If you’re like me, you might be worried that the world is being taken over by monoids. Binary operators that are associative and respect unity are popping up everywhere. Even our beloved monads are being dismissively called “just” monoids in the category of endofunctors. Personally, I have nothing against associativity, but when it’s not there, it’s not there. Did you know, for instance, that addition is non-associative? Of course, I’m talking about floating-point addition. Taking averages is non-associative. Lie algebras are non-associative. At a recent FARM (Functional Art, Music, Modelling and Design) Workshop, Paul Hudak spoke of a non-associative way of tiling temporal media (the %\ — percent backslash — operator).
In mathematics, there exists a simpler notion of magma, which is a structure with a binary operator with no additional conditions imposed on it. So monoid is really “just” a magma with some additional structure. Take that monoid!
There was a recent interesting post about free magmas and their relation to binary trees, which shows that magmas might be relevant outside of geology. All this encouraged me to revisit some of the areas traditionally associated with monoids and give them another look-see. If nothing else, it’s an opportunity to have a little refresher in monoidal and enriched categories. Hopefully, this will also help in understanding Edward Kmett’s latest experiments with encoding category theory in Hask.
Let’s start by slimming down the usual definition of a monoidal category: A magmatic category is a category 𝒱0 with a functor ⨁ : 𝒱0 ⨯ 𝒱0 → 𝒱0 from the cartesian product of 𝒱0 with itself to 𝒱0. This is our “attach” operator that works on objects and can be lifted to morphisms.
That’s it! A monoidal category would also have the unit object I, the associator, the left unitor, the right unitor, and two coherence axioms. But if magma is all we need, we’re done.
In what follows I’m not going to be interested in associativity until much later, but I will consider including the unit object I (a magma with a unit is called a unital magma). What defines the unit object are two natural (in X) isomorphisms: lX: I ⨁ X → X, and rX: X ⨁ I → X, called, respectively, the left and the right unitors. Still, because of no associativity, there are no coherence axioms one has to worry about in a monoidal category.
Having a unit object allows us to define an interesting functor that maps any object X to the hom-set 𝒱0(I, X) — the set of morphisms from I to X (obviously, this is a set only if 𝒱0 is locally small). We’ll call this functor V, to make sure you pay attention to fonts, and not just letters. Why is it interesting? Because it’s as close as we can get, in category theory, to defining an element of an object. Remember: in category theory you don’t look inside objects — all you have at your disposal are morphisms between amorphous objects.
So how would you define an element of, let’s say, a set, without looking inside it? You’d define it as a function (morphism) from a one-element set. Such a morphism picks a single element in the target set. But what’s a one-element set (remember: we can’t look inside sets)? Well, it’s a terminal object in the category of sets. Again, a terminal object is defined entirely in terms of morphisms. It’s the object that has a unique morphism coming to it from any other object. (Exercise: Show that a one element set is terminal among sets.)
So an element of a set can be defined as a morphism from the terminal set. Nothing stops us from extending this definition of an “element” to an arbitrary category that has a terminal object. But what does a terminal object have to do with a unit object in the monoidal (or unital-magmatic) category? In the archetypal monoidal category — the cartesian monoidal category — the terminal object is the unit. The binary operator is the product, defined by a universal construction. That’s why we can call a morphism f from the hom-set 𝒱0(I, X) an element of X. In this language, V maps X into a set of elements of X — the underlying set.
Exponential Objects and Currying
The usual universal construction of an exponential object YX can be easily reproduced in a magmatic category. We just replace the cartesian product in that construction with the “attach” operator we have at our disposal.
Here’s how you do it: Start with a pair of objects, X and Y. Consider all pairs <object Z, morphism g> in which g is a morphism from Z ⨁ X to Y. We are looking for one of these pairs, call it <YX, app>, such that any other pair “factorizes” through it (YX is just an object with a funny symbol). It means that, for each <Z, g> there is a unique morphism λg : Z→ YX, such that:
g = app ∘ (λg ⨁ id)
Notice that we are using the fact that ⨁ is a (bi-)functor, which means it can be applied to morphisms as well as objects. Here it’s applied to λg and id (the identity morphism on X).
The universal construction of an exponential object
There is an interesting class of categories in which there is an exponential object for every pair of objects. Or even better, an exponential objects that is defined by an adjunction. An adjunction relates two functors. In this case, one functor is defined by the action of the binary operator: Z→ Z ⨁ X. The exponentiation functor Y→ YX is then its right adjoint. The adjunction is a natural bijection of hom-sets:
λ: 𝒱0(Z ⨁ X, Y) ≅ 𝒱0(Z, YX)
This equation is just the re-statement of the condition that for every g : Z ⨁ X→ Y there is a unique λg : Z→ YX, except that we now require this mapping to be natural in both Y and Z.
Here’s the interesting part: This definition can be viewed as generalized currying. Normal currying establishes the relationship between functions of two arguments and functions returning functions. A function of two arguments can be thought of as a function defined on a cartesian product of those arguments. Here, we are replacing the cartesian product with our magmatic operation, and the adjunction tells us that for every morphism g from Z ⨁ X to Y there is a morphism λg from Z to the exponential object YX. The currying intuition is that YX objectifies morphisms from X to Y; it objectifies the hom-set 𝒱0(X, Y) inside 𝒱0. In fact YX is often called the internal hom-set (in which case 𝒱0(X, Y) is called the external hom-set).
This intuition is reinforced by the observation that, if we replace Z with I, the magmatic unit, in the adjunction, we get:
𝒱0(I ⨁ X, Y) ≅ 𝒱0(I, YX)
I ⨁ X is naturally isomorphic to X (through the left unitor), so we get:
𝒱0(X, Y) ≅ 𝒱0(I, YX)
Now recall what we have said about morphisms from the unit: they define elements of an object through the functor we called V. The equation:
𝒱0(X, Y) ≅ V YX
tells us that elements of the (external) hom-set 𝒱0(X, Y) are in one-to-one correspondence with the “elements” of the exponential YX. This is not exactly the isomorphism of external and internal hom-sets, because of the intervening functor V, which might not even be faithful. However, this is exactly one of the conditions that define a closed category, hinting at the possibility that the adjunction defining the exponential object in a unital magmatic category might be enough to make it closed.
All this was done without any mention of associativity. So what is associativity good for? Here’s an example. Let’s replace X with (A ⨁ B) in the adjunction:
𝒱0(Z ⨁ (A ⨁ B), Y) ≅ 𝒱0(Z, Y(A ⨁ B))
If we have associativity, the above is naturally isomorphic to:
𝒱0((Z ⨁ A) ⨁ B, Y) ≅ 𝒱0(Z ⨁ A, YB) ≅ 𝒱0(Z, YBA)
Putting it all together and replacing Z with I, we get the natural isomorphism between “elements” of two objects:
V Y(A ⨁ B) ≅ V YBA
This looks just like currying inside the internal hom-set, except for the functor V. So to curry from the external to the internal hom-set it’s enough to have a unital magmatic category, but to curry within the internal hom-set we need the full monoidal category.
A word about notation: Traditionally, the exponential object from X to Y is denoted as YX. That’s the convention I’ve been following so far. Arguably, this notation is somewhat awkward. In his book, Basic Concepts of Enriched Category Theory, Kelly uses a more convenient notation: [X, Y]. In his notation, the above equation takes a slightly more readable form:
V [(A ⨁ B), Y] ≅ V [A, [B, Y]]
Magmatically Enriched Category
A (locally small) category consists of object and sets of morphisms, hom-sets, between every pair of objects. All we need to assume about morphisms is that they compose nicely and that there exist morphisms that act as units of composition. Think of it this way: Each hom-set is an object from the Set category. Composition is a mapping from a cartesian product of two such objects to a third object.
But why restrict ourselves to Set to draw our hom-sets from? Why not pick an arbitrary category? The only question is: How should we define composition of morphisms? Morphisms are elements of hom-sets. In an arbitrary category we might not be able to define elements of objects. What we could do, however, is to pick a category that has all the products and define composition as a mapping (morphism) from a product of two objects to a third object. This is analogous to defining composition in terms of cartesian products of hom-sets, rather than defining it point-wise on individual morphisms.
But once we made a bold decision to abandon sets, why restrict ourselves to cartesian monoidal categories (the ones with a product defined by a universal construction)? All we really need is some notion of a binary operation, and that’s given by a magmatic category. So let’s define a magmatically enriched category 𝒜 as a collection of objects together with an additional magmatic category (𝒱0, ⨁) from which we pick our hom-objects (no longer called hom-sets). For any two objects A and B in the category 𝒜, a hom-object 𝒜(A, B) is an object of 𝒱0. Composition is defined for any triple of objects A, B, and C. It’s a morphism between objects in 𝒱0:
M: 𝒜(B, C) ⨁ 𝒜(A, B) → 𝒜(A, C)
Of course I stole this definition from a monoidally enriched category. Another part of it, which might be useful in the future, is the definition of a unit element. In a normal category, a unit morphism idA is an element of the hom-set 𝒜(A, A) of morphisms that go from A back to A. In an enriched category, the only way to define an element of a hom-object is though a mapping from a unit object I, so here’s the generalization of the identity morphism:
jA: I → 𝒜(A, A)
We have to make sure that there is appropriate interplay between composition, left and right unitors, and the unit element. These are expressed through commutative diagrams:
Commuting diagrams relating composition M, unitors l and r, and the unit element I
As before, since we don’t have to worry about associativity, we can omit the rest of the axioms.
Functors and Natural Transformations
A functor maps objects to objects and morphisms to morphisms. The mapping of objects translates directly to enriched settings, but we don’t want to deal with morphisms individually. We don’t want to look inside hom objects. Fortunately, we can define the action of a functor on a hom-object as a whole. The mappings of hom-objects are just morphisms in 𝒱0.
To define a functor T from an enriched category 𝒜 to ℬ, we have to define its action on any hom-object 𝒜(A, B):
TAB : 𝒜(A, B) → ℬ(T A, T B)
Notice that both categories must be enriched over the same 𝒱0 for TAB to be a morphism.
A natural transformation is trickier. Normally it is defined as a family of morphisms parameterized by objects. In the enriched setting, we have a slight problem: For a given object A we have to pick a single morphism from the hom-object ℬ(T A, S A). Here T and S are functors between two enriched categories 𝒜 and ℬ. Luckily, we have this trick of replacing element selection with a mapping from the unit object. We’ll use it to define the components of a natural transformation:
αA: I → ℬ(T A, S A).
The fact that the existence of the unit is necessary to define natural transformations in enriched categories is a little surprising. It also means that adjointness, which is defined through natural isomorphism between hom-objects, cannot be defined without a unit.
The generalization of the naturality condition is also tricky. Normally we just pick a morphism f : A → B in the category 𝒜 and lift it using two functors, T and S, to the target category ℬ. We get two morphisms,
T f : T A → T B
S f : S A → S B
We can then compose them with two components of the natural transformation,
αA : T A → S A
αB : T B → S B
and demand that the resulting diagram commute:
αB ∘ T f = S f ∘ αA
In an enriched category we can’t easily pick individual morphisms, so we have to operate on whole hom-objects. Here’s what we have at our disposal. The components of the natural transformation at A and B:
αB: I → ℬ(T B, S B)
αA: I → ℬ(T A, S A)
The action of the two functors on the hom-object connecting A to B:
TAB : 𝒜(A, B) → ℬ(T A, T B)
SAB : 𝒜(A, B) → ℬ(S A, S B)
Also, the composition of morphisms is replaced with the action of our binary operator on hom-objects. If you look at the hom-sets involved in the standard naturality condition, their composition corresponds to these two mappings under M:
ℬ(T B, S B) ⨁ ℬ(T A, T B) → ℬ(T A, S B)
ℬ(S A, S B) ⨁ ℬ(T A, S A) → ℬ(T A, S B)
The right hand sides are identical, the left hand sides can be obtained by either acting with α on the unit object, or by lifting the hom-object 𝒜(A, B) using S or T. There aren’t that many choices in this jigsaw puzzle. In particular, we get:
α ⨁ T : I ⨁ 𝒜(A, B) → ℬ(T B, S B) ⨁ ℬ(T A, T B)
S ⨁ α : 𝒜(A, B) ⨁ I → ℬ(S A, S B) ⨁ ℬ(T A, S A)
Finally, both of the left hand sides can be obtained from 𝒜(A, B) by applying the inverse of, respectively, l and r — the left and right unitors (remember, they are isomorphisms, so they are invertible). That gives us the naturality hexagon:
Naturality condition in enriched categories
Representable functors and the Yoneda lemma
On the surface, the enriched version of the Yoneda lemma looks deceptively simple. But there are some interesting nuances that should be mentioned. So let me recap what Yoneda is in the more traditional setting (see my blog post about it). It’s about functors from an arbitrary category to a much simpler category of sets — a category where we can actually look at elements, and where morphisms are familiar functions. In particular we are practically given for free those very convenient representable functors. These are the functors that map objects to hom-sets. You just fix one object, say K, in the category 𝒜, and for any other object X you get the set 𝒜(K, X) in Set.
This mapping of X to 𝒜(K, X) is called a representable functor because, in a way, it represents the object X as a set. It is a functor because it’s also defined on morphisms. Indeed, take a morphism f: X → Y. Its image must be a function from 𝒜(K, X) to 𝒜(K, Y). So pick an h in 𝒜(K, X) and map it to the composition f∘h, which is indeed in 𝒜(K, Y).
Now take any functor F from 𝒜 to Set, not necessarily representable. Yoneda tells us that natural transformations from any representable functor 𝒜(K, _) to F are in one to one correspondence with the elements of F K. In other words, there is a bijection:
Nat(𝒜(K, _), F) ≅ F K
which, additionally, is natural in both K and F (naturality if F requires you to look at F as an object in the category of functors, where morphisms are natural transformations).
Enriched representable functors
So how do we go about generalizing Yoneda to an enriched category. We start with a representable functor, except that now 𝒜(K, X) is not a hom-set but a hom-object in 𝒱0. The mapping from the enriched category 𝒜 to 𝒱0 cannot be treated as an enriched functor, because enriched functors are only defined between categories enriched over the same base category, and 𝒱0 is not enriched. So the trick is to enrich 𝒱0 over itself.
What does it mean? We want to keep the objects of 𝒱0 but somehow replace the hom-sets with objects from — again — 𝒱0. Now, if 𝒱0 is closed, it contains objects that are perfect for this purpose: the exponential objects. We’ll call the category resulting from enriching 𝒱0 over 𝒱0 simply 𝒱, hoping not to cause too much confusion. So we define a hom-object 𝒱(X, Y) as [X, Y] (the new notation for YX). We can apply our magmatic binary operator to two such hom-objects and map the resulting object to another hom-object:
M: [Y, Z] ⨁ [X, Y] → [X, Z]
This defines composition of hom-objects for us. The mapping is induced by the adjunction that defines the exponential object, and in fact requires associativity, so we can relax and admit that we are now in a closed monoidal (rather than magmatic) category. Oh well!
We’ve seen before that there is a mapping between hom-sets and exponentials, which we can rewrite in this form:
𝒱0(X, Y) ≅ V [X, Y]
(with V on the right hand side the functor that defines “elements” of an object). So 𝒱 not only has the same objects as 𝒱0 but its hom-objects are lifted by the functor V from hom-sets of 𝒱0. That makes 𝒱 a perfect stand-in for 𝒱0 in the enriched realm.
We are now free to define (enriched) functors that go from any category 𝒜 enriched over 𝒱0 to 𝒱. In particular, the mapping 𝒜(K, _): 𝒜 → 𝒱, which assigns the object 𝒜(K, X) to every X, defines a representable enriched functor. This way an object X in some exotic enriched category 𝒜 is represented by an object in a presumably better understood category 𝒱 which, being closed and monoidal, has more structure to play with.
Enriched Yoneda lemma
We now have all the tools to formulate the Yoneda lemma for enriched categories. Let’s pick an object K in an enriched category 𝒜. It defines an enriched representable functor, 𝒜(K, _) from 𝒜 to 𝒱. Let’s take another functor F from 𝒜 to 𝒱. The Yoneda lemma states that all natural transformations between these two functors:
α: 𝒜(K, _) → F
are in one-to-one correspondence with “elements” of the object F K, where by “elements” we mean mappings:
η: I → F K
More precisely, there is a bijection (one-to-one mapping that is onto) between the set of natural transformations Nat(𝒜(K, _)) and the set 𝒱0(I, F K).
Nat(𝒜(K, _)) ≅ 𝒱0(I, F K).
Notice that the bijection here is between sets: a set of natural transformations and a set of morphisms. There is a stronger version of the Yoneda lemma, which established a bijection between objects rather than sets. But for that one needs to objectify natural transformations, something that can indeed be done using ends. But that’s a topic for a whole new post.
My main motivation in this blog post was to try to figure out how far one can get with magmas before being forced to introduce a unit, and how far one can get with unital magmas before being forced to introduce associativity. These results are far from rigorous, since I might not have been aware of some hidden assumptions, and because the fact that some property is used in the proof of another property doesn’t mean that it is necessary.
So here’s the short summary. The following things are possible to define:
- Magmatic category
- Exponential objects and currying in a magmatic category
- Adjunction between magmatic product and exponentiation — the internal hom
- In unital magmatic category, the adjunction leads to isomorphism between external hom-set and “elements” of the internal hom
- Currying in the internal hom requires a monoidal category
Having a magmatic category one can define a magmatically enriched category and the following things:
- Natural transformations — only over unitally magmatic category
- Enriched Yoneda lemma — only over monoidal category
I’m grateful to Gershom Bazerman for stimulating discussions and helpful pointers.
- G. M. Kelly, Basic Concepts of Enriched Category Theory
- nLab, Closed category
I’m not fond of arguments based on lack of imagination. “There’s no way this code may fail!” might be a sign of great confidence or the result of ignorance. The inability to come up with a counterexample doesn’t prove a theorem. And yet there is one area of programming where such arguments work, and are quite useful. These are parametricity arguments: free theorems about polymorphic functions. Fortunately, there is solid theory behind parametricity. Free theorems are not based on ignorance. So I decided to read the relevant papers (see bibliography at the end of this post) and write a blog about it. How hard could it be? A few months and several failed attempts later I realized how naive I was. But I think I finally understand the basics enough to explain them in relatively simple terms.
Here’s a classic example — a function that takes a list of arbitrary type
a and returns a list of the same type:
r :: [a] -> [a]
What can this function do? Since it has to work with any type of list element, it can’t do anything type-specific. It can’t modify the elements or invent new ones. So all it can do is rearrange them, duplicate, or remove. Can you think of anything else?
The questions it a little tricky because it all depends on the kind of polymorphism your language supports. In Haskell, where we have parametric polymorphism, the above statement is for the most part true (modulo termination worries). In C++, which supports ad-hoc polymorphism, a generic function like:
can do all kinds of weird things.
Parametric polymorphism means that a function will act on all types uniformly, so the above declaration of
r indeed drastically narrows down the possibilities.
For instance, consider what happens when you
map any function of the type:
f :: a -> b
over a list of
a. You can either apply
map before or after acting on it with
r. It shouldn’t matter whether you first modify the elements of the list and then rearrange them, or first rearrange and then modify them. The result should be the same:
r (map f as) = map f (r as)
But is it true just because we can’t imagine how it may fail, or can we make a formal argument to prove it?
Let’s Argue (Denotational) Semantics
One way to understand polymorphism is to have a good model for types. At first approximation types can be modeled as sets of values (strictly speaking, as shown by Reynolds, the set-theoretical model fails in the case of polymorphic lambda calculus, but there are ways around it).
Bool is a two-element set of
Integer is a set of integers, and so on. Composite types can also be defined set-theoretically. For instance, a pair type is a cartesian product of two sets. A list of
a is a set of lists with elements from the set
a. A function type
a->b is a set of functions between two sets.
For parametric polymorphism you need to first be able to define functions on types: functions that take a type and produce a new type. In other words, you should be able to define a family of types that is parametrized by another type. In Haskell, we call such things type constructors.
For instance, given some type
a, produce a type of pairs:
(a, a). This can be formally written (not in Haskell) as:
Λa . (a, a)
Notice the capital lambda for defining functions on types (sets), as opposed to the lowercase lambda used for functions on values (set elements).
To turn a family of types into a family of values — a polymorphic value — you put the universal quantifier
forall in front of it. Don’t read too much into the quantifier aspect of it — it makes sense in the Curry-Howard isomorphism, but here it’s just a piece of syntax. It means that you use the type constructor to pick a type, and then you pick a specific value of that type.
You may recall the Axiom of Choice (AoC) from set theory. This axiom says that if you have a set of sets then there always exists a set of samples created by picking one element from each set. It’s like going to a chocolate store and ordering one of each. It’s a controversial axiom, and mathematicians are very careful in either using or avoiding it. The controversy is that, for infinite sets of sets, there may be no constructive way of picking elements. And in computer science we are not so much interested in proofs of existence, as in actual algorithms that produce tangible results.
Here’s an example:
forall a . (a, a)
This is a valid type signature, but you’d be hard pressed to implement it. You’d have to provide a pair of concrete values for every possible type. You can’t do it uniformly across all types. (Especially that some types are uninhabited, as Gershom Bazerman kindly pointed out to me.)
Interestingly enough, you can sometimes define polymorphic values if you constrain polymorphism to certain typeclasses. For instance, when you define a numeric constant in Haskell:
x = 1
its type is polymorphic:
x :: forall a. Num a => a
(using the language extension
x represents a whole family of values, including:
1.0 :: Float
1 :: Int
1 :: Integer
with potentially different representations.
But there are some types of values that can be specified wholesale. These are function values. Functions are first class values in Haskell (although you can’t compare them for equality). And with one formula you can define a whole family of functions. The following signature, for instance, is perfectly implementable:
forall a . a -> a
Let’s analyze it. It consists of a type function, or a type constructor:
Λa . a -> a
which, for any type
a, returns a function type
a->a. When universally quantified with
forall, it becomes a family of concrete functions, one per each type. This is possible because all these functions can be defined with one closed term (see Appendix 2). Here’s this term:
\x -> x
In this case we actually have a constructive way of picking one element — a function — for each type
a. For instance, if
a is a
String, we pick a function that takes any
String and returns the same string. It’s a particular
String->String function, one of many possible
String->String functions. And it’s different from the
Int->Int function that takes an
Int and returns the same
Int. But all these identity functions are encoded using the same lambda expression. It’s that generic formula that allows us to chose a representative function from each set of functions
a->a: one from the set
String->String, one from the set
In Haskell, we usually omit the
forall quantifier when there’s no danger of confusion. Any signature that contains a type variable is automatically universally quantified over it. (You’ll have to use explicit
forall, however, with higher-order polymorphism, where a polymorphic function can be passed as an argument to another function.)
So what’s the set-theoretic model for polymorphism? You simply replace types with sets. A function on types becomes a function on sets. Notice that this is not the same as a function between sets. The latter assigns elements of one set to elements of another. The former assigns sets to sets — you could call it a set constructor. As in: Take any set
a and return a cartesian product of this set with itself.
Or take any set
a and return the set of functions from this set to itself. We have just seen that for this one we can easily build a polymorphic function — one which for every type
a produces an actual function whose type is
(a->a). Now, with ad-hoc polymorphism it’s okay to code the
String function separately from the
Int function; but in parametric polymorphism, you’ll have to use the same code for all types.
This uniformity — one formula for all types — dramatically restricts the set of polymorphic functions, and is the source of free theorems.
Any language that provides some kind of pattern-matching on types (e.g., template specialization in C++) automatically introduces ad-hoc polymorphism. Ad-hoc polymorphism is also possible in Haskell through the use of type classes and type families.
Preservation of Relations
Let’s go to our original example and rewrite it using the explicit universal quantifier:
r :: forall a. [a] -> [a]
It defines a family of functions parametrized by the type
a. When used in Haskell code, a particular member of this family will be picked automatically by the type inference system, depending on the context. In what follows, I’ll use explicit subscripting for the same purpose. The free theorem I mentioned before can be rewritten as:
rb (map f as) = map f (ra as)
with the function:
f :: a -> b
serving as a bridge between the types
f relates values of type
a to values of type
b. This relation happens to be functional, which means that there is only one value of type
b corresponding to any given value of type
But the correspondence between elements of two lists may, in principle, be more general. What’s more general than a function? A relation. A relation between two sets
b is defined as a set of pairs — a subset of the cartesian product of
b. A function is a special case of a relation, one that can be represented as a set of pairs of the form
(x, f x), or in relational notation
x <=> f x. This relation is often called the graph of the function, since it can be interpreted as coordinates of points on a 2-d plane that form the plot the function.
The key insight of Reynolds was that you can abstract the shape of a data structure by defining relations between values. For instance, how do we know that two pairs have the same shape — even if one is a pair of integers, say
(1, 7), and the other a pair of colors, say
(Red, Blue)? Because we can relate
Blue. This relation may be called: “occupying the same position”.
Notice that the relation doesn’t have to be functional. The pair
(2, 2) can be related to the pair
(Black, White) using the non-functional relation:
(2 <=> Black),
(2 <=> White)
This is not a function because 2 is not mapped to a single value.
Conversely, given any relation between integers and colors, you can easily test which integer pairs are related to which color pairs. For the above relation, for instance, these are all the pairs that are related:
((2, 2) <=> (Black, Black)),
((2, 2) <=> (Black, White)),
((2, 2) <=> (White, Black)),
((2, 2) <=> (White, White))
Thus a relation between values induces a relation between pairs.
This idea is easily extended to lists. Two lists are related if their corresponding elements are related: the first element of one list must be related to the first element of the second list, etc.; and empty lists are always related.
In particular, if the relationship between elements is established by a function
f, it’s easy to convince yourself that the lists
bs are related if
bs = map f as
With this in mind, our free theorem can be rewritten as:
rb bs = map f (ra as)
In other words, it tells us that the two lists
are related through
Fig 1. Polymorphic function r rearranges lists but preserves relations between elements
r transforms related lists into related lists. It may change the shape of the list, but it never touches the values in it. When it acts on two related lists, it rearranges them in exactly the same way, without breaking any of the relations between corresponding elements.
Reading Types as Relations
The above examples showed that we can define relations between values of composite types in terms of relations between values of simpler types. We’ve seen this with the pair constructor and with the list constructor. Continuing this trend, we can state that two functions:
f :: a -> b
g :: a' -> b'
are related iff, for related
f x is related to
g y. In other words, related functions map related arguments to related values.
Notice what we are doing here: We are consistently replacing types with relations in type constructors. This way we can read complex types as relations. The type constructor
-> acts on two types,
b. We extend it to act on relations: The “relation constructor”
-> in A->B takes two relations A (between
a') and B (between
b') and produces a relation between functions
But what about primitive types? Let’s consider an example. Two functions from lists to integers that simply calculate the lengths of the lists:
lenStr :: [Char] -> Int
lenBool :: [Bool] -> Int
What happens when we call them with two related lists? The first requirement for lists to be related is that they are of equal length. So when called with related lists the two functions will return the same integer value . It makes sense for us to consider these two functions related because they don’t inspect the values stored in the lists — just their shapes. (They also look like components of the same parametrically polymorphic function,
It therefore makes sense to read a primitive type, such as
Int, as an identity relation: two values are related if they are equal. This way our two functions,
lenBool are indeed related, because they turn related lists to related (equal) results.
Notice that for non-polymorphic functions the relationship that follows from their type is pretty restrictive. For instance, two functions
Int->Int are related if and only if their outputs are equal for equal inputs. In other words, the functions must be (extensionally) equal.
All these relations are pretty trivial until we get to polymorphic functions. The type of a polymorphic function is specified by universally quantifying a function on types (a type constructor).
f :: forall a. φa
The type constructor
φ maps types to types. In our set-theoretical model it maps sets to sets, but we want to read it in terms of relations.
Functions on relations
A general relation is a triple: We have to specify three sets,
a', and a set of pairs — a subset of the cartesian product
a × a'. It’s not at all obvious how to define functions that map relations to relations. What Reynolds chose is a definition that naturally factorizes into three mappings of sets, or to use the language of programming, three type constructors.
First of all, a function on relations Φ (or a “relation constructor”) is defined by two type constructors,
ψ. When Φ acts on a relation A between sets
a', it first maps those sets, so that
b'=ψa'. ΦA then establishes a relation between the sets
b'. In other words, ΦA is a subset of
b × b'.
Fig 2. Φ maps relations to relations. The squarish sets represent cartesian products (think of a square as a cartesian product of two segments). Relations A and ΦA are subsets of these products.
Relations between polymorphic functions
Given that Φ maps relations to relations, a universally quantified version of it:
forall A. ΦA
maps pairs of sets to pairs of values.
Now suppose that you have two polymorphic functions
g :: forall a . φa
g' :: forall a'. ψa'
They both map types (sets) to values.
- We can instantiate
g at some type
a, and it will return a value
ga of the type
- We can instantiate
g' at some type
a', and it will return a value
g'a' of the type
We can do this for any relation A between two arbitrary sets
We will say that
g' are related through the relation induced by the type
(forall A. ΦA) iff the results
g'a' are related by ΦA.
Fig 3. Relation between two polymorphic functions. The pair
(g a, g' a') falls inside the relation ΦA.
In other words, polymorphic functions are related if they map related types to related values. Notice that in the interesting examples these values are themselves functions.
With these definitions, we can now reinterpret any type signature as a relation between values.
The Parametricity Theorem
Reynolds’ second key insight was that any term is in a relation with itself — the relation being induced by the term’s type. We have indeed defined the mapping of types to relations to make this work. Primitive types turn into identity relations, so obviously a primitive value is in relation with itself. A function between primitive types is in relation with itself because it maps related (equal) arguments into related (equal) results. A list or a pair of primitive types is in relation with itself because each element of it is equal to itself. You can recurse and consider a list of functions, or a pair of lists, etc., building the proof inductively, proceeding from simpler types to more and more complex types. The proof goes over all possible term construction rules and typing rules in a given theory.
Formally, this kind of proof is called “structural induction,” because you’re showing that more complex structures will satisfy the theorem as long as the simpler ones, from which they are constructed, do. The only tricky part is dealing with polymorphic functions, because they are quantified over all types (including polymorphic types). In fact, this is the reason why the naive interpretation of types as sets breaks down (see, however, Pitts’ paper). It is possible, however, to prove the parametricity theorem in a more general setting, for instance, using frames, or in the framework of operational semantics, so we won’t worry about it here.
Wadler’s key insight was to interpret Reynolds’ theorem not only as a way of identifying different implementations of the same type — for instance, cartesian and polar representations of complex numbers — but also as a source of free theorems for polymorphic types.
Let’s try applying parametricity theorem to some simple examples. Take a constant term: an integer like 5. Its type
Int can be interpreted as a relation, which we defined to be the identity relation (it’s one of the primitive types). And indeed, 5 is in this relation with 5.
Take a function like:
ord :: Char -> Int
Its type defines a relation between functions: Two functions of the type
Char->Int are related if they return equal integers for equal characters. Obviously,
ord is in this relation with itself.
Parametricity in Action
Those were trivial examples. The interesting ones involve polymorphic functions. So let’s go back to our starting example. The term now is the polymorphic function
r whose type is:
r :: forall a . [a] -> [a]
Parametricity tells us that
r is in relation with itself. However, comparing a polymorphic function to itself involves comparing the instantiations of the same function at two arbitrary types, say
a'. Let’s go through this example step by step.
We are free to pick an arbitrary relation A between elements of two arbitrary input sets
a'. The type of
r induces a mapping Φ on relations. As with every function on relations, we have to first identify the two type constructors
ψ, one mapping
a and one mapping
a'. In our case they are identical, because they are induced by the same polymorphic function. They are equal to:
Λ a. [a]->[a]
It’s a type constructor that maps an arbitrary type
a to the function type
The universal quantifier
forall means that
r lets us pick a particular value of the type
[a]->[a] for each
a. This value is a function that we call
ra. We don’t care how this function is picked by
r, as long as it’s picked uniformly, using a single formula for all
a, so that our parametricity theorem holds.
Fig 4. Polymorphic function r maps related types to related values, which themselves are functions on lists
Parametricity means that, if
a is related to
ra <=> ra'
This particular relation is induced by the function type
[a]->[a]. By our definition, two functions are related if they map related arguments to related results. In this case both the arguments and the results are lists. So if we have two related lists,
as :: [a]
as' :: [a']
they must, by parametricity, be mapped to two related lists,
bs = ra as
bs' = ra' as'
This must be true for any relation A, so let’s pick a functional relation generated by some function:
f :: a -> a'
This relation induces a relation on lists:
as' = map f as
The results of applying
r, therefore, must be related through the same relation:
bs' = map f bs
Combining all these equalities, we get our expected result:
ra' (map f as) = map f (ra as)
Parametricity and Natural Transformations
The free theorem I used as the running example is interesting for another reason: The list constructor is a functor. You may think of functors as generalized containers for storing arbitrary types of values. You can imagine that they have shapes; and for two containers of the same shape you may establish a correspondence between “positions” at which the elements are stored. This is quite easy for traditional containers like lists or trees, and with a leap of faith it can be stretched to non-traditional “containers” like functions. We used the intuition of relations corresponding to the idea of “occupying the same position” within a data structure. This notion can be readily generalized to any polymorphic containers. Two trees, for instance, are related if they are both empty, or if they have the same shape and their corresponding elements are related.
Let’s try another functor: You can also think of
Maybe as having two shapes:
Nothings are always related, and two
Justs are related if their contents are related.
This observation immediately gives us a free theorem about polymorphic functions of the type:
r :: forall a. [a] -> Maybe a
an example of which is
safeHead. The theorem is:
fmap h . safeHead == safeHead . fmap h
Notice that the
fmap on the left is defined by the
Maybe functor, whereas the one on the right is the list one.
If you accept the premise that an appropriate relation can be defined for any functor, then you can derive a free theorem for all polymorphic functions of the type:
r :: forall a. f a -> g a
g are functors. This type of function is known as a natural transformation between the two functors, and the free theorem:
fmap h . r == r . fmap h
is the naturality condition. That’s how naturality follows from parametricity.
I’d like to thank all the people I talked to about parametricity at the ICFP in Gothenburg, and Edward Kmett for reading and commenting on the draft of this blog.
Appendix 1: Other Examples
Here’s a list of other free theorems from Wadler’s paper. You might try proving them using parametricity.
r :: [a] -> a -- for instance, head
f . r == r . fmap f
r :: [a] -> [a] -> [a] -- for instance, (++)
fmap f (r as bs) == r (fmap f as) (fmap f bs)
r :: [[a]] -> [a] -- for instance, concat
fmap f . r == r . fmap (fmap f)
r :: (a, b) -> a -- for instance, fst
f . r == r . mapPair (f, g)
r :: (a, b) -> b -- for instance, snd
g . r == r . mapPair (f, g)
r :: ([a], [b]) -> [(a, b)] -- for instance, uncurry zip
fmap (mapPair (f, g)) . r == r . mapPair (fmap f, fmap g)
r :: (a -> Bool) -> [a] -> [a] -- for instance, filter
fmap f . r (p . f) = r p . fmap f
r :: (a -> a -> Ordering) -> [a] -> [a] -- for instance, sortBy
-- assuming: f is monotone (preserves order)
fmap f . r cmp == r cmp' . fmap f
r :: (a -> b -> b) -> b -> [a] -> b -- for instance, foldl
-- assuming: g (acc x y) == acc (f x) (g y)
g . foldl acc zero == foldl acc (g zero) . fmap f
r :: a -> a -- id
f . r == r . f
r :: a -> b -> a -- for instance, the K combinator
f (r x y) == r (f x) (g y)
mapPair :: (a -> c, b -> d) -> (a, b) -> (c, d)
mapPair (f, g) (x, y) = (f x, g y)
Appendix 2: Identity Function
Let’s prove that there is only one polymorphic function of the type:
r :: forall a. a -> a
and it’s the identity function:
id x = x
We start by picking a particular relation. It’s a relation between the unit type
() and an arbitrary (inhabited) type
a. The relation consists of just one pair
((), c), where
() is the unit value and
c is an element of
a. By parametricity, the function
r() :: () -> ()
must be related to the function
ra :: a -> a
There is only one function of the type
()->() and it’s
id(). Related functions must map related argument to related values. We know that
r() maps unit value
() to unit value
ra must map
c is arbitrary,
ra must be an identity for all (inhabited)
- John C Reynolds, Types, Abstraction and Parametric Polymorphism
- Philip Wadler, Theorems for Free!
- Claudio Hermida, Uday S. Reddy, Edmund P. Robinson, Logical Relations and Parametricity – A Reynolds Programme for Category Theory and Programming Languages
- Derek Dreyer, Paremetricity and Relational Reasoning, Oregon Programming Languages Summer School
- Janis Voigtländer, Free Theorems Involving Type Constructor Classes