I spent some time trying to figure out co- and contra-variance in Scala, and it turns out to be both interesting enough to be worth blogging about, and subtle enough that doing so will test my understanding!

So, you’ve probably seen classes in Scala that look a bit like this:

1 2 3 4 5 | sealed abstract class List[+A] { def head : A def ::[B >: A] (x:B) : List[B] = ... ... } |

And you’ve probably heard that the +A means that A is a “covariant type parameter”, whatever that means. And if you’ve tried to use classes with co- or contra-variant type parameters, you’ve probably run into cryptic errors about “covariant positions” and other such gibberish. Hopefully, by the end of this post, you’ll have some idea what that all means.

The first thing that’s going on there is that List is a “generic” type. That is, you can have lots of List types. You can have List[Int], and List[MyClass] or whatever. To put this in another way, List[_] is a *type constructor*; it’s like a function that takes another concrete type and produces a new one. So if you already have a type X, you can use the List type constructor to make a new type, List[X].

## A little bit of category theory

To get the cool stuff in all its generality, we’re going to need to start thinking about things in terms of categories. Fortunately, it’s pretty non-scary categories stuff. Recall that a category C is just some objects and some arrows (which we usually gloss as “functions”). Arrows go from one object to another, and the only requirements for being a category are that you have some binary operation on arrows (usually glossed as “composition”), that makes new arrows that go from and to the right places; and that you have an “identity” arrow on every object that does just what you’d expect.^{1} The category we’re mostly interested in is the category of *types*: types like Int, Person, Map[Foo, Bar] are the objects, and arrows are precisely functions.

The other concept we’re going to need is that of a functor. A functor F: C -> D is a mapping *between* categories. However, there’s no reason you can’t have functors from categories to themselves (“helpfully” called “endofunctors”), and those are the ones we’re going to be interested in. Functors have to turn objects in the source category into objects in the target category, and they also have to turn arrows into new arrows. Again, functors have to obey certain laws, but don’t worry too much about that.^{2}

Okay, so who cares about functors? The answer is that type constructors are basically functors on the category of types. How is that? Well, they turn types (which are our objects) into other types: check! But what about the arrows (i.e. functions). Don’t functors have to map those over as well? Yes, they do, but in Scala we don’t call the function that comes out of the List functor List[f], we call it map(f).^{3}

One final concept and then I promise this will start to get relevant. Some mappings between categories look a lot like functors, except that they reverse the direction of arrows. So instead of getting F(f): FX -> FY. So these got a special name, they’re called *contravariant* functors. To distiguish them, normal functors are called *covariant* functors.

Look at that, there are those funny words again. But what on *earth* do contravariant functors have to do with Scala?

Good question.

## Subtyping

The key feature of Scala, for our purposes, is that it’s a language with *subtyping*. Classes (types) can be sub- or super- types of other classes. This gives us the familiar idea of a class hierarchy. Looking at it mathematically, we can say that we have a relation <: between types that acts as a partial order. Here comes neat Category Theory Trick no. 1: we can view any partially ordered set as a category! The objects are the objects, and we have an arrow A ->B iff A <: B. This is a bit weird, because we’re only ever going to have one arrow between objects, and they’re not really “functions” any more, but all the formal machinery still works.^{4}

Now some type constructors on this category still look like functors. They map objects to other objects, and if one of those objects is a subtype of the other, then they may or may not impose a relationship between the mapped objects.

This is where the Scala type annotations come in. When we declare List[A+], we are saying that List is covariant in the parameter A.^{5} What that means is that it takes a type, say Parent, to a new type List[Parent], and if Child is a subtype of Parent, then List[Child] will be a subtype of List[Parent]. If we’d declared List to be *contravariant* (List[-A]), then List[Child] would be a *supertype* of List[Parent].

There’s one final possibility. Since subtyping is a partial order, we can have two types where neither one is a subtype of the other. There’s no reason in principle why a type constructor T couldn’t take Parent and Child to new types which were completely unrelated. In Scala, this is the case when you don’t provide an annotation for the type in the declaration; such a constructor is said to be *invariant* in that parameter. Arrays, for example, have this property.

And that, fundamentally, is it. That’s what those little +s and -s on type paramters mean. You can go home now.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | class GParent class Parent extends GParent class Child extends Parent class Box[+A] class Box2[-A] def foo(x : Box[Parent]) : Box[Parent] = identity(x) def bar(x : Box2[Parent]) : Box2[Parent] = identity(x) foo(new Box[Child]) // success foo(new Box[GParent]) // type error bar(new Box2[Child]) // type error bar(new Box2[GParent]) // success |

## But what about those cryptic errors?

1 2 3 4 5 | class Box[+A] { def set(x : A) : Box[A] } // won't compile |

You get these kinds of errors in Scala because of the subtleties of how variance relates to *functions* (and later, methods). We can see that there’s something weird going on if we look at the declaration of the Function trait:

1 2 3 4 5 6 7 | trait Function1[-T1, +R] { def apply(t : T1) : R ... } |

Whoa. That’s pretty strange. Not only does it have two type parameters, one of them is contravariant. Weird. Let’s work through this methodically.

We have Function1[A,B], which is a *type* of one-parameter functions that go from type A to type B. It can therefore be a sub- or super-type of other (function) types. For example,

Function1[GParent, Child] <: Function1[Parent, Parent]

How do I know this? Because of the variance annotations on Function1. The first parameter is contravariant, so can vary upwards, and the second parameter is covariant, so can vary downwards.

The reason why Function1 behaves in this way is a bit subtle, but makes sense if you think about the way substitution has to work when you have subtyping. If you have a function from A to B, what can you substitue for it? Anything you put in its place must make *fewer* requirements on it’s input type; since the function can’t, for example, get away with calling a method that only exists on subtypes of A. On the other hand, it must return a type at least as specialised as B, since the *caller* of the function may be expecting all the methods on B to be available.

## Function Functors

There’s actually a nice category theory justification for why things have to be this way. In general, for any category C we can also construct a category of the Hom-sets of C. Functions between these sets will just be higher-order functions that turn functions into different functions. There is then an obvious functor, Hom(-, -) that takes two objects A and B and produces Hom(A, B). The Hom-functor is a bit tricky because it’s a *bifunctor*: it takes two arguments. The easiest way to deal with it is to sort of “partially apply” it and look at how it behaves on each of its arguments individually.

So Hom(A, -) takes an object B to the set of functions from A to B. How does it act on functions? If we have a morphism f:B B’ we need a function Hom(A, f): Hom(A, B) -> Hom(A, B’). The obvious definition is

Hom(A, f)(g) = f . g

That is, you do g first, to get from A to B, and then f to get from B to B’. So Hom(A, -) acts as a covariant functor.

On the other hand, if you try and make Hom(-, B) into a covariant functor, good luck! The types just don’t line up if you try and do composition. What *does* work is the following:

Hom(f, B)(g) = g . f

where g is in Hom(B’, B), rather than Hom(A, B). So Hom(-, B) acts as a contravariant functor.^{6} Which makes Hom(A, B) contravariant in A, and covariant in B — just like Function1!^{7}

This is actually a more general result, since it applies in any category, and not just in the category of types with subtyping. Cool!

## Back to Earth

Okay, so functions in Scala have these weird variance properties. But from a theoretical point of view, methods are just functions, and so they ought to have the same variance properties, even though we can’t see them (methods don’t have a trait in Scala!).

So we can now see why we got that cryptic compile error. We declared that A was covariant in our class, and also that set takes a parameter of type A. But then, for some B <: A we could replace an instance of Box[A] with an instance of Box[B], and hence an instance of Box[A].set(x) with Box[B].set(x), where x:B. But set[A] *can’t* be replaced by set[B] as an argument, for the reasons we disucussed above; at best it can be contravariant. So this would allow us to do stuff we shouldn’t be able to do. Likewise, if we declared A as *contravariant* then we would run into conflict with the *return* type of set. So it looks like we have to make A invariant.

As an aside, this is why it’s an absolutely terrible idea that Java’s arrays are covariant. That means that you can write code like the following:

1 2 3 4 5 | Integer[] ints = [1,2] Object[] objs = ints objs[0] = "I'm an integer!" |

Which will *compile*, but throw an ArrayStoreException at runtime. Nice.

Actually, we don’t *have* to make container types with an “append”-like method invariant. Scala also lets us put type bounds on things. So if we modify Box as follows:

1 2 3 4 5 | class BoundedBox[+A] { set[B >: A](x:B) : Box[B] } |

then it will compile. This ensures that the input type of the set method is properly contravariant.

And that’s about it. The thing to remember with Scala is that *everything* is a method. So if you’re getting surprising variance errors, it might be that you have a sneaky method somewhere that needs a lower bound.

- In full, the requirements are:A class of objects: CFor every pair of objects, a class of morphisms between them: Hom(A, B)A binary operation . : Hom(B, C) x Hom(A, B) -> Hom(A, C), which is associative and has the identity morphism as its identity.
- These are:F(id{X}) = id{FX}F(f.g) = F(f).F(g)
- The astute reader will have noticed that not all type constructors come with a map function. This does indeed mean that not all type constructors are functors. But pretend that they are for now.
- Crucially, we can use the relation to give us our arrows because it’s transitive, and hence composition will work properly.
- Yes, there can be more than one parameter. Don’t worry about it for now.
- If you’re wondering whether there couldn’t be some other way of mapping the functions that would work, it turns out that there can’t be one that also makes the functor laws work. You can try it yourself if you don’t believe me!
- We actually need to do a little bit more work to show that Hom(-, -) is a true bifunctor (functor on the product category), but it’s not terribly interesting.

## Comments (3)

Pingback:Covariância e Contravariância em Scala | Blog de Programação LiteratiBy Vivek Kumar on April 4, 2013

By alex on June 24, 2013