Resources for learning practical category theory - MathOverflow

Resources for learning practical category theory

2009-10-17T18:03:56Z

I've been doing functional programming, primarily in OCaml, for a couple years now, and have recently ventured into the land of monads. I'm able to work them now, and understand how to use them, but I'm interested in understanding more about their mathematical foundations. These foundations are usually presented as coming from category theory. So we get explanations such as the following:

A monad is a monoid in the category of endofunctors.

Now, my goal (partially) is to understand what that means. Can anyone suggest a gentle introduction to category theory, particularly one aimed at programmers already familiar with a functional language such as ML or Haskell, with references for further reading? Resources not necessarily aimed at programmers but accessible to readers with a background in discrete math and first-order logic would be quite acceptable as well.

Answer by Quadrescence for Resources for learning practical category theory

2009-10-17T18:13:49Z

Category Theory by Steve Awodey from Carnegie Mellon University has been a favorite text of mine. It's 266 pages, and while it's not a "practical theory book" (e.g., one aimed at programmers), it is very easy to follow and very well explained.

Answer by sdcvvc for Resources for learning practical category theory

2009-10-17T18:15:44Z

Online resources:

The Catsters channel
MATH198 course notes - examples in Haskell
Rydehard, Burstall: Computional Category Theory - examples in ML
MAGIC course
Barr, Wells: Category theory for computing science
Jaap van Oosten: basic category theory
Tom Leinster
Eugenia Cheng
Steve Awodey - very similar to the book mentioned by Quadrescence
Daniele Turi
Thomas Streicher

Books:

"Basic category theory for computer scientists" by Benjamin Pierce
MacLane - solid mathematical foundations, but hardly any references to computing
Abstract and concrete categories - might be considered too verbose, but it's full of examples

Category theory in Haskell:

Wikibooks introductory text
sigfpe's blog has a lot of category theory articles - (di)natural transformations, monads, Yoneda lemma...
Comonad.Reader
The Monad.Reader - check "Calculating monads with category theory"

Another list

Answer by Grétar Amazeen for Resources for learning practical category theory

2009-10-17T18:22:12Z

The book "Abelian Categories, an Introduction to the Theory of Functors" by Peter J. Freyd is a very good introduction to category theory. It is not aimed at computer scientists, but it is a good introduction to the subject.

Answer by Reid Barton for Resources for learning practical category theory

2009-10-17T18:25:24Z

TWF Week 89 for your specific question, and This Week's Finds in Mathematical Physics in general for wonderful presentations of all sorts of things category-related.

Answer by John D. Cook for Resources for learning practical category theory

2009-10-17T19:43:24Z

I'm skeptical of "practical category theory," especially books by mathematicians telling computer scientists how important category theory is. I'm not saying there aren't connections -- I know there are -- but there's a tremendous temptation for mathematicians to include gratuitous material.

Answer by Scott Morrison for Resources for learning practical category theory

2009-10-17T21:00:42Z

Categories for the working mathematician, by Maclane, is probably good for some working computer scientists too. No disputing taste, of course.

Answer by Justin Hilburn for Resources for learning practical category theory

2009-10-22T21:31:13Z

Since sdvccv already pointed out a number of good sources for learning category theory as applied to CS, I will try and provide some guide posts.

My favorite book on the subject is Practical Foundations of Mathematics by Paul Taylor since he does a really good job of giving you a big picture (unfortunately he doesn't always give you enough details if you don't already have a logic background). Bart Jacobs book Categorical Logic and Type Theory is also very readable.

In general I think the most important thing to understand in order to apply categories to computer science is the Curry-Howard-Lambek correspondence which loosely states that lambda calculii, intuitionist logics, and cartesian closed categories (categories where you have products and function spaces) are the same thing. Proofs and Types which was transcribed from some of Girard's lecture notes is an excellent source for the Curry-Howard part of the correspondence. Steve Awodey's book and these notes by Samson Abramsky are good places to see this translated into categorical language. For connections to Topoi Mac Lane's Sheaves in Geometry and Logic looks good.

Next you will probably want to learn about categorical and universal algebra. One of the more immediate and accessible applications of these ideas is the theory of algebraic data types (categorically: initial algebras for polynomial functors) and maps and folds between them. Monads are also a part of this subject since they are type constructors (endofunctors) that also have a multiplication and unit. Haskell do notation corresponds to forming the Kliesli category for a monad.

The nascent field of universal coalgebra has been very useful for formalizing notions of state and observation. There are also some emerging connections between coalgebra and modal logic.

Finally, if you aren't worn out you may want to learn about Stone duality which is a way of relating "logics of observable properties" and topology. For computer scientist Stone duality is primarily useful for giving a logical interpretation to domain theory, but mathematicians may recognize the duality between commutative rings and Zariski spectra as a special case.

Answer by Wayne D for Resources for learning practical category theory

2009-10-26T15:32:41Z

You may find Computational Category Theory by Rydeheard and Burstall interesting. It's available online at Rydeheard's page: http://www.cs.manchester.ac.uk/~david/categories/

The Manual there is the textbook and The Programs provide a great deal of the ML code in the text in its development of category theory.

For someone new to CT it's very different than most introductions but the code helps make some of the ideas much more concrete.