Since some computer scientists use category theory, I was wondering if there are any programming languages that use it extensively.




Haskell is a purely functional language. However sideeffects are (almost by definition) difficult to incorporate into a functional language. This is an important problem since I/O is a very important sideeffect for most computer programs. Haskell's method of incorporating sideeffects is to use monads. One of the simplest ways to get a monad is from a pair of adjoint functors. For more on monads see: (1) Embûches tissues blog: Monads in Mathematics 1: examples (2) A series of lectures on youtube by TheCatsters: Pairs of adjoint functors are fairly common and I've found they provide a useful way for seeing part of the "bigpicture" in many different branches of mathematics. Here is one of several introductions to pairs of adjoint functors from the Concrete Nonsense blog. 


There's Charity. 


ML is used in Apparently there exists at least one "categorical programming language", namely Hagino A Categorical Programming Language (Hagino's Thesis) 


All of the answers so far are based on Cartesian closed categories. There are a few languages based on Cartesian categories, which reject use of higherorder functions in the base language:



See also the answers to Resources for learning practical category theory and What is λcalculus related to? 


From the Categorical_abstract_machine entry on wikipedia, which I'm not allowed to link to:
Caml (aka the basis for Microsoft's F#) is an acronym for Categorical Abstract Machine Language. Other interesting reading is A Categorical Manifesto by Joseph Goguen. His language project is the OBJ family of languages. Which I can't link to because I'm a new user and am restricted to 1 hyperlink. 


Here's another one based on Hagino's thesis, asides charity. Implemented both in Haskell and Ruby. 


Also see here. 


CAML definitely uses it, but not extensively. 


CAML by definition is Categorical Abstract Machine Language, however I am not cetain that you can say that an language explicitly uses category theory. Perhaps you are asking "Are there languages that allow Category Theory Concepts to be easily represented?" or perhaps you are asking if the compilation or interpretation of a particular programming language uses Category Theory in its implementation? While technically, all Turingcomplete capable languages should be equivalently able to express the same set of computations, some languages do so more elegantly than others, allowing the programmer or mathematician to be more eloquent. I would say LISP and SCHEME, even though based on lambdacalculus, are more connected to the spirit of category theory in concept. While the numbers and integers are conceptually defined as atomic and can be built up from primitives in concept and in theory; in practice, the implementations of SCHEME and LISP and (CLU) tend to take shortcuts to speed up implementation. The hierarchical ability to pass functions and functions of functions (etc.) as firstclass parameters to functions in LISP and SCHEME let you be able to emulate the actions or morphisms of category theory better in that language than others. You just have to start from the ground up, as I have not yet seen a library or package in LISP or SCHEME for category theory. 


I'd say every statically typed functional language is such. Why? There's a wellknown relation between propositions and types, or more precisely between certain theories of logic and certain type theories going by the name of CurryHoward correspondence. This relation has a less wellknown cousin known as CurryHowardLambek correspondence that extends it to categories. Thus, because every reasonable statically typed functional language is based on type theory, it has direct connection to category theory as well. You simply cannot escape it, try as you might. Bob Harper likes to call this The Holy Trinity  categories, languages and logic. The list of such languages follows: Haskell, ML, OCaml, Idris, Coq, Agda, etc. Of course, you have to take all of this with a grain of salt. Safe for Coq and Agda  both of which are foremost theorem provers  these languages contain many industrydriven quirks that make them less susceptible to the formal reasoning mentioned in the first paragraph. In a bit different direction, because type theory is so expressive (for example, it can replace set theory as foundations of mathematics), it's fairly straightforward to express many algebraic structures in it. All of the above languages use this feature profoundly, with e.g. monads in Haskell playing a very prominent role. 


See A HigherOrder Calculus for Categories by by Glynn Winskel and his student Mario Jose Cáccamo. And also my related question. 

