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
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 monoidEvery 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
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
mappendis based on combination, just as
<*>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
Even the choice for
(->) rmakes 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
Maybe, and not like
To be entirely modern, this would be the
Alternativemonoid. Despite the possibilities for equivocation, this monoid is just as good as any other.
Alternative(a subclass of
Applicativeand a superclass of the more well-known
MonadPlus) gives rise to a monoid that is universal over the argument, no
-- 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.
Alternative, and indeed this is the monoid of choice for
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
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 monoidThis 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
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
instance Monad m => Monoid (ReaderT a m a) where mempty = ask ReaderT f `mappend` ReaderT g = ReaderT $ f <=< g
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
Moreover, should you discover it for
Sem, its applicability to any category-ish thing should still give you pause.
The burden of proofIn 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.