it makes a huge impact, and follow your own rythm. Even one lecture per week would be fine. If you have to pause for a while, do so. We won't blame you, we thank you for everything you already gave. And good luck with the gym! Mens sana in corpore sano.`

I have unfortunately fallen rather behind thanks to my full-time internship this summer -- I haven't had the energy to consistently keep up with things when i get back from work. :/ To echo everyone else, this course has already been an incredible resource; I think I've already mentioned to you that this is the first time I've really understood what natural transformations are! Category theory no longer feels quite so impenetrable, and I'm starting to see the right footholds to cling to.

It's also been a problem that the forum notifications no longer seem to be hitting my inbox. Being able to see how much activity has been happening from my phone's lock screen is a powerful motivator to come back and catch up on things, and without the notifications... well, out of sight, out of mind, as they say. I'm not sure if this is a problem on my end, but I don't seem to have any control over it.

`I have unfortunately fallen rather behind thanks to my full-time internship this summer -- I haven't had the energy to consistently keep up with things when i get back from work. :/ To echo everyone else, this course has already been an incredible resource; I think I've already mentioned to you that this is the first time I've really understood what natural transformations are! Category theory no longer feels quite so impenetrable, and I'm starting to see the right footholds to cling to. It's also been a problem that the forum notifications no longer seem to be hitting my inbox. Being able to see how much activity has been happening from my phone's lock screen is a powerful motivator to come back and catch up on things, and without the notifications... well, out of sight, out of mind, as they say. I'm not sure if this is a problem on my end, but I don't seem to have any control over it.`

Dare I demand that you spend more of your time, when I have been so fortunate as to have had months of free -- free! -- tuition in my favourite area of study by one of the world's foremost experts? The temerity! - but here I am. I can only apologise for being so inactive in the comment threads - like others, I will make time my excuse. The course, for me, has been on that razor edge of productive learning between the trivial and the impossible that maths seems to skate, and I'm enjoying it immensely.

Some hits:

My intuition around constructing "free" things has really improved

The intuition about adjunctions being a "best guess" where there's no exact answer - I've had the technical definition memorised for years, but it took the course for the intuition to drop

Enriched categories as glorified posets! They are much easier to understand this way

String diagrams have clicked for me - the monoidy ones I'd seen previously had seemed abstruse, but the lemon meringue diagram just nailed it

The way the monoidal structure interplays with posets - I could have taken a stab to guess the laws, but working it out with real-ish examples really makes it click

Using free categories & functors to model database migrations was enlightening, and is enormously relevant to my work.

I already had an intuition for profunctors as "pipeline shaped things with input & output" from programming, but working through constructing the enriched composition & categorical structure has really fleshed out the concept.

So here's a fun, and slightly embarrassing one. I think seeing the connection between feasibility relation composition & matrices was supposed to help me understand profunctors, but it actually helped me get.....

ahemmatrices. So I never looked at matrices after high school, and never really understood why they were composed end-on-end with these specific rote-learned multiplication rules. The feasibility matrices got me there.This is from a combination of the book & the course, but I never would have gotten this far into the book in this level of detail without the course.

I fervently hope that you will continue the course, but will be nothing but grateful if you understandably decide to wrap it up here. Thank you John. That said, operads are my favourite thing in the world, and I might cry if we miss out.

`Dare I demand that you spend more of your time, when I have been so fortunate as to have had months of free -- free! -- tuition in my favourite area of study by one of the world's foremost experts? The temerity! - but here I am. I can only apologise for being so inactive in the comment threads - like others, I will make time my excuse. The course, for me, has been on that razor edge of productive learning between the trivial and the impossible that maths seems to skate, and I'm enjoying it immensely. Some hits: - My intuition around constructing "free" things has really improved - The intuition about adjunctions being a "best guess" where there's no exact answer - I've had the technical definition memorised for years, but it took the course for the intuition to drop - Enriched categories as glorified posets! They are much easier to understand this way - String diagrams have clicked for me - the monoidy ones I'd seen previously had seemed abstruse, but the lemon meringue diagram just nailed it - The way the monoidal structure interplays with posets - I could have taken a stab to guess the laws, but working it out with real-ish examples really makes it click - Using free categories & functors to model database migrations was enlightening, and is enormously relevant to my work. - I already had an intuition for profunctors as "pipeline shaped things with input & output" from programming, but working through constructing the enriched composition & categorical structure has really fleshed out the concept. - So here's a fun, and slightly embarrassing one. I think seeing the connection between feasibility relation composition & matrices was supposed to help me understand profunctors, but it actually helped me get..... *ahem* matrices. So I never looked at matrices after high school, and never really understood why they were composed end-on-end with these specific rote-learned multiplication rules. The feasibility matrices got me there. This is from a combination of the book & the course, but I never would have gotten this far into the book in this level of detail without the course. I fervently hope that you will continue the course, but will be nothing but grateful if you understandably decide to wrap it up here. Thank you John. That said, operads are my favourite thing in the world, and I might cry if we miss out.`

I have been enjoying the course immensely, and feel like I am finally getting my grounding in category theory. I've explored all sorts of advanced topics, but coming back to base has really made me feel like I understand, not just parrot these ideas.

I do need to do my homework for this course though. I keep trying to compleatly formalize and getting distracted by fun but mathematically trivial proof system issues.

(For example it takes a page of tactics to prove that direct image with powerset on objects is a endo-functor on Set)

`I have been enjoying the course immensely, and feel like I am finally getting my grounding in category theory. I've explored all sorts of advanced topics, but coming back to base has really made me feel like I understand, not just parrot these ideas. I do need to do my homework for this course though. I keep trying to compleatly formalize and getting distracted by fun but mathematically trivial proof system issues. (For example it takes a page of tactics to prove that direct image with powerset on objects is a endo-functor on Set)`

Thanks, everyone! I'm back in action, revitalized and re-energized thanks to the cuisine of Singapore.

`Thanks, everyone! I'm back in action, revitalized and re-energized thanks to the cuisine of Singapore.`

Thank you so much for the lectures, John!

I've been following the course since day 1 and it has been a tremendous help. I'm just studying a master in computer science so I've been lacking a lot of the mathematical maturity people seem to have here - which is why I was only marginally commenting and starting new threads.

It's been a great help reading the lectures as a supplement to the Seven sketches and trying to come up with answers to puzzles. What has been great is also just seeing how

otherpeople come up with answers and understanding what their thought processes are. I feel like I'm just now starting to get some solid foundations in CT. I also feel like there's so much more stuff to learn in CT than I originally anticipated :)I hope you do manage to finish the lectures, in one form or another. Personally, I'd be okay with having fewer lectures per chapter, as long as we get to hear a bit about everything that the book talks about. The last two chapters seem especially interesting and I'd hate not hearing your take on them!

`Thank you so much for the lectures, John! I've been following the course since day 1 and it has been a tremendous help. I'm just studying a master in computer science so I've been lacking a lot of the mathematical maturity people seem to have here - which is why I was only marginally commenting and starting new threads. It's been a great help reading the lectures as a supplement to the Seven sketches and trying to come up with answers to puzzles. What has been great is also just seeing how *other* people come up with answers and understanding what their thought processes are. I feel like I'm just now starting to get some solid foundations in CT. I also feel like there's so much more stuff to learn in CT than I originally anticipated :) I hope you do manage to finish the lectures, in one form or another. Personally, I'd be okay with having fewer lectures per chapter, as long as we get to hear a bit about everything that the book talks about. The last two chapters seem especially interesting and I'd hate not hearing your take on them!`

Okay, I'll try to say

somethingabout all these chapters. I'm actually in the mood for becoming a bit less technical, though it's hard to do without just saying a bunch of mush.`Okay, I'll try to say _something_ about all these chapters. I'm actually in the mood for becoming a bit less technical, though it's hard to do without just saying a bunch of mush.`

Welcome back John :) Usually everybody finds something in "a bunch of mush", sometimes only a single from 3-4 analogies clicks. I'm going to start profunctors closer to this weekend, hopefully will be able to quickly catch up.

`Welcome back John :) Usually everybody finds something in "a bunch of mush", sometimes only a single from 3-4 analogies clicks. I'm going to start profunctors closer to this weekend, hopefully will be able to quickly catch up.`

@John , I feel you are offering us a great roadmap..my gratitude about your decision on sharing your deep and very inspiring math Knowledge..I´m assuming that it´s not a intro course..then, for me, some barriers are natural and personnal effort, persistency and avoid anxiety are going to allow me to transpose they and, luckly, enjoy the content a 100% ..For me your course is just fine and very inspiring..Please keep it running on line as long as possible..I wish more people (worldwide) could have enough english/math skills to profit of it also..best regards

`@John , I feel you are offering us a great roadmap..my gratitude about your decision on sharing your deep and very inspiring math Knowledge..I´m assuming that it´s not a intro course..then, for me, some barriers are natural and personnal effort, persistency and avoid anxiety are going to allow me to transpose they and, luckly, enjoy the content a 100% ..For me your course is just fine and very inspiring..Please keep it running on line as long as possible..I wish more people (worldwide) could have enough english/math skills to profit of it also..best regards`

Another not-quite-so-energetic student chiming in. I joined the course late May and just finished the book exercises of chapter 4. At some point my progress in the book got ahead of my progress reading through the lectures (just finishing chapter 2). I think working through the book is a lot more "comfortable" in two senses. First, it lends itself to the "shut up and calculate" mentality and you don't really need to digest the material as much. Second, you don't risk looking dumb in public! But I'm realizing now that the lectures and puzzles contain so many deep, intuitive explanations (both from yourself and the others who've been working through the puzzles "out loud"). So I'm committed to catch up and also interact more on the forums. I agree with the above posters who suggested that the later lectures could be more condensed. In particular, I'm really looking forward to your perspective on toposes. I've been slowly studying Mac Lane + Moerdijk for awhile now, and it's been very encouraging to find it dramatically clearer as I progress in this course. Thank you very much for your effort John. :)

`Another not-quite-so-energetic student chiming in. I joined the course late May and just finished the book exercises of chapter 4. At some point my progress in the book got ahead of my progress reading through the lectures (just finishing chapter 2). I think working through the book is a lot more "comfortable" in two senses. First, it lends itself to the "shut up and calculate" mentality and you don't really need to digest the material as much. Second, you don't risk looking dumb in public! But I'm realizing now that the lectures and puzzles contain so many deep, intuitive explanations (both from yourself and the others who've been working through the puzzles "out loud"). So I'm committed to catch up and also interact more on the forums. I agree with the above posters who suggested that the later lectures could be more condensed. In particular, I'm really looking forward to your perspective on toposes. I've been slowly studying Mac Lane + Moerdijk for awhile now, and it's been very encouraging to find it dramatically clearer as I progress in this course. Thank you very much for your effort John. :)`

Great, Dennis! I've been taking a little break from lectures and enjoying life in Singapore, but I will spring back into action soon - maybe tomorrow.

I agree, the lectures, the puzzles, and especially the online discussion take the material in the book and expand on it a lot... which should be helpful for people who want to deeply understand this material. I hope you join our discussions!

`Great, Dennis! I've been taking a little break from lectures and enjoying life in Singapore, but I will spring back into action soon - maybe tomorrow. I agree, the lectures, the puzzles, and especially the online discussion take the material in the book and expand on it a lot... which should be helpful for people who want to deeply understand this material. I hope you join our discussions!`

Dear John, Thank you very much for this course. It's a real gift to have category theory made more intuitive with such a supportive environment. I have so many other things I'm researching but this is a subject that I want to learn. Also, I prefer to relate the subject to my own particular interests rather than work through a textbook linearly. But I hope that I can be active here nevertheless. Thank you.

`Dear John, Thank you very much for this course. It's a real gift to have category theory made more intuitive with such a supportive environment. I have so many other things I'm researching but this is a subject that I want to learn. Also, I prefer to relate the subject to my own particular interests rather than work through a textbook linearly. But I hope that I can be active here nevertheless. Thank you.`

I think the course is going well! I wish I had more time to spend on the puzzles. Here's what I like

poset reflectionto think about hypergraph categories in a more simplified way.One thing I think is a little lacking is what new ideas category theory brings to the table when talking about resource theories/codesign diagrams/electric circuits, etc. Like if I notice that a digrammatic language can be formalized as a hypergraph category, what do I learn about the network/diagramming language? It took me a while to understand how group theory was useful, because I could see how groups encoded symmetry and formalized reversible actions in my initial readings, but it wasn't until I learned the orbit-stabilizer theorem and burnside lemma that I could see how thinking in terms of groups was useful for all kinds of previous problems.

But I still really like the course!

`I think the course is going well! I wish I had more time to spend on the puzzles. Here's what I like * Starting with posets and monoidal posets, and adjoint monotone functions really helped me grasp the later ideas. Just today I was using the *poset reflection* to think about hypergraph categories in a more simplified way. * The applied examples provides a nice scaffold for the category theory (and helps me think of ways that the category theory might be useful elsewhere). One thing I think is a little lacking is what new ideas category theory brings to the table when talking about resource theories/codesign diagrams/electric circuits, etc. Like if I notice that a digrammatic language can be formalized as a hypergraph category, what do I learn about the network/diagramming language? It took me a while to understand how group theory was useful, because I could see how groups encoded symmetry and formalized reversible actions in my initial readings, but it wasn't until I learned the orbit-stabilizer theorem and burnside lemma that I could see how thinking in terms of groups was useful for all kinds of previous problems. But I still really like the course!`

The main thing you instantly learn is that you can stick together diagrams in a whole bunch of ways and get diagrams that make sense. This 'whole bunch of ways' is the definition of 'hypergraph category'.

The second thing you instantly learn is a whole bunch of equations saying 'if you stick together diagrams like this, and then stick on some more like that, then you get the same thing as if you did... something else'.

The third thing you instantly learn is that the semantics of your diagram language should be a hypergraph functor: in other words, a functor that's compatible with all these ways of sticking together diagrams. That's indeed how it works in all the examples I've studied so far.

Furthermore, Brendan's theorems on decorated corelation categories give recipes for actually defining choices of semantics that will be hypergraph functors. So, you don't need to do this 'from scratch'.

`> Like if I notice that a digrammatic language can be formalized as a hypergraph category, what do I learn about the network/diagramming language? The main thing you instantly learn is that you can stick together diagrams in a whole bunch of ways and get diagrams that make sense. This 'whole bunch of ways' is the definition of 'hypergraph category'. The second thing you instantly learn is a whole bunch of equations saying 'if you stick together diagrams like this, and then stick on some more like that, then you get the same thing as if you did... something else'. The third thing you instantly learn is that the semantics of your diagram language should be a hypergraph functor: in other words, a functor that's compatible with all these ways of sticking together diagrams. That's indeed how it works in all the examples I've studied so far. Furthermore, Brendan's theorems on decorated corelation categories give recipes for actually defining choices of semantics that will be hypergraph functors. So, you don't need to do this 'from scratch'.`

This course is extraordinary! Thank you, John! I know what it means to run such a lectures. You spent a lot of energy in a selfless manner.

If I understand correctly, the course is temporarily over. I was especially looking forward to the next chapter, where diagrams are understood as morphisms between their input and output sets. I deal a lot with Causal Loop Diagrams (CLD) that explicitly have no input and no output sets and only consist of loops. Therefore, the view of seeing a diagram as a morphism between two sets can not quite satisfy me. I would like to have discussed this in the lessons to the next chapter.

But maybe I can find answers elsewhere in your very rich offers.

Thanks once again. It was fun!

Peter

`This course is extraordinary! Thank you, John! I know what it means to run such a lectures. You spent a lot of energy in a selfless manner. If I understand correctly, the course is temporarily over. I was especially looking forward to the next chapter, where diagrams are understood as morphisms between their input and output sets. I deal a lot with Causal Loop Diagrams (CLD) that explicitly have no input and no output sets and only consist of loops. Therefore, the view of seeing a diagram as a morphism between two sets can not quite satisfy me. I would like to have discussed this in the lessons to the next chapter. But maybe I can find answers elsewhere in your very rich offers. Thanks once again. It was fun! Peter`

Yes, Peter, the course is over. I don't think "temporarily" is a good adverb to insert here. Sorry!

However, the book by Fong and Spivak is still there for you to read, along with Tai-Danae Bradley's excellent short book. I also have zillions of blog articles and papers on categories where the morphisms are diagrams, available here:

If you have questions, just email me.

`Yes, Peter, the course is over. I don't think "temporarily" is a good adverb to insert here. Sorry! However, the book by Fong and Spivak is still there for you to read, along with Tai-Danae Bradley's excellent short book. I also have zillions of blog articles and papers on categories where the morphisms are diagrams, available here: * [Network theory](http://math.ucr.edu/home/baez/networks/). If you have questions, just email me.`