Universe Polymorphism

Introduction to universes

Russell's paradox implies that the collection of all sets is not itself a set. Namely, if there were such a set `U`, then one could form the subset `A ⊆ U` of all sets that do not contain themselves. Then we would have `A ∈ A` if and only if `A ∉ A`, a contradiction.

For similar reasons, not every Agda type is a `Set`. For example, we have

``` Bool : Set
Nat : Set
```

but not `Set : Set`. However, it often convenient for `Set` to have a type of its own, and so in Agda, it is given the type `Set₁`:

``` Set : Set₁
```

In many ways, expressions of type `Set₁` behave just like expressions of type `Set`; for example, they can be used as types of other things. However, the elements of `Set₁` are potentially "larger"; when `A : Set₁`, then `A` is sometimes called a large set. In turn, we have

``` Set₁ : Set₂
Set₂ : Set₃
```

and so on. A type whose elements are types is called a universe; Agda provides an infinite number of universes `Set`, `Set₁`, `Set₂`, `Set₃`, ..., each of which is an element of the next one. In fact, `Set` itself is just an abbreviation for `Set₀`. The subscript `n` is called the level of the universe `Setn`.

A note on syntax: you can also write `Set1`, `Set2`, etc., instead of `Set₁`, `Set₂`. To enter a subscript in the Emacs mode, type "`\_1`".

Universe example

So why are universes useful? Because sometimes it is necessary to define, and prove theorems about, functions that operate not just on sets but on large sets. In fact, most Agda users sooner or later experience an error message where Agda complains that `Set₁ != Set`. These errors usually mean that a small set was used where a large one was expected, or vice versa.

For example, suppose you have defined the usual datatypes for lists and cartesian products:

``` data List (A : Set) : Set where
[] : List A
_::_ : A -> List A -> List A

data _×_ (A B : Set) : Set where
_,_ : A -> B -> A × B

infixr 5 _::_
infixr 4 _,_
infixr 2 _×_
```

Now suppose you would like to define an operator `Prod` that inputs a list of n sets and takes their cartesian product, like this:

``` Prod (A :: B :: C :: []) = A × B × C
```

There is only one small problem with this definition. The type of `Prod` should be

``` Prod : List Set -> Set
```

However, the definition of `List A` specified that `A` had to be a set. Therefore, `List Set` is not a valid type. The solution is to define a special version of the `List` operator that works for large sets:

``` data List₁ (A : Set₁) : Set₁ where
[]₁ : List₁ A
_::₁_ : A -> List₁ A -> List₁ A
```

With this, we can indeed define:

``` Prod : List₁ Set -> Set
Prod []₁ = Unit
Prod (A ::₁ As) = A × Prod As
```

Universe polymorphism

Although we were able to give a type to the `Prod` operator by defining a special notion of large list, this quickly gets tiresome. Sooner or later, we find that we require yet another list type `List₂`, and it doesn't stop there. Also every function on lists (such as `append`) must be re-defined, and every theorem about such functions must be re-proved, for every possible level.

The solution to this problem is universe polymorphism. Agda provides a special primitive type `Level`, whose elements are possible levels of universes. In fact, the notation for the nth universe, `Setn`, is just an abbreviation for `Set n`, where `n : Level` is a level. We can use this to write a polymorphic `List` operator that works at any level. The library Agda.Primitive must be imported to access the `Level` type. The definition then looks like this:

``` open import Agda.Primitive

data List {n : Level} (A : Set n) : Set n where
[] : List A
_::_ : A -> List A -> List A
```

This new operator works at all levels; for example, we have

``` List Nat : Set
List Set : Set₁
List Set₁ : Set₂
```

Level arithmetic

It is not specified how many levels there are, but we know that there is a lowest level `lzero`, and for each level `n`, there exists some higher level `lsuc n`; therefore, the set of levels is infinite. In addition, we can also take the least upper bound `n ⊔ m` of two levels. In summary, the following (and only the following) operations on levels are provided:

```  lzero : Level
lsuc  : (n : Level) → Level
_⊔_   : (n m : Level) → Level
```

This is sufficient for most purposes; for example, we can define the cartesian product of two types of arbitrary (and not necessarily equal) levels like this:

``` data _×_ {n m : Level} (A : Set n) (B : Set m) : Set (n ⊔ m) where
_,_ : A -> B -> A × B
```

With this definition, we have, for example:

```  Nat × Nat : Set
Nat x Set : Set₁
Set × Set : Set₁
```

∀-Notation

From the fact that we write `Set n`, it can always be inferred that `n` is a level. Therefore, when defining universe-polymorphic functions, it is common to use ∀-notation. For example, the type of the universe-polymorphic `map` operator on lists can be written

``` map : ∀ {n m} {A : Set n} {B : Set m} → (A → B) → List A → List B
```

which is equivalent to

``` map : {n m : Level} {A : Set n} {B : Set m} → (A → B) → List A → List B
```

Expressions of kind `Setω`

In a sense, universes were introduced to ensure that every Agda expression has a type, including expressions such as `Set`, `Set₁`, etc. However, the introduction of universe polymorphism inevitably breaks this property again, by creating some new terms that have no type. Consider the polymorphic singleton set `Unit n : Setn`, defined by

``` data Unit (n : Level) : Set n where
<> : Unit n
```

It is well-typed, and has type

``` Unit : (n : Level) -> Set n
```

However, the type `(n : Level) -> Set n`, which is a valid Agda expression, does not belong to any universe. Indeed, the expression denotes a function mapping levels to universes, so if it had a type, it should be something like `Level -> Universe`, where `Universe` is the collection of all universes. But since the elements of `Universe` are types, `Universe` is itself a universe, so we have `Universe : Universe`. This leads to circularity and inconsistency. In other words, just as we cannot have a set of all sets, we also can't have a universe of all universes.

As a consequence, although the expression `(n : Level) -> Set n` is a type, it does not have a type. It does, however, have a "kind", which Agda calls `Setω`. There is no way to refer to `Setω` explicitly, but you may sometimes encounter it in error messages. For example, consider the following attempted definition:

``` badSet : ?
badSet = (n : Level) -> Set n
```

This results in the error message "`Setω` is not a valid type". For another example, consider trying to form the singleton list `[ Unit ]`:

``` badList: List ((n : Level) -> Set n)
This generates another error message about `Setω`. Again, the problem is that `((n : Level) -> Set n)` does not belong to any universe, and therefore even the polymorphic `List` operator, which requires an argument of type `Setn` for some `n`, can't be applied to it. The only type constructor that can be applied to expressions of kind `Setω` is `->`.
• Universe polymorphism is now enabled by default. Use `--no-universe-polymorphism` to disable it.
• Universe levels are no longer defined as a data type. The level combinators are defined in `~/.cabal/share/<GHC-version>/<Agda-version>/lib/prim/Agda/Primitive.agda`.
• `trustMe` is now universe-polymorphic.