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 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
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.
