## Tuesday, October 13, 2015

### The uninteresting monoids of certain monads

Suppose there is some structure from which arises a monad. Let’s call one `Sem`.

```data Sem a = ... -- doesn't matter
```

In the spirit of defining every typeclass instance you can think of—a spirit that I share, believe me—you discover a monoid, and suggest that it be included with `Sem`.

```instance ??? => Monoid (Sem a) where
-- definition here
```

But then, you are surprised to encounter pessimism and waffling, from me!

I’m so skeptical of your monoid because it is “common”; many monoids simply fall out of numerous monads, to greater or lesser degree, but that doesn’t make them “good” monoids. Having rediscovered a common, uninteresting monoid, you need to provide more justification of why it should be “the” monoid for this data type.

## The lifted monoid

Every applicative functor gives rise to a monoid that lifts their arguments’ monoid.

```instance Monoid a => Monoid (Sem a) where
mempty = pure mempty
mappend = liftA2 mappend
```

This is “the” monoid for `(->) r` and `Maybe`. It is decidedly not the monoid for `[]`. For in that universe,

```> [Sum 2] `mappend` [Sum 3, Sum 7]
[Sum 5, Sum 9]
> [Sum 42] `mappend` []
[]
```

Maybe you reaction is “but that’s not a legal monoid!” Sure it is. The `mappend` is based on combination, just as `Applicative []`’s `<*>` is. And, in the example above, the left and right identity is `[Sum 0]`, not `[]`.

It’s just not the monoid you’re used to.

Moreover, it isn’t quite right for `Maybe`! The constraint generalizes to `Semigroup a`. It is an unfortunate accident of history that the constraint on Haskell `Maybe`’s monoid is also `Monoid`.

Even the choice for `(->) r` makes many people unhappy, though we’re not quite ready to explore the reason for that.

So, what makes you think this is a good choice for `Sem`? It’s not enough justification that it can be written; that is always the case. There must be something that makes `Sem` like `(->) r` or `Maybe`, and not like `[]`.

## The `MonadPlus` monoid

To be entirely modern, this would be the `Alternative` monoid. Despite the possibilities for equivocation, this monoid is just as good as any other.

Simply: every `Alternative` (a subclass of `Applicative` and a superclass of the more well-known `MonadPlus`) gives rise to a monoid that is universal over the argument, no `Monoid` constraint required.

```-- supposing Alternative Sem,
instance Monoid (Sem a) where
mempty = empty
mappend = (<|>)
```

You would not be surprised at this having prepared by reading the haddock for `Alternative`: “a monoid on applicative functors”, it says.

`[]` is `Alternative`, and indeed this is the monoid of choice for `[]`. But `Maybe` is also `Alternative`. Why is this one good for `[]`, but not `Maybe`? Let’s take a peek through the looking glass.

```> Just 1 `mappend` Just 4
Just 1
> Nothing `mappend` Just 3
Just 3
```

I happen to agree with the monoid of choice for `Maybe`. But I’m sure many have been surprised it’s not “just take the leftmost `Just`, or give `Nothing`”.

Except where phantom `Const`-style functors are involved, the two preceding monoids always have incompatible behavior. One sums the underlying values, the other never touchs them, only rearranging them. So, if both are available to `Sem`, to define a monoid, we must give up at least one of these.

Alternatively, we could put off the decision until someone comes up with a convincing argument for “the” monoid.

## The category endomorphism monoid

This monoid hasn’t let the lack of a pithy name handicap it; despite the stunning blow of losing the prized `(->)` to the lifted monoid (the commit), this one probably has even more fans eager for a rematch today than it did back then.

I’m referring to this one, still thought of as “the” monoid for `(->)` by some.

```instance Monoid (a -> a) where
mempty = id
mappend = (.)
```

The elegance of this kind of “summing” of functions is undeniable. Moreover, it applies to every `Category`, not just `(->)`. Even more, it works for anything sufficiently `Category`-ish, such as `ReaderT`.

```instance Monad m => Monoid (ReaderT a m a) where
```

Its fatal flaw is that twin appearance of `a`; it requires `FlexibleInstances`, so can’t be written in portable Haskell 2010. As such, it will probably remain in the minor leagues of newtypes like `Endo`.

Moreover, should you discover it for `Sem`, its applicability to any category-ish thing should still give you pause.

## The burden of proof

In Haskell, hacking until it compiles is a great way to work. It is tempting to rely on its conclusions in ever more cases, once you have discovered its effectiveness. However, in the cases above, it is very easy to be led astray by the facile promises of the typechecker.

Introducing one of these monoids is risky. It precludes the later introduction of the “right” monoid for a datatype, for want of compatibility. If you really must offer one of these monoids as “the” monoid for a datatype, the responsibility falls to you: demonstrate that this is a good monoid, not just an easy one.