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POWhat is this special functor structure called?
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Suppose that F is an applicative functor with the additional laws (with Haskell syntax): <ol> <li><code>pure (const ()) <*> m</code> === <code>pure ()</code></li> <li><code>pure (\a b -> (a, b)) <*> m <*> n</code> === <code>pure (\a b -> (b, a)) <*> n <*> m</code></li> <li><code>pure (\a b -> (a, b)) <*> m <*> m</code> === <code>pure (\a -> (a, a)) <*> m</code></li> </ol> What is the structure called if we omit (3.)? Where can I find more info on these laws/structures? <h2>Comments on comments</h2> Functors which satisfy (2.) are often called commutative. The question is now, whether (1.) implies (2.) and how these structures can be described. I am especially interested in structures which satisfies (1-2.) but not (3.) Examples: <ul> <li>The reader monad satisfies (1-3.)</li> <li>The writer monad on a commutative monoid satisfies only (2.)</li> <li>The monad <code>F</code> given below satisfies (1-2.) but not (3.)</li> </ul> Definition of <code>F</code>: <pre><code>{-# LANGUAGE GeneralizedNewtypeDeriving #-} {-# LANGUAGE RankNTypes #-} import Control.Monad.State newtype X i = X Integer deriving (Eq) newtype F i a = F (State Integer a) deriving (Monad) new :: F i (X i) new = F $ modify (+1) >> gets X evalF :: (forall i . F i a) -> a evalF (F m) = evalState m 0 </code></pre> We export only the types <code>X</code>, <code>F</code>, <code>new</code>, <code>evalF</code>, and the instances. Check that the following holds: <ul> <li><code>liftM (const ()) m</code> === <code>return ()</code></li> <li><code>liftM2 (\a b -> (a, b)) m n</code> === <code>liftM2 (\a b -> (b, a)) n m</code></li> </ul> On the other hand, <code>liftM2 (,) new new</code> cannot be replaced by <code>liftM (\a -> (a,a)) new</code>: <pre><code>test = evalF (liftM (uncurry (==)) $ liftM2 (,) new new) /= evalF (liftM (uncurry (==)) $ liftM (\a -> (a,a)) new) </code></pre> <h2>Comments on C. A. McCann's answer</h2> I have a sketch of proof that (1.) implies (2.) <pre><code>pure (,) <*> m <*> n </code></pre> = <pre><code>pure (const id) <*> pure () <*> (pure (,) <*> m <*> n) </code></pre> = <pre><code>pure (const id) <*> (pure (const ()) <*> n) <*> (pure (,) <*> m <*> n) </code></pre> = <pre><code>pure (.) <*> pure (const id) <*> pure (const ()) <*> n <*> (pure (,) <*> m <*> n) </code></pre> = <pre><code>pure const <*> n <*> (pure (,) <*> m <*> n) </code></pre> = ... = <pre><code>pure (\_ a b -> (a, b)) <*> n <*> m <*> n </code></pre> = see below = <pre><code>pure (\b a _ -> (a, b)) <*> n <*> m <*> n </code></pre> = ... = <pre><code>pure (\b a -> (a, b)) <*> n <*> m </code></pre> = <pre><code>pure (flip (,)) <*> n <*> m </code></pre> Observation For the missing part first consider <pre><code>pure (\_ _ b -> b) <*> n <*> m <*> n </code></pre> = ... = <pre><code>pure (\_ b -> b) <*> n <*> n </code></pre> = ... = <pre><code>pure (\b -> b) <*> n </code></pre> = ... = <pre><code>pure (\b _ -> b) <*> n <*> n </code></pre> = ... = <pre><code>pure (\b _ _ -> b) <*> n <*> m <*> n </code></pre> Lemma We use the following lemma: <pre><code>pure f1 <*> m === pure g1 <*> m pure f2 <*> m === pure g2 <*> m </code></pre> implies <pre><code>pure (\x -> (f1 x, f2 x)) m === pure (\x -> (g1 x, g2 x)) m </code></pre> I could prove this lemma only indirectly. The missing part With this lemma and the first observation we can prove <pre><code>pure (\_ a b -> (a, b)) <*> n <*> m <*> n </code></pre> = <pre><code>pure (\b a _ -> (a, b)) <*> n <*> m <*> n </code></pre> which was the missing part. Questions Is this proved already somewhere (maybe in a generalized form)? <h2>Remarks</h2> (1.) implies (2.) but otherwise (1-3.) are independent. To prove this, we need two more examples: <ul> <li>The monad <code>G</code> given below satisfies (3.) but not (1-2.)</li> <li>The monad <code>G'</code> given below satisfies (2-3.) but not (1.)</li> </ul> Definition of <code>G</code>: <pre><code>newtype G a = G (State Bool a) deriving (Monad) putTrue :: G () putTrue = G $ put True getBool :: G Bool getBool = G get evalG :: G a -> a evalG (G m) = evalState m False </code></pre> We export only the type <code>G</code>, <code>putTrue</code>, <code>getBool</code>, <code>evalG</code>, and the <code>Monad</code> instance. The definition of <code>G'</code> is similar to the definition of <code>G</code> with the following differences: We define and export <code>execG</code>: <pre><code>execG :: G' a -> Bool execG (G m) = execState m False </code></pre> We do not export <code>getBool</code>.
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<haskell><category-theory><applicative>
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What is this special functor structure called?
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