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In a previous answer in the Theoretical Computer Science site, I said that category theory is the "foundation" for type theory. Here, I would like to say something stronger. Category theory is type theory. Conversely, type theory is category theory. Let me expand on these points.

Category theory is type theory

In any typed formal language, and even in normal mathematics using informal notation, we end up declaring functions with types $f : A \to B$. Implicit in writing that is the idea that $A$ and $B$ are some things called "types" and $f$ is a "function" from one type to another. Category theory is the algebraic theory of such "types" and "functions". (Officially, category theory calls them "objects" and "morphisms" so as to avoid treading on the set-theoretic toes of the traditionalists, but increasingly I see category theorists throwing such caution to the wind and using the more intuitive terms: "type" and "function". But, be prepared for protests from the traditionalists when you do so.)

We have all been brought up on set theory from high school onwards. So, we are used to thinking of types such as $A$ and $B$ as sets, and functions such as $f$ as set-theoretic mappings. If you never thought of them that way, you are in good shape. You have escaped set-theoretic brain-washing. Category theory says that there are many kinds of types and many kinds of functions. So, the idea of types as sets is limiting. Instead, category theory axiomatizes types and functions in an algebraic way. Basically, that is what category theory is. A theory of types and functions. It does get quite sophisticated, involving high levels of abstraction. But, if you can learn it, you will acquire a deep understanding of types and functions.

Type theory is category theory

By "type theory," I mean any kind of typed formal language, based on rigid rules of term-formation which make sure that everything type checks. It turns out that, whenever we work in such a language, we are working in a category-theoretic structure. Even if we use set-theoretic notations and think set-theoretically, still we end up writing stuff that makes sense categorically. That is an amazing fact.

Historically, Dana Scott may have been the first to realize this. He worked on producing semantic models of programming languages based on typed (and untyped) lambda calculus. The traditional set-theoretic models were inadequate for this purpose, because programming languages involve unrestricted recursion which set theory lacks. Scott invented a series of semantic models that captured programming phenomena, and came to the realization that typed lambda calculus exactly represented a class of categories called cartesian closed categories. There are plenty of cartesian closed categories that are not "set-theoretic". But typed lambda calculus applies to all of them equally. Scott wrote a nice essay called "Relating theories of lambda calculus" explaining what is going on, parts of which seem to be available on the web. The original article was published in a volume called "To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism", Academic Press, 1980. Berry and Curien came to the same realization, probably independently. They defined a categorical abstract machine (CAM) to use these ideas in implementing functional languages, and the language they implemented was called "CAML" which is the underlying framework of Microsoft's F#.

Standard type constructors like $\times$, $\to$, $List$ etc. are functors. That means that they not only map types to types, but also functions between types to functions between types. Polymorphic functions preserve all such functions resulting from functor actions. Category theory was invented in 1950's by Eilenberg and MacLane precisely to formalize the concept of polymorphic functions. They called them "natural transformations", "natural" because they are the only ones that you can write in a type-correct way using type variables. So, one might say that category theory was invented precisely to formalize polymorphic programming languages, even before programming languages came into being!

A set-theoretic traditionalist has no knowledge of the functors and natural transformations that are going on under the surface when he uses set-theoretic notations. But, as long as he is using the type system faithfully, he is really doing categorical constructions without being aware of them.

All said and done, category theory is the quintessential mathematical theory of types and functions. So, all programmers can benefit from learning a bit of category theory, especially functional programmers. Unfortunately, there do not seem to be any text books on category theory targeted at programmers specifically. The "category theory for computer science" books are typically targeted at theoretical computer science students/researchers. The book by Benjamin Pierce, Basic category theory for computer scientists is perhaps the most readable of them.

However, there are plenty of resources on the web, which are targeted at programmers. The Haskellwiki page can be a good starting point. At the Midlands Graduate School, we have lectures on category theory (among others). Graham Hutton's course was pegged as a "beginner" course, and mine was pegged as an "advanced" course. But both of them cover essentially the same content, going to different depths. University of Chalmers has a nice resource page on books and lecture notes from around the world. The enthusiastic blog site of "sigfpe" also provides a lot of good intuitions from a programmer's point of view.

The basic topics you would want to learn are:

definition of categories, and some examples of categories

functors, and examples of them

natural transformations, and examples of them

definitions of products, coproducts and exponents (function spaces), initial and terminal objects.

adjunctions

monads, algebras and Kleisli categories

My own lecture notes in the Midlands Graduate School covers all these topics except for the last one (monads). There are plenty of other resources available for monads these days. So that is not a big loss.

The more mathematics you know, the easier it would be to learn category theory. Because category theory is a general theory of mathematical structures, it is helpful to know some examples to appreciate what the definitions mean. (When I learnt category theory, I had to make up my own examples using my knowledge of programming language semantics, because the standard text books only had mathematical examples, which I didn't know anything about.) Then came the brilliant book by Lambek and Scott called "Introduction to categorical logic" which related category theory to type systems (what they call "logic"). It is now possible to understand category theory just by relating it to type systems even without knowing a lot of examples. A lot of the resources I mentioned above use this approach to explain category theory.