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POWriting cojoin or cobind for n-dimensional grid type
 

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    <p>Using the typical definition of type-level naturals, I've defined an n-dimensional grid.</p> <pre><code>{-# LANGUAGE KindSignatures #-} {-# LANGUAGE DataKinds #-} {-# LANGUAGE GADTs #-} {-# LANGUAGE TypeFamilies #-} data Nat = Z | S Nat data U (n :: Nat) x where Point :: x -&gt; U Z x Dimension :: [U n x] -&gt; U n x -&gt; [U n x] -&gt; U (S n) x dmap :: (U n x -&gt; U m r) -&gt; U (S n) x -&gt; U (S m) r dmap f (Dimension ls mid rs) = Dimension (map f ls) (f mid) (map f rs) instance Functor (U n) where fmap f (Point x) = Point (f x) fmap f d@Dimension{} = dmap (fmap f) d </code></pre> <p>Now I want to make it an instance of Comonad, but I can't quite wrap my brain around it.</p> <pre><code>class Functor w =&gt; Comonad w where (=&gt;&gt;) :: w a -&gt; (w a -&gt; b) -&gt; w b coreturn :: w a -&gt; a cojoin :: w a -&gt; w (w a) x =&gt;&gt; f = fmap f (cojoin x) cojoin xx = xx =&gt;&gt; id instance Comonad (U n) where coreturn (Point x) = x coreturn (Dimension _ mid _) = coreturn mid -- cojoin :: U Z x -&gt; U Z (U Z x) cojoin (Point x) = Point (Point x) -- cojoin ::U (S n) x -&gt; U (S n) (U (S n) x) cojoin d@Dimension{} = undefined -- =&gt;&gt; :: U Z x -&gt; (U Z x -&gt; r) -&gt; U Z r p@Point{} =&gt;&gt; f = Point (f p) -- =&gt;&gt; :: U (S n) x -&gt; (U (S n) x -&gt; r) -&gt; U (S n) r d@Dimension{} =&gt;&gt; f = undefined </code></pre> <p>Using <code>cojoin</code> on an n-dimensional grid will produce an n-dimensional grid of n-dimensional grids. I'd like to provide an instance with the same idea as <a href="http://hpaste.org/76465" rel="noreferrer">this one</a>, which is that the <em>value</em> of the <em>cojoined</em> grid at (x,y,z) should be the <em>original</em> grid <em>focused</em> on (x,y,z). To adapt that code, it appears that we need to reify <code>n</code> in order to perform <code>n</code> "fmaps" and <code>n</code> "rolls". You don't have to do it that way but if that helps, then there you go.</p>
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    1. COComonads m, n compose if there's a suitable "distributive law" of type forall x. m (n x) -> n (m x). Looks to me like you're iteratively composing the element-between-left-list-and-right-list comonad whose distributive law (modulo unenforced rectangularity constraints) is transposition. So you might be able to crunch this one step at a time. (Apparently trivial lifesaving tip: use snoc-lists for left-of-the-cursor, so you keep orientation clear in your code.)
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