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POCan liftM differ from liftA?
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1634911
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2009-10-28T02:34:44.847
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2013-01-15T16:54:59.657
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2011-01-06T20:58:04.893
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Body
According to <a href="http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf" rel="noreferrer">the Typeclassopedia</a> (among other sources), <code>Applicative</code> logically belongs between <code>Monad</code> and <code>Pointed</code> (and thus <code>Functor</code>) in the type class hierarchy, so we would ideally have something like this if the Haskell prelude were written today: <pre><code>class Functor f where fmap :: (a -> b) -> f a -> f b class Functor f => Pointed f where pure :: a -> f a class Pointed f => Applicative f where (<*>) :: f (a -> b) -> f a -> f b class Applicative m => Monad m where -- either the traditional bind operation (>>=) :: (m a) -> (a -> m b) -> m b -- or the join operation, which together with fmap is enough join :: m (m a) -> m a -- or both with mutual default definitions f >>= x = join ((fmap f) x) join x = x >>= id -- with return replaced by the inherited pure -- ignoring fail for the purposes of discussion </code></pre> (Where those default definitions were re-typed by me from the <a href="http://en.wikipedia.org/wiki/Monad_%28functional_programming%29#fmap_and_join" rel="noreferrer">explanation at Wikipedia</a>, errors being my own, but if there are errors it is at least in principle possible.) As the libraries are currently defined, we have: <pre><code>liftA :: (Applicative f) => (a -> b) -> f a -> f b liftM :: (Monad m) => (a -> b) -> m a -> m b </code></pre> and: <pre><code>(<*>) :: (Applicative f) => f (a -> b) -> f a -> f b ap :: (Monad m) => m (a -> b) -> m a -> m b </code></pre> Note the similarity between these types within each pair. My question is: are <code>liftM</code> (as distinct from <code>liftA</code>) and <code>ap</code> (as distinct from <code><*></code>), simply a result of the historical reality that <code>Monad</code> wasn't designed with <code>Pointed</code> and <code>Applicative</code> in mind? Or are they in some other behavioral way (potentially, for some legal <code>Monad</code> definitions) distinct from the versions that only require an <code>Applicative</code> context? If they are distinct, could you provide a simple set of definitions (obeying the laws required of <code>Monad</code>, <code>Applicative</code>, <code>Pointed</code>, and <code>Functor</code> definitions described in the Typeclassopedia and elsewhere but not enforced by the type system) for which <code>liftA</code> and <code>liftM</code> behave differently? Alternatively, if they are not distinct, could you prove their equivalence using those same laws as premises?
Tags
<haskell><theory><typeclass><category-theory>
Title
Can liftM differ from liftA?
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1. PO
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1. USDoug McClean
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 POCan liftM differ from liftA?
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 POCan liftM differ from liftA?
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 POCan liftM differ from liftA?
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