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README.md

learn-tt

Lots of people seem curious about type theory but it's not at all clear how to go from no math background to understanding "Homotopical Patch Theory" or whatever the latest cool paper is. In this repository I've gathered links to some of the resources I've personally found helpful.

A Disclaimer

At this point learn-tt is a few years old and I can write with slightly more confidence than when I first created this document. I still stand by the fact that the links in this list have helped me learn ideas that would have been difficult to find elsewhere. At the same time, I worry that this list carries more of a ring of authority than I wish it did, particularly now that it is relatively popular. I learn more about type theory every day (admittedly, some days more slowly than I wish) and my views on what constitutes a good explanation, a good approach, or even a good type theory have changed in small and large ways since I first compiled these resources.

I toyed with the idea of deleting this repository because I have worried whether or not presenting my own learning path does more harm than good. I have decided to leave it around but to add a disclaimer instead.

What follows below should not be construed as some blessed path for learning type theory. You may find it better to skim this list or simply snort and ignore it entirely. My process for learning continues to be distinctly wandering and non-linear. Someone with different goals than me would find some of these links useless and I would not be nearly so bold as to claim that these resources are canonical, necessary, or even helpful for everyone. I can only hope that you enjoy reading them as much as I have.

Danny

The Resources

Textbooks

  • Practical Foundations of Programming Languages (PFPL)

    I reference this more than any other book. It's a very wide ranging survey of programming languages that assumes very little background knowledge. A lot people prefer the next book I mention but I think PFPL does a better job explaining the foundations it works from and then covers more topics I find interesting.

  • Types and Programming Languages (TAPL)

    Another very widely used introductory book (the one I learned with). It's good to read in conjunction with PFPL as they emphasize things differently. Notably, this includes descriptions of type inference which PFPL lacks and TAPL lacks most of PFPL's descriptions of concurrency/interesting imperative languages. Like PFPL this is very accessible and well written.

  • Advanced Topics in Types and Programming Languages (ATTAPL)

    Don't feel the urge to read this all at once. It's a bunch of fully independent but excellent chapters on a bunch of different topics. Read what looks interesting, save what doesn't. It's good to have in case you ever need to learn more about one of the subjects in a pinch.

Proof Assistants

One of the fun parts of taking in an interest in type theory is that you get all sorts of fun new programming languages to play with. Some major proof assistants are

Type Theory

  • The Works of Per Martin-Löf

    Per Martin-Löf has contributed a ton to the current state of dependent type theory. So much so that it's impossible to escape his influence. His papers on Martin-Löf Type Theory (he called it Intuitionistic Type Theory) are seminal.

    If you're confused by the papers above read the book in the next entry and try again. The book doesn't give you as good a feel for the various flavors of MLTT (which spun off into different areas of research) but is easier to follow.

  • Programming In Martin-Löf's Type Theory

    It's good to read the original papers and here things from the horses mouth, but Martin-Löf is much smarter than us and it's nice to read other people explanations of his material. A group of people at Chalmers have elaborated it into a book.

  • The Works of John Reynolds

    John Reynolds' works are similarly impressive and always a pleasure to read.

  • Homotopy Type Theory

    A new exciting branch of type theory. This exploits the connection between homotopy theory and type theory by treating types as spaces. It's the subject of a lot of active research but has some really nice introductory resources even now.

Proof Theory

Category Theory

Learning category theory is necessary to understand some parts of type theory. If you decide to study categorical semantics, realizability, or domain theory eventually you'll have to buckledown and learn a little at least. It's actually really cool math so no harm done!

  • Category Theory in Context

    A newly released textbook on category theory with a focus on using representable functors as a tool to place various concepts of category theory in a coherent framework. This has the substantial advantage of being freely available online! It's also published by Dover so the actual book itself is remarkably cheap.

  • Practical Foundations of Mathematics

    This books does an excellent job of tying together general mathematics into the framework of category theory. It is accordingly covers a large basis of math outside of the field of category theory. It contains a large amount of categorical logic which warrants its inclusion in this list and is one of the more approachable texts on categorical logic. At least for me.

  • Category Theory

    One of the better introductory books to category theory in my opinion. It's notable in assuming relatively little mathematical background and for covering quite a lot of ground in a readable way.

  • Ed Morehouse's Category Theory Lecture Notes

    Another valuable piece of reading are these lecture notes. They cover a lot of the same areas as "Category Theory" so they can help to reinforce what you learned there as well giving you some of the author's perspective on how to think about these things.

  • Categorical Logic and Type Theory

    This book is honestly quite difficult to get through, but it's an absolutely indispensable resource for folks interested in categorical logic. More generally, this book contains one of the few coherent and comprehensive accounts of how to model type theory categorically. It is not a book to learn category theory or type theory from, it demands a good understanding of both since it's focused on applying category theory, not explaining it so much. This is also the book to read if you're interested in understanding the theory of fibered categories in general (the style of categorical semantics that it uses).

  • Introduction to Higher-Order Categorical Logic

    This is a relatively short book on categorical logic that introduces all the basic concepts you needed to model simple higher-order logics in category theory. It is much easier reading than Categorical Logic and Type Theory but correspondingly less comprehensive. It focuses mainly on modeling the simply typed lambda calculus in cartesian closed categories and then on modeling a richer type theory internally to a topos. It provides a basic explanation of topos theory so it's intelligible having read an introductory category theory book.

  • Sheaves in Geometry and Logic

    This is not an ideal first book on category theory by any stretch. It merits inclusion because there are deep and interesting relationships between topos theory and type theory and this is one of the more approachable introductions. Some knowledge of topology would be helpful in understanding some of the examples in this books but I am told it is possible to muscle your way through without it.

Other Goodness

  • Gunter's "Semantics of Programming Language"

    While I'm not as big a fan of some of the earlier chapters, the math presented in this book is absolutely top-notch and gives a good understanding of how some cool fields (like domain theory) work.

  • Abramsky and Jung's "Domain Theory"

    This what I reference nowadays for domain theory. It's a very good (if a little dense) introduction covering all the basic mathematics necessary to work with domains productively. It should definitely be possible to follow if you've read some of Gunter's book.

  • Realizability: An Introduction to Its Categorical Side

    Categorical realizability is a fascinating area of overlap between type theory and category theory that, frustratingly, lacks many approachable introductions. van Oosten's book does a good job going through the basic aspects of categorical realizability. It is heavily dependent on knowledge of category theory though, I would recommend making it through Sheaves and Geometry and Logic (see above) or something equivalent first.

  • OPLSS

    The Oregon Programming Languages Summer School is a 2 week long bootcamp on PLs held annually at the university of Oregon. It's a wonderful event to attend but if you can't make it they record all their lectures anyways! They're taught be a variety of lecturers but they're all world class researchers.

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A collection of resources for learning type theory and type theory adjacent fields.

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