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<title>src/Data/Functor/Kan/Rift.hs</title>
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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE CPP #-}</span>
<a name="line-2"></a><span class='hs-comment'>{-# LANGUAGE RankNTypes #-}</span>
<a name="line-3"></a><span class='hs-comment'>{-# LANGUAGE GADTs #-}</span>
<a name="line-4"></a>
<a name="line-5"></a><span class='hs-cpp'>#if __GLASGOW_HASKELL__ >= 702 && __GLASGOW_HASKELL__ < 710</span>
<a name="line-6"></a><span class='hs-comment'>{-# LANGUAGE Trustworthy #-}</span>
<a name="line-7"></a><span class='hs-cpp'>#endif</span>
<a name="line-8"></a><span class='hs-comment'>-------------------------------------------------------------------------------------------</span>
<a name="line-9"></a><span class='hs-comment'>-- |</span>
<a name="line-10"></a><span class='hs-comment'>-- Copyright : 2013 Edward Kmett and Dan Doel</span>
<a name="line-11"></a><span class='hs-comment'>-- License : BSD</span>
<a name="line-12"></a><span class='hs-comment'>--</span>
<a name="line-13"></a><span class='hs-comment'>-- Maintainer : Edward Kmett <ekmett@gmail.com></span>
<a name="line-14"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-15"></a><span class='hs-comment'>-- Portability : rank N types</span>
<a name="line-16"></a><span class='hs-comment'>--</span>
<a name="line-17"></a><span class='hs-comment'>-- Right and Left Kan lifts for functors over Hask, where they exist.</span>
<a name="line-18"></a><span class='hs-comment'>--</span>
<a name="line-19"></a><span class='hs-comment'>-- <<a href="http://ncatlab.org/nlab/show/Kan+lift">http://ncatlab.org/nlab/show/Kan+lift</a>></span>
<a name="line-20"></a><span class='hs-comment'>-------------------------------------------------------------------------------------------</span>
<a name="line-21"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Kan</span><span class='hs-varop'>.</span><span class='hs-conid'>Rift</span>
<a name="line-22"></a> <span class='hs-layout'>(</span>
<a name="line-23"></a> <span class='hs-comment'>-- * Right Kan lifts</span>
<a name="line-24"></a> <span class='hs-conid'>Rift</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span>
<a name="line-25"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>toRift</span><span class='hs-layout'>,</span> <span class='hs-varid'>fromRift</span><span class='hs-layout'>,</span> <span class='hs-varid'>grift</span>
<a name="line-26"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>composeRift</span><span class='hs-layout'>,</span> <span class='hs-varid'>decomposeRift</span>
<a name="line-27"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>adjointToRift</span><span class='hs-layout'>,</span> <span class='hs-varid'>riftToAdjoint</span>
<a name="line-28"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>composedAdjointToRift</span><span class='hs-layout'>,</span> <span class='hs-varid'>riftToComposedAdjoint</span>
<a name="line-29"></a> <span class='hs-layout'>,</span> <span class='hs-varid'>liftRift</span><span class='hs-layout'>,</span> <span class='hs-varid'>lowerRift</span><span class='hs-layout'>,</span> <span class='hs-varid'>rap</span>
<a name="line-30"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-31"></a>
<a name="line-32"></a><span class='hs-cpp'>#if __GLASGOW_HASKELL__ < 710</span>
<a name="line-33"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Applicative</span>
<a name="line-34"></a><span class='hs-cpp'>#endif</span>
<a name="line-35"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Adjunction</span>
<a name="line-36"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Composition</span>
<a name="line-37"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Identity</span>
<a name="line-38"></a>
<a name="line-39"></a><span class='hs-comment'>-- * Right Kan Lift</span>
<a name="line-40"></a>
<a name="line-41"></a><span class='hs-comment'>-- |</span>
<a name="line-42"></a><span class='hs-comment'>--</span>
<a name="line-43"></a><span class='hs-comment'>-- @g . 'Rift' g f => f@</span>
<a name="line-44"></a><span class='hs-comment'>--</span>
<a name="line-45"></a><span class='hs-comment'>-- This could alternately be defined directly from the (co)universal propertly</span>
<a name="line-46"></a><span class='hs-comment'>-- in which case, we'd get 'toRift' = 'UniversalRift', but then the usage would</span>
<a name="line-47"></a><span class='hs-comment'>-- suffer.</span>
<a name="line-48"></a><span class='hs-comment'>--</span>
<a name="line-49"></a><span class='hs-comment'>-- @</span>
<a name="line-50"></a><span class='hs-comment'>-- data 'UniversalRift' g f a = forall z. 'Functor' z =></span>
<a name="line-51"></a><span class='hs-comment'>-- 'UniversalRift' (forall x. g (z x) -> f x) (z a)</span>
<a name="line-52"></a><span class='hs-comment'>-- @</span>
<a name="line-53"></a><span class='hs-comment'>--</span>
<a name="line-54"></a><span class='hs-comment'>-- We can witness the isomorphism between Rift and UniversalRift using:</span>
<a name="line-55"></a><span class='hs-comment'>--</span>
<a name="line-56"></a><span class='hs-comment'>-- @</span>
<a name="line-57"></a><span class='hs-comment'>-- riftIso1 :: Functor g => UniversalRift g f a -> Rift g f a</span>
<a name="line-58"></a><span class='hs-comment'>-- riftIso1 (UniversalRift h z) = Rift $ \\g -> h $ fmap (\\k -> k \<$\> z) g</span>
<a name="line-59"></a><span class='hs-comment'>-- @</span>
<a name="line-60"></a><span class='hs-comment'>--</span>
<a name="line-61"></a><span class='hs-comment'>-- @</span>
<a name="line-62"></a><span class='hs-comment'>-- riftIso2 :: Rift g f a -> UniversalRift g f a</span>
<a name="line-63"></a><span class='hs-comment'>-- riftIso2 (Rift e) = UniversalRift e id</span>
<a name="line-64"></a><span class='hs-comment'>-- @</span>
<a name="line-65"></a><span class='hs-comment'>--</span>
<a name="line-66"></a><span class='hs-comment'>-- @</span>
<a name="line-67"></a><span class='hs-comment'>-- riftIso1 (riftIso2 (Rift h)) =</span>
<a name="line-68"></a><span class='hs-comment'>-- riftIso1 (UniversalRift h id) = -- by definition</span>
<a name="line-69"></a><span class='hs-comment'>-- Rift $ \\g -> h $ fmap (\\k -> k \<$\> id) g -- by definition</span>
<a name="line-70"></a><span class='hs-comment'>-- Rift $ \\g -> h $ fmap id g -- \<$\> = (.) and (.id)</span>
<a name="line-71"></a><span class='hs-comment'>-- Rift $ \\g -> h g -- by functor law</span>
<a name="line-72"></a><span class='hs-comment'>-- Rift h -- eta reduction</span>
<a name="line-73"></a><span class='hs-comment'>-- @</span>
<a name="line-74"></a><span class='hs-comment'>--</span>
<a name="line-75"></a><span class='hs-comment'>-- The other direction is left as an exercise for the reader.</span>
<a name="line-76"></a><span class='hs-comment'>--</span>
<a name="line-77"></a><span class='hs-comment'>-- There are several monads that we can form from @Rift@.</span>
<a name="line-78"></a><span class='hs-comment'>--</span>
<a name="line-79"></a><span class='hs-comment'>-- When @g@ is corepresentable (e.g. is a right adjoint) then there exists @x@ such that @g ~ (->) x@, then it follows that</span>
<a name="line-80"></a><span class='hs-comment'>--</span>
<a name="line-81"></a><span class='hs-comment'>-- @</span>
<a name="line-82"></a><span class='hs-comment'>-- Rift g g a ~</span>
<a name="line-83"></a><span class='hs-comment'>-- forall r. (x -> a -> r) -> x -> r ~</span>
<a name="line-84"></a><span class='hs-comment'>-- forall r. (a -> x -> r) -> x -> r ~</span>
<a name="line-85"></a><span class='hs-comment'>-- forall r. (a -> g r) -> g r ~</span>
<a name="line-86"></a><span class='hs-comment'>-- Codensity g r</span>
<a name="line-87"></a><span class='hs-comment'>-- @</span>
<a name="line-88"></a><span class='hs-comment'>--</span>
<a name="line-89"></a><span class='hs-comment'>-- When @f@ is a left adjoint, so that @f -| g@ then</span>
<a name="line-90"></a><span class='hs-comment'>--</span>
<a name="line-91"></a><span class='hs-comment'>-- @</span>
<a name="line-92"></a><span class='hs-comment'>-- Rift f f a ~</span>
<a name="line-93"></a><span class='hs-comment'>-- forall r. f (a -> r) -> f r ~</span>
<a name="line-94"></a><span class='hs-comment'>-- forall r. (a -> r) -> g (f r) ~</span>
<a name="line-95"></a><span class='hs-comment'>-- forall r. (a -> r) -> Adjoint f g r ~</span>
<a name="line-96"></a><span class='hs-comment'>-- Yoneda (Adjoint f g r)</span>
<a name="line-97"></a><span class='hs-comment'>-- @</span>
<a name="line-98"></a><span class='hs-comment'>--</span>
<a name="line-99"></a><span class='hs-comment'>-- An alternative way to view that is to note that whenever @f@ is a left adjoint then @f -| 'Rift' f 'Identity'@, and since @'Rift' f f@ is isomorphic to @'Rift' f 'Identity' (f a)@, this is the 'Monad' formed by the adjunction.</span>
<a name="line-100"></a><span class='hs-comment'>--</span>
<a name="line-101"></a><span class='hs-comment'>-- @'Rift' 'Identity' m@ can be a 'Monad' for any 'Monad' @m@, as it is isomorphic to @'Yoneda' m@.</span>
<a name="line-102"></a>
<a name="line-103"></a><a name="Rift"></a><span class='hs-keyword'>newtype</span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span>
<a name="line-104"></a> <span class='hs-conid'>Rift</span> <span class='hs-layout'>{</span> <span class='hs-varid'>runRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-keyword'>forall</span> <span class='hs-varid'>r</span><span class='hs-varop'>.</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>h</span> <span class='hs-varid'>r</span> <span class='hs-layout'>}</span>
<a name="line-105"></a>
<a name="line-106"></a><a name="instance%20Functor%20(Rift%20g%20h)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Functor</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Functor</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span> <span class='hs-varid'>h</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-107"></a> <span class='hs-varid'>fmap</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>g</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-varop'>.</span><span class='hs-varid'>f</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-108"></a> <span class='hs-comment'>{-# INLINE fmap #-}</span>
<a name="line-109"></a>
<a name="line-110"></a><a name="instance%20Applicative%20(Rift%20g%20h)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Functor</span> <span class='hs-varid'>g</span><span class='hs-layout'>,</span> <span class='hs-varid'>g</span> <span class='hs-keyglyph'>~</span> <span class='hs-varid'>h</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Applicative</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span> <span class='hs-varid'>h</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-111"></a> <span class='hs-varid'>pure</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-varop'>$</span><span class='hs-varid'>a</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-112"></a> <span class='hs-comment'>{-# INLINE pure #-}</span>
<a name="line-113"></a> <span class='hs-conid'>Rift</span> <span class='hs-varid'>mf</span> <span class='hs-varop'><*></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>ma</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>ma</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mf</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-varop'>.</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-114"></a> <span class='hs-comment'>{-# INLINE (<*>) #-}</span>
<a name="line-115"></a>
<a name="line-116"></a><a name="liftRift"></a><span class='hs-comment'>-- | The natural isomorphism between @f@ and @Rift f f@.</span>
<a name="line-117"></a><span class='hs-comment'>-- @</span>
<a name="line-118"></a><span class='hs-comment'>-- 'lowerRift' '.' 'liftRift' ≡ 'id'</span>
<a name="line-119"></a><span class='hs-comment'>-- 'liftRift' '.' 'lowerRift' ≡ 'id'</span>
<a name="line-120"></a><span class='hs-comment'>-- @</span>
<a name="line-121"></a><span class='hs-comment'>--</span>
<a name="line-122"></a><span class='hs-comment'>-- @</span>
<a name="line-123"></a><span class='hs-comment'>-- 'lowerRift' ('liftRift' x) -- definition</span>
<a name="line-124"></a><span class='hs-comment'>-- 'lowerRift' ('Rift' ('<*>' x)) -- definition</span>
<a name="line-125"></a><span class='hs-comment'>-- ('<*>' x) ('pure' 'id') -- beta reduction</span>
<a name="line-126"></a><span class='hs-comment'>-- 'pure' 'id' '<*>' x -- Applicative identity law</span>
<a name="line-127"></a><span class='hs-comment'>-- x</span>
<a name="line-128"></a><span class='hs-comment'>-- @</span>
<a name="line-129"></a><span class='hs-definition'>liftRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Applicative</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>f</span> <span class='hs-varid'>a</span>
<a name="line-130"></a><span class='hs-definition'>liftRift</span> <span class='hs-varid'>fa</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varop'><*></span> <span class='hs-varid'>fa</span><span class='hs-layout'>)</span>
<a name="line-131"></a><span class='hs-comment'>{-# INLINE liftRift #-}</span>
<a name="line-132"></a>
<a name="line-133"></a><a name="lowerRift"></a><span class='hs-comment'>-- | Lower 'Rift' by applying 'pure' 'id' to the continuation.</span>
<a name="line-134"></a><span class='hs-comment'>--</span>
<a name="line-135"></a><span class='hs-comment'>-- See 'liftRift'.</span>
<a name="line-136"></a><span class='hs-definition'>lowerRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Applicative</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>g</span> <span class='hs-varid'>a</span>
<a name="line-137"></a><span class='hs-definition'>lowerRift</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>pure</span> <span class='hs-varid'>id</span><span class='hs-layout'>)</span>
<a name="line-138"></a><span class='hs-comment'>{-# INLINE lowerRift #-}</span>
<a name="line-139"></a>
<a name="line-140"></a><a name="rap"></a><span class='hs-comment'>-- | Indexed applicative composition of right Kan lifts.</span>
<a name="line-141"></a><span class='hs-definition'>rap</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Functor</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>h</span> <span class='hs-varid'>b</span>
<a name="line-142"></a><span class='hs-definition'>rap</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>mf</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>ma</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>ma</span> <span class='hs-varop'>.</span> <span class='hs-varid'>mf</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-varop'>.</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-143"></a><span class='hs-comment'>{-# INLINE rap #-}</span>
<a name="line-144"></a>
<a name="line-145"></a><a name="grift"></a><span class='hs-definition'>grift</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Adjunction</span> <span class='hs-varid'>f</span> <span class='hs-varid'>u</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span>
<a name="line-146"></a><span class='hs-definition'>grift</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>rightAdjunct</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>r</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>leftAdjunct</span> <span class='hs-layout'>(</span><span class='hs-varid'>runRift</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-varid'>id</span><span class='hs-layout'>)</span>
<a name="line-147"></a><span class='hs-comment'>{-# INLINE grift #-}</span>
<a name="line-148"></a>
<a name="line-149"></a><a name="toRift"></a><span class='hs-comment'>-- | The universal property of 'Rift'</span>
<a name="line-150"></a><span class='hs-definition'>toRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Functor</span> <span class='hs-varid'>g</span><span class='hs-layout'>,</span> <span class='hs-conid'>Functor</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>x</span><span class='hs-varop'>.</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-varid'>k</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>h</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span>
<a name="line-151"></a><span class='hs-definition'>toRift</span> <span class='hs-varid'>h</span> <span class='hs-varid'>z</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>g</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>h</span> <span class='hs-varop'>$</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-varop'><$></span> <span class='hs-varid'>z</span><span class='hs-layout'>)</span> <span class='hs-varid'>g</span>
<a name="line-152"></a><span class='hs-comment'>{-# INLINE toRift #-}</span>
<a name="line-153"></a>
<a name="line-154"></a><a name="fromRift"></a><span class='hs-comment'>-- |</span>
<a name="line-155"></a><span class='hs-comment'>-- When @f -| u@, then @f -| Rift f Identity@ and</span>
<a name="line-156"></a><span class='hs-comment'>--</span>
<a name="line-157"></a><span class='hs-comment'>-- @</span>
<a name="line-158"></a><span class='hs-comment'>-- 'toRift' . 'fromRift' ≡ 'id'</span>
<a name="line-159"></a><span class='hs-comment'>-- 'fromRift' . 'toRift' ≡ 'id'</span>
<a name="line-160"></a><span class='hs-comment'>-- @</span>
<a name="line-161"></a><span class='hs-definition'>fromRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Adjunction</span> <span class='hs-varid'>f</span> <span class='hs-varid'>u</span> <span class='hs-keyglyph'>=></span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span><span class='hs-varop'>.</span> <span class='hs-varid'>k</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>k</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>h</span> <span class='hs-varid'>b</span>
<a name="line-162"></a><span class='hs-definition'>fromRift</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>grift</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>f</span>
<a name="line-163"></a><span class='hs-comment'>{-# INLINE fromRift #-}</span>
<a name="line-164"></a>
<a name="line-165"></a><a name="adjointToRift"></a><span class='hs-comment'>-- | @Rift f Identity a@ is isomorphic to the right adjoint to @f@ if one exists.</span>
<a name="line-166"></a><span class='hs-comment'>--</span>
<a name="line-167"></a><span class='hs-comment'>-- @</span>
<a name="line-168"></a><span class='hs-comment'>-- 'adjointToRift' . 'riftToAdjoint' ≡ 'id'</span>
<a name="line-169"></a><span class='hs-comment'>-- 'riftToAdjoint' . 'adjointToRift' ≡ 'id'</span>
<a name="line-170"></a><span class='hs-comment'>-- @</span>
<a name="line-171"></a><span class='hs-definition'>adjointToRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Adjunction</span> <span class='hs-varid'>f</span> <span class='hs-varid'>u</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>u</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-conid'>Identity</span> <span class='hs-varid'>a</span>
<a name="line-172"></a><span class='hs-definition'>adjointToRift</span> <span class='hs-varid'>ua</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-conid'>Identity</span> <span class='hs-varop'>.</span> <span class='hs-varid'>rightAdjunct</span> <span class='hs-layout'>(</span><span class='hs-varop'><$></span> <span class='hs-varid'>ua</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-173"></a><span class='hs-comment'>{-# INLINE adjointToRift #-}</span>
<a name="line-174"></a>
<a name="line-175"></a><a name="riftToAdjoint"></a><span class='hs-comment'>-- | @Rift f Identity a@ is isomorphic to the right adjoint to @f@ if one exists.</span>
<a name="line-176"></a><span class='hs-definition'>riftToAdjoint</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Adjunction</span> <span class='hs-varid'>f</span> <span class='hs-varid'>u</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-conid'>Identity</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>u</span> <span class='hs-varid'>a</span>
<a name="line-177"></a><span class='hs-definition'>riftToAdjoint</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>leftAdjunct</span> <span class='hs-layout'>(</span><span class='hs-varid'>runIdentity</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varid'>id</span>
<a name="line-178"></a><span class='hs-comment'>{-# INLINE riftToAdjoint #-}</span>
<a name="line-179"></a>
<a name="line-180"></a><a name="composeRift"></a><span class='hs-comment'>-- |</span>
<a name="line-181"></a><span class='hs-comment'>--</span>
<a name="line-182"></a><span class='hs-comment'>-- @</span>
<a name="line-183"></a><span class='hs-comment'>-- 'composeRift' . 'decomposeRift' ≡ 'id'</span>
<a name="line-184"></a><span class='hs-comment'>-- 'decomposeRift' . 'composeRift' ≡ 'id'</span>
<a name="line-185"></a><span class='hs-comment'>-- @</span>
<a name="line-186"></a><span class='hs-definition'>composeRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Composition</span> <span class='hs-varid'>compose</span><span class='hs-layout'>,</span> <span class='hs-conid'>Adjunction</span> <span class='hs-varid'>g</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span> <span class='hs-varid'>h</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>compose</span> <span class='hs-varid'>g</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span>
<a name="line-187"></a><span class='hs-definition'>composeRift</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>grift</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>f</span> <span class='hs-varop'>.</span> <span class='hs-varid'>decompose</span><span class='hs-layout'>)</span>
<a name="line-188"></a><span class='hs-comment'>{-# INLINE composeRift #-}</span>
<a name="line-189"></a>
<a name="line-190"></a><a name="decomposeRift"></a><span class='hs-definition'>decomposeRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Composition</span> <span class='hs-varid'>compose</span><span class='hs-layout'>,</span> <span class='hs-conid'>Functor</span> <span class='hs-varid'>f</span><span class='hs-layout'>,</span> <span class='hs-conid'>Functor</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>compose</span> <span class='hs-varid'>g</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>g</span> <span class='hs-varid'>h</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span>
<a name="line-191"></a><span class='hs-definition'>decomposeRift</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-varop'>$</span> <span class='hs-keyglyph'>\</span><span class='hs-varid'>far</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varop'>.</span> <span class='hs-varid'>compose</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>rs</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-varid'>rs</span><span class='hs-varop'>.</span><span class='hs-layout'>)</span> <span class='hs-varid'>far</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-192"></a><span class='hs-comment'>{-# INLINE decomposeRift #-}</span>
<a name="line-193"></a>
<a name="line-194"></a>
<a name="line-195"></a><span class='hs-comment'>-- | @Rift f h a@ is isomorphic to the post-composition of the right adjoint of @f@ onto @h@ if such a right adjoint exists.</span>
<a name="line-196"></a><span class='hs-comment'>--</span>
<a name="line-197"></a><span class='hs-comment'>-- @</span>
<a name="line-198"></a><span class='hs-comment'>-- 'riftToComposedAdjoint' . 'composedAdjointToRift' ≡ 'id'</span>
<a name="line-199"></a><span class='hs-comment'>-- 'composedAdjointToRift' . 'riftToComposedAdjoint' ≡ 'id'</span>
<a name="line-200"></a><span class='hs-comment'>-- @</span>
<a name="line-201"></a>
<a name="line-202"></a><a name="riftToComposedAdjoint"></a><span class='hs-definition'>riftToComposedAdjoint</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Adjunction</span> <span class='hs-varid'>f</span> <span class='hs-varid'>u</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>u</span> <span class='hs-layout'>(</span><span class='hs-varid'>h</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-203"></a><span class='hs-definition'>riftToComposedAdjoint</span> <span class='hs-layout'>(</span><span class='hs-conid'>Rift</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>leftAdjunct</span> <span class='hs-varid'>m</span> <span class='hs-varid'>id</span>
<a name="line-204"></a><span class='hs-comment'>{-# INLINE riftToComposedAdjoint #-}</span>
<a name="line-205"></a>
<a name="line-206"></a><a name="composedAdjointToRift"></a><span class='hs-comment'>-- | @Rift f h a@ is isomorphic to the post-composition of the right adjoint of @f@ onto @h@ if such a right adjoint exists.</span>
<a name="line-207"></a><span class='hs-definition'>composedAdjointToRift</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Functor</span> <span class='hs-varid'>h</span><span class='hs-layout'>,</span> <span class='hs-conid'>Adjunction</span> <span class='hs-varid'>f</span> <span class='hs-varid'>u</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>u</span> <span class='hs-layout'>(</span><span class='hs-varid'>h</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Rift</span> <span class='hs-varid'>f</span> <span class='hs-varid'>h</span> <span class='hs-varid'>a</span>
<a name="line-208"></a><span class='hs-definition'>composedAdjointToRift</span> <span class='hs-varid'>uha</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Rift</span> <span class='hs-varop'>$</span> <span class='hs-varid'>rightAdjunct</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>b</span> <span class='hs-varop'><$></span> <span class='hs-varid'>uha</span><span class='hs-layout'>)</span>
<a name="line-209"></a><span class='hs-comment'>{-# INLINE composedAdjointToRift #-}</span>
<a name="line-210"></a>
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