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I've come across the term 'Functor' a few times while reading various articles on functional programming, but the authors typically assume the reader already understands the term. Looking around on the web has provided either excessively technical descriptions (see the Wikipedia article) or incredibly vague descriptions (see the section on Functors at this ocaml-tutorial website).

Can someone kindly define the term, explain its use, and perhaps provide an example of how Functors are created and used?

Edit: While I am interested in the theory behind the term, I am less interested in the theory than I am in the implementation and practical use of the concept.

Edit 2: Looks like there is some cross-terminoligy going on: I'm specifically referring to the Functors of functional programming, not the function objects of C++.

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See also: adit.io/posts/… –  Vlad the Impala May 11 '13 at 23:16
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11 Answers

up vote 152 down vote accepted

The word "functor" comes from category theory, which is a very general, very abstract branch of mathematics. It has been borrowed by designers of functional languages in at least two different ways.

  • In the ML family of languages, a functor is a module that takes one or more other modules as a parameter. It's considered an advanced feature, and most beginning programmers have difficulty with it.

    As an example of implementation and practical use, you could define your favorite form of balanced binary search tree once and for all as a functor, and it would take as a parameter a module that provides:

    • The type of key to be used in the binary tree

    • A total-ordering function on keys

    Once you've done this, you can use the same balanced binary tree implementation forever. (The type of value stored in the tree is usually left polymorphic—the tree doesn't need to look at values other than to copy them around, whereas the tree definitely needs to be able to compare keys, and it gets the comparison function from the functor's parameter.)

    Another application of ML functors is layered network protocols. The link is to a really terrific paper by the CMU Fox group; it shows how to use functors to build more complex protocol layers (like TCP) on type of simpler layers (like IP or even directly over Ethernet). Each layer is implemented as a functor that takes as a parameter the layer below it. The structure of the software actually reflects the way people think about the problem, as opposed to the layers existing only in the mind of the programmer. In 1994 when this work was published, it was a big deal.

    For a wild example of ML functors in action, you could see the paper ML Module Mania, which contains a publishable (i.e., scary) example of functors at work. For a brilliant, clear, pellucid explanation of the ML modules system (with comparisons to other kinds of modules), read the first few pages of Xavier Leroy's brilliant 1994 POPL paper Manifest Types, Modules, and Separate Compilation.

  • In Haskell, and in some related pure functional language, Functor is a type class. A type belongs to a type class (or more technically, the type "is an instance of" the type class) when the type provides certain operations with certain expected behavior. A type T can belong to class Functor if it has certain collection-like behavior:

    • The type T is parameterized over another type, which you should think of as the element type of the collection. The type of the full collection is then something like T Int, T String, T Bool, if you are containing integers, strings, or Booleans respectively. If the element type is unknown, it is written as a type parameter a, as in T a.

      Examples include lists (zero or more elements of type a), the Maybe type (zero or one elements of type a), sets of elements of type a, arrays of elements of type a, all kinds of search trees containing values of type a, and lots of others you can think of.

    • The other property that T has to satisfy is that if you have a function of type a -> b (a function on elements), then you have to be able to take that function and product a related function on collections. You do this with the operator fmap, which is shared by every type in the Functor type class. The operator is actually overloaded, so if you have a function even with type Int -> Bool, then

      fmap even

      is an overloaded function that can do many wonderful things:

      • Convert a list of integers to a list of Booleans

      • Convert a tree of integers to a tree of Booleans

      • Convert Nothing to Nothing and Just 7 to Just False

      In Haskell, this property is expressed by giving the type of fmap:

      fmap :: (Functor t) => (a -> b) -> t a -> t b

      where we now have a small t, which means "any type in the Functor class."

    To make a long story short, in Haskell a functor is a kind of collection for which if you are given a function on elements, fmap will give you back a function on collections. As you can imagine, this is an idea that can be widely reused, which is why it is blessed as part of Haskell's standard library.

As usual, people continue to invent new, useful abstractions, and you may want to look into applicative functors, for which the best reference may be a paper called Applicative Programming with Effects by Conor McBride and Ross Paterson.

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This is an excellent answer - thanks a bunch for the information. =) –  Erik Forbes Jan 10 '10 at 4:50
Wish I could upvote this answer more times. –  miloshadzic Jan 10 '10 at 17:03
I understand both ML-functors and Haskell-functors, but lack the insight to relate them together. What's the relationship between these two, in a category-theoretical sense? –  Wei Hu Jan 11 '10 at 21:19
@Wei Hu: Category theory has never made any sense to me. The best I can say is that all three notions involve mapping. –  Norman Ramsey Jan 12 '10 at 1:25
According to this Haskell wiki: en.wikibooks.org/wiki/Haskell/Category_theory , it's like this: A category is a collection of objects and morphisms (functions), where the morphisms are from objects in a category to other objects in that category. A functor is a function which maps objects and morphisms from one category to objects and morphisms in another. Least that's how I understand it. What that means exactly for programming I have yet to understand. –  paul Jan 18 '10 at 22:31
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There are three different meanings, not much related!

  • In Ocaml it is a parametrized module. See manual. I think the best way to grok them is by example: (written quickly, might be buggy)

    module type Order = sig
        type t
        val compare: t -> t -> bool
    module Integers = struct
        type t = int
        let compare x y = x > y
    module ReverseOrder = functor (X: Order) -> struct
        type t = X.t
        let compare x y = X.compare y x
    (* We can order reversely *)
    module K = ReverseOrder (Integers);;
    Integers.compare 3 4;;   (* this is false *)
    K.compare 3 4;;          (* this is true *)
    module LexicographicOrder = functor (X: Order) -> 
      functor (Y: Order) -> struct
        type t = X.t * Y.t
        let compare (a,b) (c,d) = if X.compare a c then true
                             else if X.compare c a then false
                             else Y.compare b d
    (* compare lexicographically *)
    module X = LexicographicOrder (Integers) (Integers);;
    X.compare (2,3) (4,5);;
    module LinearSearch = functor (X: Order) -> struct
        type t = X.t array
        let find x k = 0 (* some boring code *)
    module BinarySearch = functor (X: Order) -> struct
        type t = X.t array
    (* linear search over arrays of integers *)
    module LS = LinearSearch (Integers);;
    LS.find [|1;2;3] 2;;
    (* binary search over arrays of pairs of integers, 
       sorted lexicographically *)
    module BS = BinarySearch (LexicographicOrder (Integers) (Integers));;
    BS.find [|(2,3);(4,5)|] (2,3);;

You can now add quickly many possible orders, ways to form new orders, do a binary or linear search easily over them. Generic programming FTW.

  • In functional programming languages like Haskell, it means some type constructors (parametrized types like lists, sets) that can be "mapped". To be precise, a functor f is equipped with (a -> b) -> (f a -> f b). This has origins in category theory. The Wikipedia article you linked to is this usage.

    class Functor f where
    instance Functor [] where      -- lists are a functor
        fmap = map
    instance Functor Maybe where   -- Maybe is option in Haskell
        fmap f (Just x) = Just (f x)
        fmap f Nothing = Nothing
    fmap (+1) [2,3,4]   -- this is [3,4,5]
    fmap (+1) (Just 5)  -- this is Just 6
    fmap (+1) Nothing   -- this is Nothing

So, this is a special kind of a type constructors, and has little to do with functors in Ocaml!

  • In imperative languages, it is a pointer to function.
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Shouldn't <q>map</q> in the last 3 lines of this comment be indeed <q>fmap</q>? –  imz -- Ivan Zakharyaschev Dec 18 '10 at 9:48
Yes, thanks. __ –  sdcvvc Dec 18 '10 at 13:38
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In OCaml, it's a parameterised module.

If you know C++, think of an OCaml functor as a template. C++ only has class templates, and functors work at the module scale.

An example of functor is Map.Make; module StringMap = Map.Make (String);; builds a map module that works with String-keyed maps.

You couldn't achieve something like StringMap with just polymorphism; you need to make some assumptions on the keys. The String module contains the operations (comparison, etc) on a totally ordered string type, and the functor will link against the operations the String module contains. You could do something similar with object-oriented programming, but you'd have method indirection overhead.

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Um, C++ has functors. –  Kornel Kisielewicz Jan 8 '10 at 21:31
I got that from the ocaml website - but I don't understand what the use of a parameterized module would be. –  Erik Forbes Jan 8 '10 at 21:31
@Kornel Yeah, what I described is an OCaml concept. The other concept is just “functional value”, which is nothing particular in FP. @Erik I expanded slightly, but the reference docs are slow to load. –  Tobu Jan 8 '10 at 21:43
I think I'm beginning to see. –  Erik Forbes Jan 8 '10 at 22:10
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There is a pretty good example in the O'Reilly OCaml book that's on Inria's website (which as of writing this is unfortunately down). I found a very similar example in this book used by caltech: Introduction to OCaml (pdf link). The relevant section is the chapter on functors (Page 139 in the book, page 149 in the PDF).

In the book they have a functor called MakeSet which creates a data structure that consists of a list, and functions to add an element, determine if an element is in the list, and to find the element. The comparison function that is used to determine if it's in/not in the set has been parametrized (which is what makes MakeSet a functor instead of a module).

They also have a module that implements the comparison function so that it does a case insensitive string compare.

Using the functor and the module that implements the comparison they can create a new module in one line:

module SSet = MakeSet(StringCaseEqual);;

that creates a module for a set data structure that uses case insensitive comparisons. If you wanted to create a set that used case sensitive comparisons then you would just need to implement a new comparison module instead of a new data structure module.

Tobu compared functors to templates in C++ which I think is quite apt.

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Here's an article on functors from a programming POV, followed up by more specifically how they surface in programming languages.

The practical use of a functor is in a monad, and you can find many tutorials on monads if you look for that.

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now my brain hurts. –  Alex Brown Jan 8 '10 at 21:48
You're not alone, Alex. =X –  Erik Forbes Jan 8 '10 at 22:01
"The practical use of a functor is in a monad" -- not only. All monads are functors, but there are lots of uses for non-monad functors. –  amindfv Apr 6 '13 at 19:33
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Other answers here are complete, but I'll try another explanation of the FP use of functor. Take this as analogy: A functor is a container of type a that, when subjected to a function that maps from a->b, yields a container of type b.

Unlike the abstracted-function-pointer use in C++, here the functor is not the function; rather, it's something that behaves consistently when subjected to a function.

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Thank you! Your first paragraph there was much easier to understand and digest than the long, excellent, accepted answer. (Maybe it helps that I skimmed the long one before I read yours.) I wish I could upvote this twice for its brevity and clarity. –  LarsH Mar 24 '11 at 19:08
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The best answer to that question is found in "Typeclassopedia" by Brent Yorgey.

This issue of Monad Reader contain a precise definition of what a functor is as well as many definition of other concepts as well as a diagram. (Monoid, Applicative, Monad and other concept are explained and seen in relation to a functor).


excerpt from Typeclassopedia for Functor: "A simple intuition is that a Functor represents a “container” of some sort, along with the ability to apply a function uniformly to every element in the container"

But really the whole typeclassopedia is a highly recommended reading that is surprisingly easy. In a way you can see the typeclass presented there as a parallel to design pattern in object in the sense that they give you a vocabulary for given behavior or capability.


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You got quite a few good answers. I'll pitch in:

A functor, in the mathematical sense, is a special kind of function on an algebra. It is a minimal function which maps an algebra to another algebra. "Minimality" is expressed by the functor laws.

There are two ways to look at this. For example, lists are functors over some type. That is, given an algebra over a type 'a', you can generate a compatible algebra of lists containing things of type 'a'. (For example: the map that takes an element to a singleton list containing it: f(a) = [a]) Again, the notion of compatibility is expressed by the functor laws.

On the other hand, given a functor f "over" a type a, (that is, f a is the result of applying the functor f to the algebra of type a), and function from g: a -> b, we can compute a new functor F = (fmap g) which maps f a to f b. In short, fmap is the part of F that maps "functor parts" to "functor parts", and g is the part of the function that maps "algebra parts" to "algebra parts". It takes a function, a functor, and once complete, it IS a functor too.

It might seem that different languages are using different notions of functors, but they're not. They're merely using functors over different algebras. OCamls has an algebra of modules, and functors over that algebra let you attach new declarations to a module in a "compatible" way.

A Haskell functor is NOT a type class. It is a data type with a free variable which satisfies the type class. If you're willing to dig into the guts of a datatype (with no free variables), you can reinterpret a data type as a functor over an underlying algebra. For example:

data F = F Int

is isomorphic to the class of Ints. So F, as a value constructor, is a function that maps Int to F Int, an equivalent algebra. It is a functor. On the other hand, you don't get fmap for free here. That's what pattern matching is for.

Functors are good for "attaching" things to elements of algebras, in an algebraically compatible way.

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Given the other answers and what I'm going to post now, I'd say that it's a rather heavily overloaded word, but anyway...

For a hint regarding the meaning of the word 'functor' in Haskell, ask GHCi:

Prelude> :info Functor
class Functor f where
  (GHC.Base.<$) :: forall a b. a -> f b -> f a
        -- Defined in GHC.Base
instance Functor Maybe -- Defined in Data.Maybe
instance Functor [] -- Defined in GHC.Base
instance Functor IO -- Defined in GHC.Base

So, basically, a functor in Haskell is something that can be mapped over. Another way to say it is that a functor is something which can be regarded as a container which can be asked to use a given function to transform the value it contains; thus, for lists, fmap coincides with map, for Maybe, fmap f (Just x) = Just (f x), fmap f Nothing = Nothing etc.

The Functor typeclass subsection and the section on Functors, Applicative Functors and Monoids of Learn You a Haskell for Great Good give some examples of where this particular concept is useful. (A summary: lots of places! :-))

Note that any monad can be treated as a functor, and in fact, as Craig Stuntz points out, the most often used functors tend to be monads... OTOH, it is convenient at times to make a type an instance of the Functor typeclass without going to the trouble of making it a Monad. (E.g. in the case of ZipList from Control.Applicative, mentioned on one of the aforementioned pages.)

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Put simply, a functor, or function object, is a class object that can be called just like a function.

In C++:

This is how you write a function

void foo()
    cout << "Hello, world! I'm a function!";

This is how you write a functor

class FunctorClass
    void operator ()
        cout << "Hello, world! I'm a functor!";

Now you can do this:

foo(); //result: Hello, World! I'm a function!

FunctorClass bar;
bar(); //result: Hello, World! I'm a functor!

What makes these so great is that you can keep state in the class - imagine if you wanted to ask a function how many times it has been called. There's no way to do this in a neat, encapsulated way. With a function object, it's just like any other class: you'd have some instance variable that you increment in operator () and some method to inspect that variable, and everything's neat as you please.

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No, those functors aren't the type theory notion that is used by FP languages. –  Tobu Jan 8 '10 at 22:02
I can kind of see how one could prove that FunctorClass fulfils the first Functor Law, but could you sketch out a proof for the second Law? I don't quite see it. –  Jörg W Mittag Jan 8 '10 at 22:22
Bah, you guys are right. I took a stab at solving the "web has provided exceedingly technical descriptions" and overshot, trying to avoid, "In the ML family of languages, a functor is a module that takes one or more other modules as a parameter." This answer, however, is, well, bad. Oversimplified and underspecified. I'm tempted to ragedelete it, but I'll leave it for future generations to shake their heads at :) –  Matt Sep 9 '11 at 23:03
I'm glad you left the answer and comments, because it helps frame the problem. Thank you! I'm having trouble in that most of the answers are written in terms of Haskell or OCaml, and to me that's a little like explaining alligators in terms of crocodiles. –  Rob Y Feb 11 at 15:50
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Functor is not specifically related to functional programming. It's just a "pointer" to a function or some kind of object, that can be called as it would be a function.

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There is a specific FP concept of a functor (from category theory), but you're right that the same word is also used for other things in non-FP languages. –  Craig Stuntz Jan 8 '10 at 21:32
Are you sure that function pointers are Functors? I don't see how function pointers fulfil the two Functor Laws, especially the Second Functor Law (preservation of morphism composition). Do you have a proof for that? (Just a rough sketch.) –  Jörg W Mittag Jan 8 '10 at 22:24
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