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# Lecture 1 - Introduction

edited April 20

Welcome to the Applied Category Theory Course! You can register at left using your full real name as username.

I haven't carefully counted, but so far about 350 people have registered. A bunch of you have written self-introductions. I haven't gotten around to responding to all these yet, but it's obvious we've got a formidable team here. If we work together we can do amazing things.

I decided to teach an online course because I was getting bored of the superficial engagement I'd been getting on my social media posts, mainly on Google+, which started out great but has been going downhill. I like to explain things, discuss things and learn things by talking to people. I've been doing this for decades, first on sci.physics.research, and later on the n-Category Café, the Azimuth Blog... and the Azimuth Forum here, which began life as a platform for creating a wiki about environmental issues: the Azimuth Wiki. All these fora were great fun, but it seems each one gets a bit stale after a while, so it's time for something new.

I decided to teach a course on "applied category theory" because that's what I'm working on, and my former student Brendan Fong is doing a postdoc with David Spivak at MIT, and they taught a course on this subject and turned it into a very nice book that you can download for free:

Our course will be based on this book. I'll give "lectures" on it - in print, like this one here - from now until at most September 25th, when I have to start teaching classes at U. C. Riverside again. The book has 7 chapters, and we've got at most 6 months. I don't have a set time-table: we'll have to see how things go. We could be done much sooner, but it may be fun to go slower and get into more detail.

I will try to write a short lecture every day or so. But sometimes I'll be busy: most notably, from April 23rd to May 5th, Brendan, David and I will be in the Netherlands at Applied Category Theory 2018, along with 60 other people, and things will get very hectic. Expect a slowdown or complete silence then. With luck you can watch videos.

We should do all the Exercises in the book. I hope some of you volunteer to copy these exercises over to the Forum, one at a time, so we can talk about them here. I'll get the game started but I can't do everything.

I will also pose Puzzles. In fact I've already started!

So, your job is to read the book, ask questions, try to do the Exercises and Puzzles, write your answers here - and most importantly, discuss everything to death! There's a place to discuss each Chapter, and a place to discuss each Lecture, and these discussions can go on indefinitely.

I will take part in these discussions - that's what I love about online forums. But if all of you ask lots of questions - and I really hope you do! - I'll be unable to answer them all myself. So, it's crucial that if someone asks a question, and you think you know the answer, help them out.

With luck, we'll all learn a lot of stuff, make a lot of new friends... and some of us will go ahead to revolutionize human thought, invent amazing new technologies, and save the planet.

Getting to Know Each Other

I'd like to keep the discussions on Chapters (of the book) and Lectures (like this) fairly focused. There is always conversation drift, and some of that is okay, but if you start wanting to talk about algebraic geometry or sky-diving or something else that's not really about the course, please take it to Chat. That's a good place for free-wheeling discussions.

Discussions

You discuss the Chapters by posting a comment on the appropriate thread, like this:

You can discuss my Lectures by posting a comment on its thread. If you have questions or comments on the subject of the course that don't quite fit into any of these places, you can start a new discussion in the category Applied Category Theory Course. If you have questions about technical aspects of how the Azimuth Forum works - or why it's not working - start a discussion in the category Technical.

Writing math

You can write equations using MathJax, which is a limited version of LaTeX good for the web. For "displayed" equations, centered on the page, use double dollar signs: $$E = \sqrt{m^2 + p^2}$$ produces this: $$E = \sqrt{m^2 + p^2}$$ For "inline" equations, mixed in with your text, use this other method: \$$E = \sqrt{m^2 + p^2}\$$ produces this: $$E = \sqrt{m^2 + p^2}$$. If you don't know anything about LaTeX, you can look at the source code of other people's comments and see how they're doing stuff.

Resources

The book Seven Sketches has a bibliography with some good reference. Wikipedia is actually very good for definitions of terms used in category theory. For more advanced material, see the nLab. Also try these free books:

Be nice

It's an inevitable feature of any discussion forum that some users become rude, bully others, try to take over conversations, or try to promote irrelevant ideas. Anyone who becomes annoying in these or other ways will be banned. I won't allow discussions about these issues, either. I have been helping run online discussion forums since around 1989, back when we used modems, and I've learned that requiring politeness is the only way to keep attention focused on the actual subject matter. If you don't want to be polite and respectful of others, please leave now.

Pro tip: if someone says something idiotic, like "all categories have colimits", don't say "You're wrong." Say "Actually not all categories have colimits: consider for example the category of free groups and homomorphisms between those." It's amazing how much better people respond when they don't perceive a correction as being an attack on their person. People are incredibly sensitive things.

To read other lectures go here.

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1.

We should do all the Exercises in the book. I hope some of you volunteer to copy these exercises over to the Forum, one at a time, so we can talk about them here. I'll get the game started but I can't do everything.

Should we put these in the chapter discussion, a discussion per exercise, or a discussion per section?

Comment Source:> We should do all the Exercises in the book. I hope some of you volunteer to copy these exercises over to the Forum, one at a time, so we can talk about them here. I'll get the game started but I can't do everything. Should we put these in the chapter discussion, a discussion per exercise, or a discussion per section?
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2.
edited March 28

I need to figure out what's best! How about a discussion per exercise? That will make them easies to spot, I think. I'll do the first one in a bit. Then it'll be great if you (and other people) post others in a similar style.

Comment Source:I need to figure out what's best! How about a discussion per exercise? That will make them easies to spot, I think. I'll do the first one in a bit. Then it'll be great if you (and other people) post others in a similar style.
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3.

John - you wrote "But sometimes I'll be busy: most notably, from April 23rd to March 5th..." Shouldn't that be May 5th, instead of March 5th?

Comment Source:John - you wrote "But sometimes I'll be busy: most notably, from April 23rd to March 5th..." Shouldn't that be May 5th, instead of March 5th?
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Vishal - yes! I'll fix that.

Comment Source:Vishal - yes! I'll fix that.
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5.

Perhaps creating a slack group is useful?

Comment Source:Perhaps creating a slack group is useful?
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6.

Wow! I'll be at the Lorentz Center that same week (23-27 April) talking about architecture description for a workshop on Software Architecture. The irony is I've been interested to learn enough category theory to formalize current ideas in the field, so Applied Category Theory school would have been perfect :-(

Comment Source:Wow! I'll be at the Lorentz Center that same week (23-27 April) talking about architecture description for a workshop on Software Architecture. The irony is I've been interested to learn enough category theory to formalize current ideas in the field, so Applied Category Theory school would have been perfect :-(