Posted on December 19, 2021

Graded monads are great. In functional programming, for example, they can be used to track certain sorts of external state at the type level. 1

Graded monads generalize ordinary monads, let’s see how: An ordinary monad can be viewed as a lax 2-functor 1 → CAT from the terminal 2-category 1 to the 2-category of large categories CAT.

The later contains the (idealized) category Hask as an object. And it’s the only one I’m going to use as an example. But you may also think of the category Set of sets and functions, if that’s more familiar.

Let’s note that the terminal 2-category 1 consists of a unique object, its identity arrow, and the later’s identity 2-arrow.

First, any lax 2-functor T : 1 → CAT needs to assign values for the unique object * and for its identity (the image of the identity 2-arrow is fixed by the choice of the 1-arrow).

Note in particular that T does not need to preserve the identity arrow. That’s part of what makes it lax.

So we define T by choosing a category as the image of the unique object. Let’s suppose we chose Hask.

Now we still get to freely choose an endomorphism of Hask, i.e., an endofunctor. Let’s also call it T. So keep in mind there will be two Ts appearing, one a lax 2-functor, the other an endofunctor of Hask.

We already see some structure emerge: because T : Hask → Hask is an endofunctor, we can take its powers: T0, T, T2, T3 … by composing T with itself repeatedly.

T0 is of course the identity functor Id : Hask → Hask of Hask. By contrast, if T were a strict 2-functor, our image T = T(id) would have to be the identity functor of Hask, and equal to all its powers. Not so here.

The relationship between the identity functor of Hask and T : Hask → Hask is not completely arbitrary however: the definition of a lax 2-functor requires us to provide a 2-arrow IdHask ⇒ T, i.e., a natural transformation.

In Haskell notation this is a function eta :: a -> T a . As for the identity, so for the composition: since T is lax, it is only required to preserve composites up to 2-arrow. For our Hask endofunctor T this means we require a natural transformation T2 ⇒ T, let’s call it μ.

In Haskell notation this is a mu :: T(T a) -> T a . (The natural transformations eta and mu are better known in Haskell as return and join from the Control.Monad module)

Continuing with our story: for (T,η,μ) to actually be a monad the natural transformations η and μ have to satisfy two axioms usually called UNIT and ASSOCIATIVITY2.

Now the big shocking reveal is that the axioms for our lax 2-functor T : 1 → CAT look exactly like the monad axioms for (T:HaskHask,η,μ).

Disregarding the fact that I have not actually given a precise definition of lax 2-functors, we now take it as established that there is a bijection between such T : 1 → Hask with object value Hask and monads T on the category Hask.

Now, to obtain the concept of a graded monad we need to replace the terminal 2-category 1 in the story above with something else. That something else is going to be a monoid. Let me explain what I mean by that.

Monoids, i.e., sets with an associative and unital operation can be viewed as categories: introduce an object, let’s call it *, and let the morphisms  *  → * be the elements of the monoid. Then the monoid operation is a composition operation for this category and the monoid unit is the identity morphism of the single object *.

(The object * is not to be confused with the Type kind in Haskell.)

So a monoid in a way is a category. But we really needed to replace the terminal 2-category 1 with another 2-category, since we want to talk about lax 2-functors out of it. This is easily fixed: just add in an identity 2-arrow for each 1-arrow.

Thus, in the same vein, a monoid in a way is a 2-category, and this shifting of viewpoint does not require supplying any new data, but merely the use of a more powerful language to talk about the same thing.

## Example

As an example, let’s consider the monoid with elements 0 and 1 and the usual multiplication as the operation. Then clearly 1 is the unit. As a category it has arrows 0 :  *  → * and 1 :  *  → * and composition is just multiplication, as we have said. Let’s call this category B.

We view B as a 2-category, and consider what it means to have a lax 2-functor T : B → CAT. T(*) is some category, for concreteness, we can continue to assume that’s Hask. Since B has 2 distinct arrows we get two Hask endo functors T0 and T1 : Hask → Hask.

Note that T1 is the image of the identity 1 :  *  → * under the lax 2-functor, so laxness means there must be a natural transformation η : IdHask ⇒ T1.

Also by laxness of T, as part of its data, there are natural transformations μ : TiTj ⇒ Tk where i and j can be 0 or 1 and k is i × j. (Note that these are 4 different μ’s, but I don’t want to clutter the notation with indices.)

In Haskell notation we might write these natural transformations as eta :: a -> T One a and mu :: T i (T j a) -> T (Mult i j) a . (Note that the first type parameter of T is of a data kind with constructors Zero and One, Mult is a type family implementing multiplication for them.)

The natural transformations need to satisfy unit and associativity laws. These can be obtained from the diagrams I included earlier, by inserting indices i, j and i × j in the right places.

## Generalization

Now, to see that graded monads generalize monads we only need to consider the fact that the trivial monoid having a single element gives exactly the terminal category when promoted to a 2-category in the way I described.

Furthermore, homomorphisms of monoids, i.e., maps preserving monoid unit and composition, can be promoted to strict 2-functors in an obvious way. In particular, for an arbitrary monoid M there is always a unique homomorphism e : 1 → M picking out the unit.

If we write the homomorphism e promoted to a strict 2-functor as E, and if we have a lax 2-functor T : M → CAT then by composing these TE : 1 → CAT, we obtain an ordinary monad contained in the graded monad given by T.

## End

There is a type class interface for graded monads implemented in effect-monad by Dominic Orchard . Using the RebindableSyntax extension of GHC or the upcoming QualifiedDo extension we can even write do-blocks that use graded monads.

To my knowledge, the primordial source for lax 2-functors is Introduction to bicategories by Jean Bénabou. Although he calls them simply morphisms of bicategories. The example section shows how versatile the notion is. In particular, he describes «polyads», these are also known today as indexed monads. I might write about them some other time.