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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.

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11 Answers 11

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Online resources:

Books (not free):

  • Benjamin Pierce: Basic category theory for computer scientists, MIT Press 1991; a slight expansion/update of the earlier (and free) CMU-CS-88-203 report
  • MacLane - solid mathematical foundations, but hardly any references to computing
  • Martin Brandenburg - Einführung in die Kategorientheorie (in german)

Category theory in Haskell:

Another list

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  • $\begingroup$ I can recommend, strongly, the Barr/Wells above for application connection, and the Awodey book for the theory. And I'm glad to see my lecture notes listed! :-D $\endgroup$
    – Mikael Vejdemo-Johansson
    Commented Oct 22, 2009 at 22:56
  • $\begingroup$ I wikified the answer. Additions are welcome. $\endgroup$
    – sdcvvc
    Commented Jan 25, 2010 at 22:16
  • $\begingroup$ @MikaelVejdemo-Johansson which ones are your lecture notes? I'm not seeing them. $\endgroup$
    – Steven Shaw
    Commented Jun 4, 2015 at 11:11
  • $\begingroup$ Mine are the MATH 198 course notes. $\endgroup$
    – Mikael Vejdemo-Johansson
    Commented Jun 8, 2015 at 9:03
  • $\begingroup$ Some of the links here are now broken. If anybody wants to take a crack at updating them, I think that would be a valuable service to the community. $\endgroup$
    – Tim Campion
    Commented Feb 20, 2021 at 16:39
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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.

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  • $\begingroup$ Very good references. I think you can post using hyperlinks now. $\endgroup$
    – sdcvvc
    Commented Oct 23, 2009 at 14:09
  • $\begingroup$ Gah! Why can't I add references to this without bumping things to the first page! $\endgroup$
    – Justin Hilburn
    Commented Jan 6, 2011 at 18:14
  • $\begingroup$ I'm glad you like my (own) book, but Proofs and Types was more than "transcribed": I tried to turn Girard's weird French into natural English and Yves did a very good job of standardising the notation throughout. $\endgroup$
    – Paul Taylor
    Commented Feb 22, 2021 at 16:30
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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.

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

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    $\begingroup$ Somebody, I think it was David Rydeheard, described Mac Lane's book as the best programming manual he had ever used. $\endgroup$
    – Paul Taylor
    Commented Feb 12, 2015 at 17:42
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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.

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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.

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To get you involved with the details of categorical computations, I've written a Web-based program that calculates products, equalisers and other constructs in the category of sets. You choose which construct you want an example of, then fill in values for some of its objects and arrows, and my program will compute the remaining ones, then send back a drawing of the diagram with values for all objects and arrows, plus an explanation.

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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.

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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.

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    $\begingroup$ The Barr/Wells book is not very heavy on the gratuitious material - it puts quite some energy into the Sketch theory formalism (Wells' big thing), but otherwise does a good job of relating almost everything to various corners of computer science. $\endgroup$
    – Mikael Vejdemo-Johansson
    Commented Oct 22, 2009 at 23:00
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I enjoyed Categories and Computer Science by RFC Walters, and it is the one book that got me anywhere toward understanding what categories might be good for:

https://www.amazon.com/Categories-Computer-Science-Cambridge-Texts/dp/0521422264/ref=sr_1_7?ie=UTF8&qid=1521225555&sr=8-7&keywords=category+theory+computing+science

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There is also a book from Pedicchio and Tholen: Categorical Foundations.

If you are interested in categorical logic, I recommend to give a look to History of categorical logic, by Reyes and Marquis.

Probably a good place to start for a computer scientist might be the book from Adamek and Trnkova: Automata and Algebras in Categories.

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