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# Exercises and Puzzles 1 - Chapter 1

edited May 2018

I've finished lectures on Chapter 1. I can't tell if I've been going too slowly or too fast. I wish I knew! Some of you, including the most active students, are familiar with most of the material so far and perhaps eager to get into something deeper, or something more "applied". But others have said you don't have time to keep up. I'm guessing that's the general situation. But I don't know.

In any event, we'll have to slow down now because on Saturday I'm flying to Amsterdam to help teach a week-long school on applied category theory at the Lorentz Center in Leiden, followed by a week-long workshop.

So, for two weeks let's discuss the exercises in Chapter 1. Luckily, these have all been set up by Fredrick Eisele, Dan Oneata and other students. Here are the first few:

To see all the exercises in Chapter 1, go here.

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1.
edited April 2018

I suspect the examples, exercises are a trifle too "simple", or toylike, and, as a result, don't really lend themselves to nontrivial applications. Obviously, the intent is to introduce beginners to the material, I suspect most folks are waiting for the deeper applications---which I notice, can be found on your personal blog.

The interface of this site is rather static. It's not really "live enough". Perhaps a slack channel is in order?

Comment Source:I suspect the examples, exercises are a trifle too "simple", or toylike, and, as a result, don't really lend themselves to nontrivial applications. Obviously, the intent is to introduce beginners to the material, I suspect most folks are waiting for the deeper applications---which I notice, can be found on your personal blog. The interface of this site is rather static. It's not really "live enough". Perhaps a slack channel is in order? 
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2.
edited April 2018

Anyone who wants to create new communication channels for the course is encouraged to do so! I don't have time for that myself, and I'm an old fogey so I don't even know what a "slack channel" is, much less feel the need for one. But other people have been creating resources for the course - like typing in all the exercises! - and when they do this I am happy to advertise them. We have about 300 people registered, many of them programmers, so if we can tap into that power we'll be unstoppable.

Comment Source:Anyone who wants to create new communication channels for the course is encouraged to do so! I don't have time for that myself, and I'm an old fogey so I don't even know what a "slack channel" is, much less feel the need for one. But other people have been creating resources for the course - like typing in all the exercises! - and when they do this I am happy to advertise them. We have about 300 people registered, many of them programmers, so if we can tap into that power we'll be unstoppable.
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3.
edited April 2018

I think I'll try to put more emphasis on real applications of category theory starting in Chapter 2. That chapter is on "resource theories", and we have Tobias Fritz here, who has written at least three important papers on resource theories.

Unfortunately adding extra material that's not in the book will require me to either

1. pick up the pace,
2. skip more stuff that's in the book, or
3. not cover all seven chapters by the end of September, when I have to quit teaching this course.

So, I'm thinking about what to do.

Comment Source:I think I'll try to put more emphasis on real _applications_ of category theory starting in Chapter 2. That chapter is on "resource theories", and we have Tobias Fritz here, who has written at least three important papers on resource theories. Unfortunately adding extra material that's not in the book will require me to either 1. pick up the pace, 2. skip more stuff that's in the book, or 3. not cover all seven chapters by the end of September, when I have to quit teaching this course. So, I'm thinking about what to do. 
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4.
edited April 2018

Slack is a chat website. I have used it on various projects.

[EDIT: Slack is invite only and kind of annoying, I am switching to gitter]

I created a chat for us, and invited you John: https://gitter.im/Applied-Category-Theory-Course/Lobby

Comment Source:Slack is a chat website. I have used it on various projects. [EDIT: Slack is invite only and kind of annoying, I am switching to gitter] I created a chat for us, and invited you John: https://gitter.im/Applied-Category-Theory-Course/Lobby 
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5.

It would be very useful to have a measure of exercise difficulty. I wonder if the forum has some way to do a poll or vote, that we could put in each exercise discussion.

Comment Source:It would be very useful to have a measure of exercise difficulty. I wonder if the forum has some way to do a poll or vote, that we could put in each exercise discussion.
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edited April 2018

I don't know how to do that. What I'd really prefer is for people to talk about things here on the forum, saying stuff like "those were really easy" or "I didn't understand this one".

All the exercises in this little bunch seem very easy except for the last, which is more difficult than it strictly needs to be, since involves some confusing notation (which I tried to explain).

Comment Source:I don't know how to do that. What I'd really prefer is for people to talk about things here on the forum, saying stuff like "those were really easy" or "I didn't understand this one". All the exercises in this little bunch seem very easy except for the last, which is more difficult than it strictly needs to be, since involves some confusing notation (which I tried to explain).
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7.
edited April 2018

Hi, in my case I find the pace reasonably paced even having a "day job" but I'm lucky to have had contact with the ideas before. It must be a difficult compromise to accommodate all backgrounds.

Comment Source:Hi, in my case I find the pace reasonably paced even having a "day job" but I'm lucky to have had contact with the ideas before. It must be a difficult compromise to accommodate all backgrounds.
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8.
edited April 2018

Thanks for letting me know how you're doing, Jesus!

It must be a difficult compromise to accommodate all backgrounds.

It could be, if people complained a lot. What I'm seeing, though, is a small group of people who are engaged in the discussions and a much larger group who are not. This is typical in classes, but usually I get some understanding of people's engagement through their homework and tests! With this class I'm finding it harder to get feedback. Maybe I should offer a degree and make people pass tests to get the degree. :-)

Comment Source:Thanks for letting me know how you're doing, Jesus! > It must be a difficult compromise to accommodate all backgrounds. It could be, if people complained a lot. What I'm seeing, though, is a small group of people who are engaged in the discussions and a much larger group who are not. This is typical in classes, but usually I get some understanding of people's engagement through their homework and tests! With this class I'm finding it harder to get feedback. Maybe I should offer a degree and make people pass tests to get the degree. :-)
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9.

Given the number and diversity of the participants, surely there is no balance of (fast/slow)x(applied/foundational) that makes everybody happy. Apart from inviting people to express their preferences, I would go with the presumption that the "utterly silent" are either entirely lost or happy with the balance you have chosen so far.

Personally, I am happy with how the course is going (thanks @JohnBaez !)

I did find the pace for chapter one a bit slow and the first exercises posted above a bit too easy. But I definitely would not want the pace to speed up just when we are moving from the relatively simple (and, to me, more familiar) stuff of that chapter to more sophisticated (and, to me, novel) material of later chapters. And definitely I would not want to venture too quickly into the details of applications before having secured sound CT foundations.

On this regard, I really like Fong and Spivak's "mission statement":

The purpose of this book is to offer a self-contained tour of applied category theory. It is an invitation to discover advanced topics in category theory through concrete real-world examples. Rather than try to give a comprehensive treatment of these topics—which include adjoint functors, enriched categories, proarrow equipments, toposes, and much more—we merely provide a taste. We want to give readers some insight into how it feels to work with these structures as well as some ideas about how they might show up in practice.

For a first course on "CT and how to use it", the examples of applications provided in the book seem sufficient. I see the main added value of the course to be in Prof. Baez's (and the community of participants') further explanations of those "advanced topics in category theory" and of their connections to other topics in CT or maths more generally, as well as in the discussions of further examples, exercises, and puzzles.

At some point, hopefully, I will enjoy a deep dive into some applications (e.g., the work of @JulesHedges on compositional game theory perhaps?), but I am not yet ready for that.

Comment Source:Given the number and diversity of the participants, surely there is no balance of (fast/slow)x(applied/foundational) that makes everybody happy. Apart from inviting people to express their preferences, I would go with the presumption that the "utterly silent" are either entirely lost or happy with the balance you have chosen so far. Personally, I am happy with how the course is going (thanks @JohnBaez !) I did find the pace for chapter one a bit slow and the first exercises posted above a bit too easy. But I definitely would *not* want the pace to speed up just when we are moving from the relatively simple (and, to me, more familiar) stuff of that chapter to more sophisticated (and, to me, novel) material of later chapters. And definitely I would *not* want to venture too quickly into the details of applications before having secured sound CT foundations. On this regard, I really like Fong and Spivak's "mission statement": >> The purpose of this book is to offer a self-contained tour of applied category theory. It is an invitation to discover advanced topics in category theory through concrete real-world examples. Rather than try to give a comprehensive treatment of these topics—which include adjoint functors, enriched categories, proarrow equipments, toposes, and much more—we merely provide a taste. We want to give readers some insight into how it feels to work with these structures as well as some ideas about how they might show up in practice. For a first course on *"CT and how to use it"*, the examples of applications provided in the book seem sufficient. I see the main added value of the course to be in Prof. Baez's (and the community of participants') further explanations of those "advanced topics in category theory" and of their connections to other topics in CT or maths more generally, as well as in the discussions of further examples, exercises, and puzzles. At some point, hopefully, I will enjoy a deep dive into some applications (e.g., the work of @JulesHedges on compositional game theory perhaps?), but I am not yet ready for that.
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10.
edited April 2018

Valter wrote:

Apart from inviting people to express their preferences, I would go with the presumption that the "utterly silent" are either entirely lost or happy with the balance you have chosen so far.

Yes - but I wish I knew which. :-)

And definitely I would not want to venture too quickly into the details of applications before having secured sound CT foundations.

Okay, good. The next chapter will be on string diagrams, monoidal posets and categories enriched over various monoids - thus providing a gentle introduction to monoidal categories and categories enriched over monoidal categories. This may start seeming new to many people who are very comfortable with the material in Chapter 1. So, I won't suddenly push my foot down on the accelerator.

I am really eager to take a deep dive into various applications, but I'm not sure how to do that while also finishing the course by the end of September! Maybe I can start up some discussion threads in parallel to the main course.

I don't understand open games. I'm trying to plan a visit to Oxford to attend a workshop on that topic in early July.

Comment Source:Valter wrote: > Apart from inviting people to express their preferences, I would go with the presumption that the "utterly silent" are either entirely lost or happy with the balance you have chosen so far. Yes - but I wish I knew which. :-) > And definitely I would not want to venture too quickly into the details of applications before having secured sound CT foundations. Okay, good. The next chapter will be on string diagrams, monoidal posets and categories enriched over various monoids - thus providing a gentle introduction to monoidal categories and categories enriched over monoidal categories. This may start seeming new to many people who are very comfortable with the material in Chapter 1. So, I won't suddenly push my foot down on the accelerator. I am really eager to take a deep dive into various applications, but I'm not sure how to do that while also finishing the course by the end of September! Maybe I can start up some discussion threads in parallel to the main course. I don't understand open games. I'm trying to plan a visit to Oxford to attend a workshop on that topic in early July. 
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I am not sure if I am showing competence in the material yet because I haven't actually written out a lot of the exercises and puzzles but I know for sure I am learning tons thanks to your lectures and the interactions on this forum. It is uncomparable to the amount I would be able to do by myself. I would have probably gotten stuck somewhere along the way with no help available so much thanks to Prof Baez and everyone on this forum.

The pace seems suitable to follow although I might be missing some of the more finer mathematical details (I was trained to trust the mathematician and use the results LOL). I have been allocating at least 5 nights a week solely for this course. I think I will be able to do more exercises and practice my proofs at this pace once I get settled in a little more.

Comment Source:I am not sure if I am showing competence in the material yet because I haven't actually written out a lot of the exercises and puzzles but I know for sure I am learning tons thanks to your lectures and the interactions on this forum. It is uncomparable to the amount I would be able to do by myself. I would have probably gotten stuck somewhere along the way with no help available so much thanks to Prof Baez and everyone on this forum. The pace seems suitable to follow although I might be missing some of the more finer mathematical details (I was trained to trust the mathematician and use the results LOL). I have been allocating at least 5 nights a week solely for this course. I think I will be able to do more exercises and practice my proofs at this pace once I get settled in a little more. 
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12.

It sounds like you're doing great, Michael. It also sounds like you're working harder than many people can afford to - five nights a week is a lot. But getting used to proving things, and reading proofs, takes a lot of work. I spent about 30 hours a week on that as an undergrad: I ate, drink and breathed mathematics. It was great fun.

Since this course is supposed to be about applied category theory I want to start talking more about applications. This will force me to deviate from the book a little.

Comment Source:It sounds like you're doing great, Michael. It also sounds like you're working harder than many people can afford to - five nights a week is a lot. But getting used to proving things, and reading proofs, takes a lot of work. I spent about 30 hours a week on that as an undergrad: I ate, drink and breathed mathematics. It was great fun. Since this course is supposed to be about _applied_ category theory I want to start talking more about applications. This will force me to deviate from the book a little.