diff --git a/COPYING b/COPYING
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+++ b/COPYING
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+ GNU GENERAL PUBLIC LICENSE
+ Version 3, 29 June 2007
+
+ Copyright (C) 2007 Free Software Foundation, Inc.
+ Everyone is permitted to copy and distribute verbatim copies
+ of this license document, but changing it is not allowed.
+
+ Preamble
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+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+ END OF TERMS AND CONDITIONS
+
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+
+ Copyright (C)
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see .
+
+Also add information on how to contact you by electronic and paper mail.
+
+ If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+ Copyright (C)
+ This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+ You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+.
+
+ The GNU General Public License does not permit incorporating your program
+into proprietary programs. If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library. If this is what you want to do, use the GNU Lesser General
+Public License instead of this License. But first, please read
+.
diff --git a/Control/Applicative/Constrained.hs b/Control/Applicative/Constrained.hs
new file mode 100644
--- /dev/null
+++ b/Control/Applicative/Constrained.hs
@@ -0,0 +1,106 @@
+-- |
+-- Module : Control.Applicative.Constrained
+-- Copyright : (c) 2013 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+
+
+module Control.Applicative.Constrained (
+ module Control.Functor.Constrained
+ -- * Monoidal / applicative functors
+ , Monoidal(..)
+ , Applicative(..)
+ -- * Helper for constrained categories
+ , constrainedFZipWith
+ -- * Utility functions
+ , constPure, fzip, (<**>), liftA, liftA2, liftA3
+ ) where
+
+
+import Control.Functor.Constrained
+import Control.Arrow.Constrained
+
+import Prelude hiding (id, const, (.), ($), Functor(..), curry, uncurry)
+import qualified Control.Category.Hask as Hask
+
+
+class (Functor f r t, Cartesian r, Cartesian t) => Monoidal f r t where
+ pureUnit :: UnitObject t `t` f (UnitObject r)
+ fzipWith :: (ObjectPair r a b, Object r c, ObjectPair t (f a) (f b), Object t (f c))
+ => r (a, b) c -> t (f a, f b) (f c)
+
+constPure :: (WellPointed r, Monoidal f r t, ObjectPoint r a, Object t (f a) )
+ => a -> t (UnitObject t) (f a)
+constPure a = fmap (const a) . pureUnit
+
+fzip :: (Monoidal f r t, ObjectPair r a b, ObjectPair t (f a) (f b), Object t (f (a,b)))
+ => t (f a, f b) (f (a,b))
+fzip = fzipWith id
+
+class (Monoidal f r t, Curry r, Curry t) => Applicative f r t where
+ -- ^ Note that this tends to make little sense for non-endofunctors.
+ -- Consider using 'constPure' instead.
+ pure :: (Object r a, Object t (f a)) => a `t` f a
+
+ (<*>) :: ( ObjectMorphism r a b
+ , ObjectMorphism t (f a) (f b), Object t (t (f a) (f b))
+ , ObjectPair r (r a b) a, ObjectPair t (f (r a b)) (f a)
+ , Object r a, Object r b )
+ => f (r a b) `t` t (f a) (f b)
+ (<*>) = curry (fzipWith $ uncurry id)
+
+infixl 4 <*>
+
+(<**>) :: ( Applicative f r (->), ObjectMorphism r a b, ObjectPair r (r a b) a )
+ => f a -> f (r a b) -> f b
+(<**>) = flip $ curry (fzipWith $ uncurry id)
+
+liftA :: (Applicative f r t, Object r a, Object r b, Object t (f a), Object t (f b))
+ => a `r` b -> f a `t` f b
+liftA = fmap
+
+liftA2 :: ( Applicative f r t, Object r c, ObjectMorphism r b c
+ , Object t (f c), ObjectMorphism t (f b) (f c)
+ , ObjectPair r a b, ObjectPair t (f a) (f b) )
+ => a `r` (b `r` c) -> f a `t` (f b `t` f c)
+liftA2 = curry . fzipWith . uncurry
+
+liftA3 :: ( Applicative f r t
+ , Object r c, Object r d
+ , ObjectMorphism r c d, ObjectMorphism r b (c`r`d), Object r (r c d)
+ , ObjectPair r a b, ObjectPair r (r c d) c
+ , Object t (f c), Object t (f d), Object t(f a,f b)
+ , ObjectMorphism t (f c)(f d),ObjectMorphism t (f b)(t(f c)(f d)),Object t(t(f c)(f d))
+ , ObjectPair t (f a) (f b), ObjectPair t (t (f c) (f d)) (f c)
+ , ObjectPair t (f (r c d)) (f c)
+ ) => a `r` (b `r` (c `r` d)) -> f a `t` (f b `t` (f c `t` f d))
+liftA3 f = curry $ (<*>) . (fzipWith $ uncurry f)
+
+
+constrainedFZipWith :: ( Category r, Category t, o a, o b, o (a,b), o c
+ , o (f a, f b), o (f c) )
+ => ( r (a, b) c -> t (f a, f b) (f c) )
+ -> ConstrainedCategory r o (a, b) c -> ConstrainedCategory t o (f a, f b) (f c)
+constrainedFZipWith zf = constrained . zf . unconstrained
+
+
+instance (Hask.Applicative f) => Monoidal f (->) (->) where
+ pureUnit = Hask.pure
+ fzipWith f (p, q) = curry f Hask.<$> p Hask.<*> q
+
+instance (Hask.Applicative f) => Applicative f (->) (->) where
+ pure = Hask.pure
+ (<*>) = (Hask.<*>)
+
+
+
+
+
diff --git a/Control/Arrow/Constrained.hs b/Control/Arrow/Constrained.hs
new file mode 100644
--- /dev/null
+++ b/Control/Arrow/Constrained.hs
@@ -0,0 +1,424 @@
+-- |
+-- Module : Control.Arrow.Constrained
+-- Copyright : (c) 2013 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+-- Haskell's 'Arr.Arrow's, going back to [Hughes 2000], combine multiple ideas from
+-- category theory:
+--
+-- * They expand upon cartesian categories, by offering ways to combine arrows between
+-- simple objects to composite ones working on tuples (i.e. /products/) thereof.
+--
+-- * They constitute a \"profunctor\" interface, allowing to \"@fmap@\" both covariantly
+-- over the second parameter, as well as contravariantly over the first. As in case
+-- of "Control.Functor.Constrained", we wish the underlying category to fmap from
+-- not to be limited to /Hask/, so 'Arrow' also has an extra parameter.
+--
+-- To facilitate these somewhat divergent needs, 'Arrow' is split up in three classes.
+-- These do not even form an ordinary hierarchy, to allow categories to implement
+-- only one /or/ the other aspect.
+--
+-- That's not the only significant difference of this module, compared to "Control.Arrow":
+--
+-- * Kleisli arrows are not defined here, but in "Control.Monad.Constrained".
+-- Monads are really a much more specific concept than category arrows.
+--
+-- * Some extra utilities are included that don't apparently have much to
+-- do with 'Arrow' at all, but require the expanded cartesian-category tools
+-- and are therefore not in "Control.Category.Constrained".
+
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE RankNTypes #-}
+{-# LANGUAGE AllowAmbiguousTypes #-}
+
+
+module Control.Arrow.Constrained (
+ -- * The Arrow type classes
+ Arrow, Morphism(..), PreArrow(..), WellPointed(..),ObjectPoint, EnhancedCat(..)
+ -- * Dual / "choice" arrows
+ , ArrowChoice, MorphChoice(..), PreArrChoice(..)
+ -- * Distributive law between sum- and product objects
+ , SPDistribute(..)
+ -- * Function-like categories
+ , Function, ($)
+ -- * Alternative composition notation
+ , (>>>), (<<<)
+ -- * Proxies for cartesian categories
+ , CartesianProxy(..)
+ , genericProxyCombine, genericUnit, genericAlg1to2, genericAlg2to1, genericAlg2to2
+ , PointProxy(..), genericPoint
+ -- * Misc utility
+ -- ** Conditionals
+ , choose, ifThenElse
+ ) where
+
+import Prelude hiding (id, const, fst, snd, (.), ($), Functor(..), Monad(..), (=<<))
+import Control.Category.Constrained
+import qualified Control.Category.Hask as Hask
+
+import GHC.Exts (Constraint)
+import Data.Tagged
+import Data.Void
+
+import qualified Control.Arrow as Arr
+
+infixr 1 >>>, <<<
+infixr 3 &&&, ***
+
+(>>>) :: (Category k, Object k a, Object k b, Object k c)
+ => k a b -> k b c -> k a c
+(>>>) = flip (.)
+(<<<) :: (Category k, Object k a, Object k b, Object k c)
+ => k b c -> k a b -> k a c
+(<<<) = (.)
+
+class (Cartesian a) => Morphism a where
+ first :: ( ObjectPair a b d, ObjectPair a c d )
+ => a b c -> a (b, d) (c, d)
+ first = (***id)
+ second :: ( ObjectPair a d b, ObjectPair a d c )
+ => a b c -> a (d, b) (d, c)
+ second = (id***)
+ (***) :: ( ObjectPair a b b', ObjectPair a c c' )
+ => a b c -> a b' c' -> a (b,b') (c,c')
+
+-- | Dual to 'Morphism', dealing with sums instead of products.
+class (CoCartesian a) => MorphChoice a where
+ left :: ( ObjectSum a b d, ObjectSum a c d )
+ => a b c -> a (b+d) (c+d)
+ left = (+++id)
+ right :: ( ObjectSum a d b, ObjectSum a d c )
+ => a b c -> a (d+b) (d+c)
+ right = (id+++)
+ (+++) :: ( ObjectSum a b b', ObjectSum a c c' )
+ => a b c -> a b' c' -> a (b+b') (c+c')
+
+
+
+-- | Unlike 'first', 'second', '***' and 'arr', '&&&' has an intrinsic notion
+-- of \"direction\": it is basically equivalent to precomposing the result
+-- of '***' with a @b -> (b,b)@, but that is in general only available
+-- for arrows that generalise ordinary functions, in their native direction.
+-- (@(b,b) ->b@ is specific to semigroups.) It is for this reason the only constituent
+-- class of 'Arrow' that actually has \"arrow\" in its name.
+--
+-- In terms of category theory, this \"direction\" reflects the distinction
+-- between /initial-/ and /terminal objects/. The latter are more interesting,
+-- basically what 'UnitObject' is useful for. It gives rise to the tuple
+-- selector morphisms as well.
+class (Morphism a) => PreArrow a where
+ (&&&) :: ( Object a b, ObjectPair a c c' )
+ => a b c -> a b c' -> a b (c,c')
+ terminal :: ( Object a b ) => a b (UnitObject a)
+ fst :: (ObjectPair a x y) => a (x,y) x
+ snd :: (ObjectPair a x y) => a (x,y) y
+
+infixr 2 |||
+-- | Dual to 'PreArrow', this class deals with the vacuous initial (zero) objects,
+-- but also more usefully with choices / sums.
+-- This represents the most part of 'Hask.ArrowChoice'.
+class (MorphChoice k) => PreArrChoice k where
+ (|||) :: ( ObjectSum k b b', Object k c )
+ => k b c -> k b' c -> k (b+b') c
+ -- | This is basically 'absurd'.
+ initial :: ( Object k b ) => k (ZeroObject k) b
+ -- | Perhaps @lft@ and @rgt@ would be more consequent names, but likely more confusing as well.
+ coFst :: (ObjectSum k a b) => k a (a+b)
+ coSnd :: (ObjectSum k a b) => k b (a+b)
+
+
+-- | Like in arithmetics, the distributive law
+-- @a ⋅ (b + c) ≈ (a ⋅ b) + (a ⋅ c)@
+-- holds for Haskell types – in the usual isomorphism sense. But like many such
+-- isomorphisms that are trivial to inline in /Hask/, this is not necessarily the case
+-- for general categories.
+class (PreArrow k, PreArrChoice k) => SPDistribute k where
+ distribute :: ( ObjectSum k (a,b) (a,c), ObjectPair k a (b+c)
+ , ObjectSum k b c, PairObjects k a b, PairObjects k a c )
+ => k (a, b+c) ((a,b)+(a,c))
+ unDistribute :: ( ObjectSum k (a,b) (a,c), ObjectPair k a (b+c)
+ , ObjectSum k b c, PairObjects k a b, PairObjects k a c )
+ => k ((a,b)+(a,c)) (a, b+c)
+ boolAsSwitch :: ( ObjectSum k a a, ObjectPair k Bool a ) => k (Bool,a) (a+a)
+ boolFromSwitch :: ( ObjectSum k a a, ObjectPair k Bool a ) => k (a+a) (Bool,a)
+-- boolFromSwitch = (boolFromSum <<< terminal +++ terminal) &&& (id ||| id)
+
+instance ( SPDistribute k
+ , ObjectSum k (a,b) (a,c), ObjectPair k a (b+c)
+ , ObjectSum k b c, PairObjects k a b, PairObjects k a c
+ ) => Isomorphic k (a, b+c) ((a,b)+(a,c)) where
+ iso = distribute
+instance ( SPDistribute k
+ , ObjectSum k (a,b) (a,c), ObjectPair k a (b+c)
+ , ObjectSum k b c, PairObjects k a b, PairObjects k a c
+ ) => Isomorphic k ((a,b)+(a,c)) (a, b+c) where
+ iso = unDistribute
+instance ( SPDistribute k
+ , ObjectSum k a a, ObjectPair k Bool a
+ ) => Isomorphic k (Bool, a) (a+a) where
+ iso = boolAsSwitch
+instance ( SPDistribute k
+ , ObjectSum k a a, ObjectPair k Bool a
+ ) => Isomorphic k (a+a) (Bool, a) where
+ iso = boolFromSwitch
+
+
+
+-- | 'WellPointed' expresses the relation between your category's objects
+-- and the values of the Haskell data types (which is, after all, what objects are
+-- in this library). Specifically, this class allows you to \"point\" on
+-- specific objects, thus making out a value of that type as a point of the object.
+--
+-- Perhaps easier than thinking about what that's supposed to mean is noting
+-- this class contains 'const'. Thus 'WellPointed' is /almost/ the
+-- traditional 'Hask.Arrow': it lets you express all the natural transformations
+-- and inject constant values, only you can't just promote arbitrary functions
+-- to arrows of the category.
+--
+-- Unlike with 'Morphism' and 'PreArrow', a literal dual of 'WellPointed' does
+-- not seem useful.
+class (PreArrow a, ObjectPoint a (UnitObject a)) => WellPointed a where
+ {-# MINIMAL unit, (globalElement | const) #-}
+ type PointObject a x :: Constraint
+ type PointObject a x = ()
+ globalElement :: (ObjectPoint a x) => x -> a (UnitObject a) x
+ globalElement = const
+ unit :: CatTagged a (UnitObject a)
+ const :: (Object a b, ObjectPoint a x)
+ => x -> a b x
+ const x = globalElement x . terminal
+
+type ObjectPoint k a = (Object k a, PointObject k a)
+
+-- -- | 'WellPointed' does not have a useful literal dual.
+-- class (PreArrChoice a, ObjectPoint a (ZeroObject a)) => WellChosen a where
+-- type ChoiceObject a x :: Constraint
+-- type ChoiceObject a x = ()
+-- localElement :: (ObjectChoice a x) => a x (ZeroObject a) -> (x -> b
+-- zero :: CatTagged a (ZeroObject a)
+-- doomed :: (Object a b, ObjectChoice a x)
+-- => x -> a x b
+-- doomed x = globalElement x . initial
+--
+-- type ObjectChoice k a = (Object k a, ChoiceObject k x)
+--
+value :: forall f x . (WellPointed f, Function f, Object f x)
+ => f (UnitObject f) x -> x
+value f = f $ untag(unit :: Tagged (f (UnitObject f) (UnitObject f)) (UnitObject f))
+
+
+class (Category k) => EnhancedCat a k where
+ arr :: (Object k b, Object k c, Object a b, Object a c)
+ => k b c -> a b c
+instance (Category k) => EnhancedCat k k where
+ arr = id
+
+
+-- | Many categories have as morphisms essentially /functions with extra properties/:
+-- group homomorphisms, linear maps, continuous functions...
+--
+-- It makes sense to generalise the notion of function application to these
+-- morphisms; we can't do that for the simple juxtaposition writing @f x@,
+-- but it is possible for the function-application operator @$@.
+--
+-- This is particularly useful for 'ConstrainedCategory' versions of Hask,
+-- where after all the morphisms are /nothing but functions/.
+type Function f = EnhancedCat (->) f
+
+infixr 0 $
+($) :: (Function f, Object f a, Object f b) => f a b -> a -> b
+f $ x = arr f x
+
+instance (Function f) => EnhancedCat (->) (ConstrainedCategory f o) where
+ arr (ConstrainedMorphism q) = arr q
+
+
+
+type Arrow a k = (WellPointed a, EnhancedCat a k)
+type ArrowChoice a k = (WellPointed a, PreArrChoice a, EnhancedCat a k)
+
+instance Morphism (->) where
+ first = Arr.first
+ second = Arr.second
+ (***) = (Arr.***)
+instance MorphChoice (->) where
+ left = Arr.left
+ right = Arr.right
+ (+++) = (Arr.+++)
+instance PreArrow (->) where
+ (&&&) = (Arr.&&&)
+ fst (a,_) = a
+ snd (_,b) = b
+ terminal = const ()
+instance PreArrChoice (->) where
+ (|||) = (Arr.|||)
+ coFst a = Left a
+ coSnd b = Right b
+ initial = absurd
+instance SPDistribute (->) where
+ distribute (a, Left b) = Left (a,b)
+ distribute (a, Right c) = Right (a,c)
+ unDistribute (Left (a,b)) = (a, Left b)
+ unDistribute (Right (a,c)) = (a, Right c)
+ boolAsSwitch (False, a) = Left a
+ boolAsSwitch (True, a) = Right a
+ boolFromSwitch (Left a) = (False, a)
+ boolFromSwitch (Right a) = (True, a)
+instance WellPointed (->) where
+ globalElement = Hask.const
+ unit = Hask.pure ()
+ const = Hask.const
+
+constrainedArr :: (Category k, Category a, o b, o c )
+ => ( k b c -> a b c )
+ -> k b c -> ConstrainedCategory a o b c
+constrainedArr ar = constrained . ar
+
+constrainedFirst :: ( Category a, Cartesian a, ObjectPair a b d, ObjectPair a c d )
+ => ( a b c -> a (b, d) (c, d) )
+ -> ConstrainedCategory a o b c -> ConstrainedCategory a o (b, d) (c, d)
+constrainedFirst fs = ConstrainedMorphism . fs . unconstrained
+
+constrainedSecond :: ( Category a, Cartesian a, ObjectPair a d b, ObjectPair a d c )
+ => ( a b c -> a (d, b) (d, c) )
+ -> ConstrainedCategory a o b c -> ConstrainedCategory a o (d, b) (d, c)
+constrainedSecond sn = ConstrainedMorphism . sn . unconstrained
+
+
+instance (Morphism a, o (UnitObject a)) => Morphism (ConstrainedCategory a o) where
+ first = constrainedFirst first
+ second = constrainedSecond second
+ ConstrainedMorphism a *** ConstrainedMorphism b = ConstrainedMorphism $ a *** b
+
+instance (PreArrow a, o (UnitObject a)) => PreArrow (ConstrainedCategory a o) where
+ ConstrainedMorphism a &&& ConstrainedMorphism b = ConstrainedMorphism $ a &&& b
+ terminal = ConstrainedMorphism terminal
+ fst = ConstrainedMorphism fst
+ snd = ConstrainedMorphism snd
+
+instance (WellPointed a, o (UnitObject a)) => WellPointed (ConstrainedCategory a o) where
+ type PointObject (ConstrainedCategory a o) x = PointObject a x
+ globalElement x = ConstrainedMorphism $ globalElement x
+ unit = cstrCatUnit
+ const x = ConstrainedMorphism $ const x
+
+cstrCatUnit :: forall a o . (WellPointed a, o (UnitObject a))
+ => CatTagged (ConstrainedCategory a o) (UnitObject a)
+cstrCatUnit = retag (unit :: CatTagged a (UnitObject a))
+
+instance (Arrow a k, o (UnitObject a)) => EnhancedCat (ConstrainedCategory a o) k where
+ arr = constrainedArr arr
+
+
+constrainedLeft :: ( CoCartesian k, ObjectSum k b d, ObjectSum k c d )
+ => ( k b c -> k (b+d) (c+d) )
+ -> ConstrainedCategory k o b c -> ConstrainedCategory k o (b+d) (c+d)
+constrainedLeft fs = ConstrainedMorphism . fs . unconstrained
+
+constrainedRight :: ( CoCartesian k, ObjectSum k b c, ObjectSum k b d )
+ => ( k c d -> k (b+c) (b+d) )
+ -> ConstrainedCategory k o c d -> ConstrainedCategory k o (b+c) (b+d)
+constrainedRight fs = ConstrainedMorphism . fs . unconstrained
+
+instance (MorphChoice k, o (ZeroObject k)) => MorphChoice (ConstrainedCategory k o) where
+ left = constrainedLeft left
+ right = constrainedRight right
+ ConstrainedMorphism a +++ ConstrainedMorphism b = ConstrainedMorphism $ a +++ b
+
+instance (PreArrChoice k, o (ZeroObject k)) => PreArrChoice (ConstrainedCategory k o) where
+ ConstrainedMorphism a ||| ConstrainedMorphism b = ConstrainedMorphism $ a ||| b
+ initial = ConstrainedMorphism initial
+ coFst = ConstrainedMorphism coFst
+ coSnd = ConstrainedMorphism coSnd
+
+instance (SPDistribute k, o (ZeroObject k), o (UnitObject k))
+ => SPDistribute (ConstrainedCategory k o) where
+ distribute = ConstrainedMorphism distribute
+ unDistribute = ConstrainedMorphism unDistribute
+ boolAsSwitch = ConstrainedMorphism boolAsSwitch
+ boolFromSwitch = ConstrainedMorphism boolFromSwitch
+
+
+
+-- | Basically 'ifThenElse' with inverted argument order, and
+-- \"morphismised\" arguments.
+choose :: (Arrow f (->), Function f, Object f Bool, Object f a)
+ => f (UnitObject f) a -- ^ \"'False'\" value
+ -> f (UnitObject f) a -- ^ \"'True'\" value
+ -> f Bool a
+choose fv tv = arr $ \c -> if c then value tv else value fv
+
+ifThenElse :: ( EnhancedCat f (->), Function f
+ , Object f Bool, Object f a, Object f (f a a), Object f (f a (f a a))
+ ) => Bool `f` (a `f` (a `f` a))
+ifThenElse = arr $ \c -> arr $ \tv -> arr $ \fv -> if c then tv else fv
+
+
+
+
+genericProxyCombine :: ( HasProxy k, PreArrow k
+ , Object k a, ObjectPair k b c, Object k d )
+ => k (b,c) d -> GenericProxy k a b -> GenericProxy k a c -> GenericProxy k a d
+genericProxyCombine m (GenericProxy v) (GenericProxy w)
+ = GenericProxy $ m . (v &&& w)
+
+genericUnit :: ( PreArrow k, HasProxy k, Object k a )
+ => GenericProxy k a (UnitObject k)
+genericUnit = GenericProxy terminal
+
+
+class (Morphism k, HasProxy k) => CartesianProxy k where
+ alg1to2 :: ( Object k a, ObjectPair k b c
+ ) => (forall q . Object k q
+ => ProxyVal k q a -> (ProxyVal k q b, ProxyVal k q c) )
+ -> k a (b,c)
+ alg2to1 :: ( ObjectPair k a b, Object k c
+ ) => (forall q . Object k q
+ => ProxyVal k q a -> ProxyVal k q b -> ProxyVal k q c )
+ -> k (a,b) c
+ alg2to2 :: ( ObjectPair k a b, ObjectPair k c d
+ ) => (forall q . Object k q
+ => ProxyVal k q a -> ProxyVal k q b -> (ProxyVal k q c, ProxyVal k q d) )
+ -> k (a,b) (c,d)
+
+genericAlg1to2 :: ( PreArrow k, u ~ UnitObject k
+ , Object k a, ObjectPair k b c
+ ) => ( forall q . Object k q
+ => GenericProxy k q a -> (GenericProxy k q b, GenericProxy k q c) )
+ -> k a (b,c)
+genericAlg1to2 f = runGenericProxy b &&& runGenericProxy c
+ where (b,c) = f $ GenericProxy id
+genericAlg2to1 :: ( PreArrow k, u ~ UnitObject k
+ , ObjectPair k a u, ObjectPair k a b, ObjectPair k b u, ObjectPair k b a
+ ) => ( forall q . Object k q
+ => GenericProxy k q a -> GenericProxy k q b -> GenericProxy k q c )
+ -> k (a,b) c
+genericAlg2to1 f = runGenericProxy $ f (GenericProxy fst) (GenericProxy snd)
+genericAlg2to2 :: ( PreArrow k, u ~ UnitObject k
+ , ObjectPair k a u, ObjectPair k a b, ObjectPair k c d
+ , ObjectPair k b u, ObjectPair k b a
+ ) => ( forall q . Object k q
+ => GenericProxy k q a -> GenericProxy k q b
+ -> (GenericProxy k q c, GenericProxy k q d) )
+ -> k (a,b) (c,d)
+genericAlg2to2 f = runGenericProxy c &&& runGenericProxy d
+ where (c,d) = f (GenericProxy fst) (GenericProxy snd)
+
+
+class (HasProxy k, ProxyVal k a x ~ p a x)
+ => PointProxy p k a x | p -> k where
+ point :: (Object k a, Object k x) => x -> p a x
+
+genericPoint :: ( WellPointed k, Object k a, ObjectPoint k x )
+ => x -> GenericProxy k a x
+genericPoint x = GenericProxy $ const x
+
diff --git a/Control/Category/Constrained.hs b/Control/Category/Constrained.hs
new file mode 100644
--- /dev/null
+++ b/Control/Category/Constrained.hs
@@ -0,0 +1,402 @@
+-- |
+-- Module : Control.Category.Constrained
+-- Copyright : (c) 2013 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+--
+-- The most basic category theory tools are included partly in this
+-- module, partly in "Control.Arrow.Constrained".
+
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE RankNTypes #-}
+{-# LANGUAGE AllowAmbiguousTypes #-}
+{-# LANGUAGE TypeOperators #-}
+
+module Control.Category.Constrained (
+ -- * The category class
+ Category (..)
+ -- * Monoidal categories
+ , Cartesian (..), ObjectPair
+ , Curry (..), ObjectMorphism
+ -- * Monoidal with coproducts
+ , (+)()
+ , CoCartesian (..), ObjectSum
+ -- * Isomorphisms
+ , Isomorphic (..)
+ -- * Constraining a category
+ , ConstrainedCategory (ConstrainedMorphism)
+ , constrained, unconstrained
+ -- * Global-element proxies
+ , HasProxy (..)
+ , genericAlg, genericProxyMap
+ , GenericProxy (..)
+ -- * Utility
+ , inCategoryOf
+ , CatTagged
+ ) where
+
+import Prelude hiding (id, (.), curry, uncurry)
+import qualified Prelude
+import GHC.Exts (Constraint)
+import Data.Tagged
+import Data.Monoid
+import Data.Void
+
+-- | In mathematics, a category is defined as a class of /objects/, plus a class of
+-- /morphisms/ between those objects. In Haskell, one traditionally works in
+-- the category @(->)@ (called /Hask/), in which /any Haskell type/ is an object.
+-- But of course
+-- there are lots of useful categories where the objects are much more specific,
+-- e.g. vector spaces with linear maps as morphisms. The obvious way to express
+-- this in Haskell is as type class constraints, and the @ConstraintKinds@ extension
+-- allows quantifying over such object classes.
+--
+-- Like in "Control.Category", \"the category @k@\" means actually @k@ is the
+-- /morphism type constructor/. From a mathematician's point of view this may
+-- seem a bit strange way to define the category, but it just turns out to
+-- be quite convenient for practical purposes.
+class Category k where
+ type Object k o :: Constraint
+ type Object k o = ()
+ id :: Object k a => k a a
+ (.) :: (Object k a, Object k b, Object k c)
+ => k b c -> k a b -> k a c
+
+infixr 9 .
+
+instance Category (->) where
+ id = Prelude.id
+ (.) = (Prelude..)
+
+-- | Analogue to 'asTypeOf', this does not actually do anything but can
+-- give the compiler type unification hints in a convenient manner.
+inCategoryOf :: (Category k) => k a b -> k c d -> k a b
+m `inCategoryOf` _ = m
+
+
+-- | A given category can be specialised, by using the same morphisms but adding
+-- extra constraints to what is considered an object.
+--
+-- For instance, @'ConstrainedCategory' (->) 'Ord'@ is the category of all
+-- totally ordered data types (but with arbitrary functions; this does not require
+-- monotonicity or anything).
+newtype ConstrainedCategory (k :: * -> * -> *) (o :: * -> Constraint) (a :: *) (b :: *)
+ = ConstrainedMorphism { unconstrainedMorphism :: k a b }
+
+-- | Cast a morphism to its equivalent in a more constrained category,
+-- provided it connects objects that actually satisfy the extra constraint.
+constrained :: (Category k, o a, o b) => k a b -> ConstrainedCategory k o a b
+constrained = ConstrainedMorphism
+
+-- | \"Unpack\" a constrained morphism again (forgetful functor).
+--
+-- Note that you may often not need to do that; in particular
+-- morphisms that are actually 'Function's can just be applied
+-- to their objects with '$' right away, no need to go back to
+-- Hask first.
+unconstrained :: (Category k) => ConstrainedCategory k o a b -> k a b
+unconstrained = unconstrainedMorphism
+
+instance (Category k) => Category (ConstrainedCategory k isObj) where
+ type Object (ConstrainedCategory k isObj) o = (Object k o, isObj o)
+ id = ConstrainedMorphism id
+ ConstrainedMorphism f . ConstrainedMorphism g = ConstrainedMorphism $ f . g
+
+
+-- | Apart from /the/ identity morphism, 'id', there are other morphisms that
+-- can basically be considered identies. For instance, in any cartesian
+-- category (where it makes sense to have tuples and unit @()@ at all), it should be
+-- possible to switch between @a@ and the isomorphic @(a, ())@. 'iso' is
+-- the method for such \"pseudo-identities\", the most basic of which
+-- are required as methods of the 'Cartesian' class.
+--
+-- Why it is necessary to make these morphisms explicit: they are needed
+-- for a couple of general-purpose category-theory methods, but even though
+-- they're normally trivial to define there is no uniform way to do so.
+-- For instance, for vector spaces, the baseis of @(a, (b,c))@ and @((a,b), c)@
+-- are sure enough structurally equivalent, but not in the same way the spaces
+-- themselves are (sum vs. product types).
+{-# DEPRECATED iso "This generic method, while looking nicely uniform, relies on OverlappingInstances and is therefore probably a bad idea. Use the specialised methods in classes like 'SPDistribute' instead." #-}
+class (Category k) => Isomorphic k a b where
+ iso :: k a b
+
+instance (Cartesian k, Object k a, u ~ UnitObject k, ObjectPair k a u) => Isomorphic k a (a,u) where
+ iso = attachUnit
+instance (Cartesian k, Object k a, u ~ UnitObject k, ObjectPair k a u) => Isomorphic k (a,u) a where
+ iso = detachUnit
+instance (Cartesian k, Object k a, u ~ UnitObject k, ObjectPair k a u, ObjectPair k u a, Object k (u, a), Object k (a, u) )
+ => Isomorphic k a (u,a) where
+ iso = swap . attachUnit
+instance (Cartesian k, Object k a, u ~ UnitObject k, ObjectPair k a u, ObjectPair k u a, Object k (u, a), Object k (a, u) )
+ => Isomorphic k (u,a) a where
+ iso = detachUnit . swap
+instance ( Cartesian k, Object k a, ObjectPair k a b, ObjectPair k b c
+ , ObjectPair k a (b,c), ObjectPair k (a,b) c, Object k c )
+ => Isomorphic k (a,(b,c)) ((a,b),c) where
+ iso = regroup
+instance ( Cartesian k, Object k a, ObjectPair k a b, ObjectPair k b c
+ , ObjectPair k a (b,c), ObjectPair k (a,b) c, Object k c )
+ => Isomorphic k ((a,b),c) (a,(b,c)) where
+ iso = regroup'
+
+
+instance (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u) => Isomorphic k a (a+u) where
+ iso = attachZero
+instance (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u) => Isomorphic k (a+u) a where
+ iso = detachZero
+instance (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u, ObjectSum k u a, Object k (u+a), Object k (a+u) )
+ => Isomorphic k a (u+a) where
+ iso = coSwap . attachZero
+instance (CoCartesian k, Object k a, u ~ ZeroObject k, ObjectSum k a u, ObjectSum k u a, Object k (u+a), Object k (a+u) )
+ => Isomorphic k (u+a) a where
+ iso = detachZero . coSwap
+instance ( CoCartesian k, Object k a, ObjectSum k a b, ObjectSum k b c
+ , ObjectSum k a (b+c), ObjectSum k (a+b) c, Object k c )
+ => Isomorphic k (a+(b+c)) ((a+b)+c) where
+ iso = coRegroup
+instance ( CoCartesian k, Object k a, ObjectSum k a b, ObjectSum k b c
+ , ObjectSum k a (b+c), ObjectSum k (a+b) c, Object k c )
+ => Isomorphic k ((a+b)+c) (a+(b+c)) where
+ iso = coRegroup'
+
+
+-- | Quite a few categories (/monoidal categories/) will permit \"products\" of
+-- objects as objects again – in the Haskell sense those are tuples – allowing
+-- for \"dyadic morphisms\" @(x,y) ~> r@.
+--
+-- Together with a unique 'UnitObject', this makes for a monoidal
+-- structure, with a few natural isomorphisms. Ordinary tuples may not
+-- always be powerful enough to express the product objects; we avoid
+-- making a dedicated associated type for the sake of simplicity,
+-- but allow for an extra constraint to be imposed on objects prior
+-- to consideration of pair-building.
+--
+-- The name 'Cartesian' is disputable: in category theory that would rather
+-- Imply /cartesian closed category/ (which we represent with 'Curry').
+-- 'Monoidal' would make sense, but we reserve that to 'Functors'.
+class ( Category k
+ , Monoid (UnitObject k), Object k (UnitObject k)
+ -- , PairObject k (UnitObject k) (UnitObject k), Object k (UnitObject k,UnitObject k)
+ ) => Cartesian k where
+ -- | Extra properties two types @a, b@ need to fulfill so @(a,b)@ can be an
+ -- object of the category. This need /not/ take care for @a@ and @b@ themselves
+ -- being objects, we do that seperately: every function that actually deals
+ -- with @(a,b)@ objects should require the stronger @'ObjectPair' k a b@.
+ --
+ -- If /any/ two object types of your category make up a pair object, then
+ -- just leave 'PairObjects' at the default (empty constraint).
+ type PairObjects k a b :: Constraint
+ type PairObjects k a b = ()
+
+ -- | Defaults to '()', and should normally be left at that.
+ type UnitObject k :: *
+ type UnitObject k = ()
+
+ swap :: ( ObjectPair k a b, ObjectPair k b a ) => k (a,b) (b,a)
+
+ attachUnit :: ( Object k a, u ~ UnitObject k, ObjectPair k a u ) => k a (a,u)
+ detachUnit :: ( Object k a, u ~ UnitObject k, ObjectPair k a u ) => k (a,u) a
+ regroup :: ( Object k a, Object k c, ObjectPair k a b, ObjectPair k b c
+ , ObjectPair k a (b,c), ObjectPair k (a,b) c
+ ) => k (a, (b, c)) ((a, b), c)
+ regroup' :: ( Object k a, Object k c, ObjectPair k a b, ObjectPair k b c
+ , ObjectPair k a (b,c), ObjectPair k (a,b) c
+ ) => k ((a, b), c) (a, (b, c))
+
+-- | Use this constraint to ensure that @a@, @b@ and @(a,b)@ are all \"fully valid\" objects
+-- of your category (meaning, you can use them with the 'Cartesian' combinators).
+type ObjectPair k a b = ( Category k, Object k a, Object k b
+ , PairObjects k a b, Object k (a,b) )
+
+instance Cartesian (->) where
+ swap = \(a,b) -> (b,a)
+ attachUnit = \a -> (a, ())
+ detachUnit = \(a, ()) -> a
+ regroup = \(a, (b, c)) -> ((a, b), c)
+ regroup' = \((a, b), c) -> (a, (b, c))
+
+instance (Cartesian f, o (UnitObject f)) => Cartesian (ConstrainedCategory f o) where
+ type PairObjects (ConstrainedCategory f o) a b = (PairObjects f a b)
+ type UnitObject (ConstrainedCategory f o) = UnitObject f
+
+ swap = ConstrainedMorphism swap
+ attachUnit = ConstrainedMorphism attachUnit
+ detachUnit = ConstrainedMorphism detachUnit
+ regroup = ConstrainedMorphism regroup
+ regroup' = ConstrainedMorphism regroup'
+
+
+type (+) = Either
+
+-- | Monoidal categories need not be based on a cartesian product.
+-- The relevant alternative is coproducts.
+--
+-- The dual notion to 'Cartesian' replaces such products (pairs) with
+-- sums ('Either'), and unit '()' with void types.
+--
+-- Basically, the only thing that doesn't mirror 'Cartesian' here
+-- is that we don't require @CoMonoid ('ZeroObject' k)@. Comonoids
+-- do in principle make sense, but not from a Haskell viewpoint
+-- (every type is trivially a comonoid).
+--
+-- Haskell of course uses sum types, /variants/, most often without
+-- 'Either' appearing. But variants are generally isomorphic to sums;
+-- the most important (sums of unit) are methods here.
+class ( Category k, Object k (ZeroObject k)
+ ) => CoCartesian k where
+ type SumObjects k a b :: Constraint
+ type SumObjects k a b = ()
+ -- | Defaults to 'Void'.
+ type ZeroObject k :: *
+ type ZeroObject k = Void
+
+ coSwap :: ( ObjectSum k a b, ObjectSum k b a ) => k (a+b) (b+a)
+
+ attachZero :: ( Object k a, z ~ ZeroObject k, ObjectSum k a z ) => k a (a+z)
+ detachZero :: ( Object k a, z ~ ZeroObject k, ObjectSum k a z ) => k (a+z) a
+ coRegroup :: ( Object k a, Object k c, ObjectSum k a b, ObjectSum k b c
+ , ObjectSum k a (b+c), ObjectSum k (a+b) c
+ ) => k (a+(b+c)) ((a+b)+c)
+ coRegroup' :: ( Object k a, Object k c, ObjectSum k a b, ObjectSum k b c
+ , ObjectSum k a (b+c), ObjectSum k (a+b) c
+ ) => k ((a+b)+c) (a+(b+c))
+
+ maybeAsSum :: ( ObjectSum k u a, u ~ UnitObject k, Object k (Maybe a) )
+ => k (Maybe a) (u + a)
+ maybeFromSum :: ( ObjectSum k u a, u ~ UnitObject k, Object k (Maybe a) )
+ => k (u + a) (Maybe a)
+ boolAsSum :: ( ObjectSum k u u, u ~ UnitObject k, Object k Bool )
+ => k Bool (u + u)
+ boolFromSum :: ( ObjectSum k u u, u ~ UnitObject k, Object k Bool )
+ => k (u + u) Bool
+
+type ObjectSum k a b = ( Category k, Object k a, Object k b
+ , SumObjects k a b, Object k (a+b) )
+
+
+instance CoCartesian (->) where
+ coSwap (Left a) = Right a
+ coSwap (Right a) = Left a
+ attachZero = Left
+ detachZero (Left a) = a
+ detachZero (Right void) = absurd void
+ coRegroup (Left a) = Left $ Left a
+ coRegroup (Right (Left a)) = Left $ Right a
+ coRegroup (Right (Right a)) = Right a
+ coRegroup' (Left (Left a)) = Left a
+ coRegroup' (Left (Right a)) = Right $ Left a
+ coRegroup' (Right a) = Right $ Right a
+ maybeAsSum Nothing = Left ()
+ maybeAsSum (Just x) = Right x
+ maybeFromSum (Left ()) = Nothing
+ maybeFromSum (Right x) = Just x
+ boolAsSum False = Left ()
+ boolAsSum True = Right ()
+ boolFromSum (Left ()) = False
+ boolFromSum (Right ()) = True
+-- boolAsSwitch (False,x) = Left x
+-- boolAsSwitch (True,x) = Right x
+-- boolFromSwitch (Left x) = (False,x)
+-- boolFromSwitch (Right x) = (True,x)
+--
+instance (CoCartesian f, o (ZeroObject f)) => CoCartesian (ConstrainedCategory f o) where
+ type SumObjects (ConstrainedCategory f o) a b = (SumObjects f a b)
+ type ZeroObject (ConstrainedCategory f o) = ZeroObject f
+
+ coSwap = ConstrainedMorphism coSwap
+ attachZero = ConstrainedMorphism attachZero
+ detachZero = ConstrainedMorphism detachZero
+ coRegroup = ConstrainedMorphism coRegroup
+ coRegroup' = ConstrainedMorphism coRegroup'
+ maybeAsSum = ConstrainedMorphism maybeAsSum
+ maybeFromSum = ConstrainedMorphism maybeFromSum
+ boolAsSum = ConstrainedMorphism boolAsSum
+ boolFromSum = ConstrainedMorphism boolFromSum
+-- boolAsSwitch = ConstrainedMorphism boolAsSwitch
+-- boolFromSwitch = ConstrainedMorphism boolFromSwitch
+
+
+
+
+
+-- | Tagged type for values that depend on some choice of category, but not on some
+-- particular object / arrow therein.
+type CatTagged k x = Tagged (k (UnitObject k) (UnitObject k)) x
+
+
+
+
+class (Cartesian k) => Curry k where
+ type MorphObjects k b c :: Constraint
+ type MorphObjects k b c = ()
+ uncurry :: (ObjectPair k a b, ObjectMorphism k b c)
+ => k a (k b c) -> k (a, b) c
+ -- uncurry f = apply . (f &&& id)
+ curry :: (ObjectPair k a b, ObjectMorphism k b c)
+ => k (a, b) c -> k a (k b c)
+ apply :: (ObjectMorphism k a b, ObjectPair k (k a b) a)
+ => k (k a b, a) b
+ apply = uncurry id
+
+-- | Analogous to 'ObjectPair': express that @k b c@ be an exponential object
+-- representing the morphism.
+type ObjectMorphism k b c = (Object k b, Object k c, MorphObjects k b c, Object k (k b c))
+
+
+instance Curry (->) where
+ uncurry = Prelude.uncurry
+ curry = Prelude.curry
+ apply (f,x) = f x
+
+
+instance (Curry f, o (UnitObject f)) => Curry (ConstrainedCategory f o) where
+ type MorphObjects (ConstrainedCategory f o) a c = ( MorphObjects f a c, f ~ (->) )
+ uncurry (ConstrainedMorphism f) = ConstrainedMorphism $ \(a,b) -> unconstrained (f a) b
+ curry (ConstrainedMorphism f) = ConstrainedMorphism $ \a -> ConstrainedMorphism $ \b -> f (a, b)
+
+
+
+infixr 0 $~
+
+-- | A proxy value is a \"general representation\" of a category's
+-- values, i.e. /global elements/. This is useful to define certain
+-- morphisms (including ones that can't just \"inherit\" from '->'
+-- with 'Control.Arrow.Constrained.arr') in ways other than point-free
+-- composition pipelines. Instead, you can write algebraic expressions
+-- much as if dealing with actual values of your category's objects,
+-- but using the proxy type which is restricted so any function
+-- defined as such a lambda-expression qualifies as a morphism
+-- of that category.
+class (Category k) => HasProxy k where
+ type ProxyVal k a v :: *
+ type ProxyVal k a v = GenericProxy k a v
+ alg :: ( Object k a, Object k b
+ ) => (forall q . Object k q
+ => ProxyVal k q a -> ProxyVal k q b) -> k a b
+ ($~) :: ( Object k a, Object k b, Object k c
+ ) => k b c -> ProxyVal k a b -> ProxyVal k a c
+
+data GenericProxy k a v = GenericProxy { runGenericProxy :: k a v }
+
+genericAlg :: ( HasProxy k, Object k a, Object k b )
+ => ( forall q . Object k q
+ => GenericProxy k q a -> GenericProxy k q b ) -> k a b
+genericAlg f = runGenericProxy . f $ GenericProxy id
+
+genericProxyMap :: ( HasProxy k, Object k a, Object k b, Object k c )
+ => k b c -> GenericProxy k a b -> GenericProxy k a c
+genericProxyMap m (GenericProxy v) = GenericProxy $ m . v
+
+
+
+instance HasProxy (->) where
+ type ProxyVal (->) a b = b
+ alg f = f
+ ($~) = ($)
+
+
+
diff --git a/Control/Category/Constrained/Prelude.hs b/Control/Category/Constrained/Prelude.hs
new file mode 100644
--- /dev/null
+++ b/Control/Category/Constrained/Prelude.hs
@@ -0,0 +1,32 @@
+-- |
+-- Module : Control.Category.Constrained.Prelude
+-- Copyright : (c) 2013 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+
+module Control.Category.Constrained.Prelude (
+ -- * The constrained-categories facilities
+ module Control.Category.Constrained
+ , module Control.Functor.Constrained
+ , module Control.Applicative.Constrained
+ , module Control.Monad.Constrained
+ , module Control.Arrow.Constrained
+ -- * The compatible part of the standard Prelude
+ , module Prelude
+ ) where
+
+import Prelude hiding ( id, const, fst, snd, (.), ($), curry, uncurry
+ , Functor(..), Monad(..), (=<<), filter
+ , mapM, mapM_, sequence, sequence_ )
+
+import Control.Category.Constrained hiding (ConstrainedMorphism)
+import Control.Functor.Constrained
+import Control.Applicative.Constrained
+import Control.Monad.Constrained hiding
+ (MonadPlus(..), MonadZero(..), (>=>), (<=<), guard, forever, void)
+import Control.Arrow.Constrained (Function, ($), ifThenElse, fst, snd, const)
+
diff --git a/Control/Category/Hask.hs b/Control/Category/Hask.hs
new file mode 100644
--- /dev/null
+++ b/Control/Category/Hask.hs
@@ -0,0 +1,36 @@
+-- |
+-- Module : Control.Category.Hask
+-- Copyright : (c) 2013 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+-- Re-exports of all the common category-theory inspired classes from the
+-- "base" package, i.e. basically endofunctors in the Hask category (with
+-- functions @(->)@ as morphisms).
+-- The module is thus intended to be imported @qualified as Hask@.
+--
+-- Main use case would be defining new such functors / monads etc.
+-- yourself; even if you only intend to use them through the more
+-- general category-agnostic interface established in this package
+-- then the /instances/ should still be defined for the plain old
+-- Hask-specific classes, i.e. for some
+--
+-- > data F a = ...
+-- > fmapF :: (a->b) -> F a->F b@
+-- >
+-- > instance Hask.Functor F where
+-- > Hask.fmap = fmapF
+--
+-- An instance of 'Control.Functor.Constrained.Functor' arises automatically
+-- from this, as defined generically for all @(->)@ functors in that
+-- module.
+
+module Control.Category.Hask( module Prelude
+ , module Control.Category
+ , module Control.Applicative
+ , module Control.Monad
+ ) where
+import Prelude hiding ((.), id)
+import Control.Category
+import Control.Applicative
+import Control.Monad
diff --git a/Control/Functor/Constrained.hs b/Control/Functor/Constrained.hs
new file mode 100644
--- /dev/null
+++ b/Control/Functor/Constrained.hs
@@ -0,0 +1,95 @@
+-- |
+-- Module : Control.Functor.Constrained
+-- Copyright : (c) 2014 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE UndecidableInstances #-}
+
+
+module Control.Functor.Constrained
+ ( module Control.Category.Constrained
+ -- * Functors
+ , Functor(..)
+ , (<$>)
+ , constrainedFmap
+ -- * [Co]product mapping
+ , SumToProduct(..)
+ ) where
+
+
+import Control.Category.Constrained
+
+import Prelude hiding (id, (.), Functor(..), filter)
+import qualified Prelude
+
+import Data.Void
+
+class ( Category r, Category t, Object t (f (UnitObject r)) )
+ => Functor f r t | f r -> t, f t -> r where
+ fmap :: (Object r a, Object t (f a), Object r b, Object t (f b))
+ => r a b -> t (f a) (f b)
+
+instance (Prelude.Functor f) => Functor f (->) (->) where
+ fmap = Prelude.fmap
+
+-- | It is fairly common for functors (typically, container-like) to map 'Either'
+-- to tuples in a natural way, thus \"separating the variants\".
+-- This is related to 'Data.Foldable.Constrained.Foldable'
+-- (with list and tuple monoids), but rather more effective.
+class ( CoCartesian r, Cartesian t, Functor f r t, Object t (f (ZeroObject r)) )
+ => SumToProduct f r t where
+ -- | @
+ -- sum2product ≡ mapEither id
+ -- @
+ sum2product :: ( ObjectSum r a b, ObjectPair t (f a) (f b) )
+ => t (f (a+b)) (f a, f b)
+ -- | @
+ -- mapEither f ≡ sum2product . fmap f
+ -- @
+ mapEither :: ( Object r a, ObjectSum r b c
+ , Object t (f a), ObjectPair t (f b) (f c) )
+ => r a (b+c) -> t (f a) (f b, f c)
+ filter :: ( Object r a, Object r Bool, Object t (f a) )
+ => r a Bool -> t (f a) (f a)
+
+instance SumToProduct [] (->) (->) where
+ sum2product [] = ([],[])
+ sum2product (Left x : l) = (x:xs, ys) where ~(xs,ys) = sum2product l
+ sum2product (Right y : l) = (xs ,y:ys) where ~(xs,ys) = sum2product l
+ mapEither _ [] = ([],[])
+ mapEither f (a:l) = case f a of
+ Left x -> (x:xs, ys)
+ Right y -> (xs ,y:ys)
+ where ~(xs,ys) = mapEither f l
+ filter = Prelude.filter
+
+(<$>) :: (Functor f r (->), Object r a, Object r b)
+ => r a b -> f a -> f b
+(<$>) = fmap
+
+
+constrainedFmap :: (Category r, Category t, o a, o b, o (f a), o (f b))
+ => ( r a b -> t (f a) (f b) )
+ -> ConstrainedCategory r o a b -> ConstrainedCategory t o (f a) (f b)
+constrainedFmap q = constrained . q . unconstrained
+
+instance (Functor [] k k, o [UnitObject k])
+ => Functor [] (ConstrainedCategory k o) (ConstrainedCategory k o) where
+ fmap (ConstrainedMorphism f) = ConstrainedMorphism $ fmap f
+
+instance (o (), o [()], o Void, o [Void]) => SumToProduct []
+ (ConstrainedCategory (->) o) (ConstrainedCategory (->) o) where
+ sum2product = ConstrainedMorphism sum2product
+ mapEither (ConstrainedMorphism f) = ConstrainedMorphism $ mapEither f
+ filter (ConstrainedMorphism f) = ConstrainedMorphism $ filter f
+
+
+
diff --git a/Control/Monad/Constrained.hs b/Control/Monad/Constrained.hs
new file mode 100644
--- /dev/null
+++ b/Control/Monad/Constrained.hs
@@ -0,0 +1,274 @@
+-- |
+-- Module : Control.Monad.Constrained
+-- Copyright : (c) 2013 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE LambdaCase #-}
+
+
+module Control.Monad.Constrained( module Control.Applicative.Constrained
+ -- * Monads
+ , Monad(..), return, (>>=), (=<<), (>>), (<<)
+ -- * Kleisli arrows
+ , (>=>), (<=<)
+ , Kleisli(..)
+ -- * Monoid-Monads
+ , MonadZero(..), mzero, MonadPlus(..), mplus
+ , MonadFail(..)
+ -- * Utility
+ , mapM, mapM_, forM, forM_, sequence, sequence_
+ , guard, when, unless
+ , forever, void
+ ) where
+
+
+import Control.Applicative.Constrained
+import Data.Foldable.Constrained
+import Data.Traversable.Constrained
+import Data.Tagged
+
+import Prelude hiding (
+ id, const, fst, snd, (.), ($)
+ , Functor(..), Monad(..), (=<<)
+ , uncurry, curry
+ , mapM, mapM_, sequence, sequence_
+ )
+import qualified Control.Category.Hask as Hask
+
+import Control.Arrow.Constrained
+
+
+class ( Applicative m k k
+ , Object k (m (UnitObject k)), Object k (m (m (UnitObject k)))
+ ) => Monad m k where
+ join :: (Object k a, Object k (m a), Object k (m (m a)))
+ => m (m a) `k` m a
+
+-- | This is monomorphic in the category /Hask/, thus exactly the same as 'Hask.return'
+-- from the standard prelude. This allows writing expressions like
+-- @'return' '$' case x of ...@, which would always be ambiguous with the more general
+-- signature @Monad m k => k a (m a)@.
+--
+-- Use 'pure' when you want to \"return\" in categories other than @(->)@; this always
+-- works since 'Applicative' is a superclass of 'Monad'.
+return :: Monad m (->) => a -> m a
+return = pure
+
+
+
+infixr 1 =<<
+(=<<) :: ( Monad m k, Object k a, Object k b
+ , Object k (m a), Object k (m b), Object k (m (m b)) )
+ => k a (m b) -> k (m a) (m b)
+(=<<) q = join . fmap q
+
+infixl 1 >>=
+(>>=) :: ( Function f, Monad m f, Object f a, Object f b
+ , Object f (m a), Object f (m b), Object f (m (m b)) )
+ => m a -> f a (m b) -> m b
+g >>= h = (=<<) h $ g
+
+infixr 1 <<
+(<<) :: ( Monad m k, WellPointed k
+ , Object k a, Object k b, Object k (m a), ObjectPoint k (m b), Object k (m (m b))
+ ) => m b -> k (m a) (m b)
+(<<) b = join . fmap (const b)
+
+infixl 1 >>
+(>>) :: ( WellPointed k, Monad m k
+ , ObjectPair k b (UnitObject k), ObjectPair k (m b) (UnitObject k)
+ , ObjectPair k (UnitObject k) (m b), ObjectPair k b a
+ , ObjectPair k a b, Object k (m (a,b)), ObjectPair k (m a) (m b)
+ , ObjectPoint k (m a)
+ ) => m a -> k (m b) (m b)
+(>>) a = fmap snd . fzip . first (globalElement a) . swap . attachUnit
+ -- where result = arr $ \b -> (join . fmap (const b)) `inCategoryOf` result $ a
+
+
+instance (Hask.Applicative m, Hask.Monad m) => Monad m (->) where
+ join = Hask.join
+
+
+-- | 'Hask.MonadPlus' cannot be adapted quite analogously to 'Monad',
+-- since 'mzero' is just a value with no way to indicate its morphism
+-- category. The current implementation is probably not ideal, mainly
+-- written to give 'MonadFail' ('fail' being needed for @RebindableSyntax@-@do@
+-- notation) a mathematically reasonable superclass.
+--
+-- Consider these classes provisorial, avoid relying on them explicitly.
+class (Monad m k) => MonadZero m k where
+ fmzero :: (Object k a, Object k (m a)) => UnitObject k `k` m a
+
+mzero :: (MonadZero m (->)) => m a
+mzero = fmzero ()
+
+class (MonadZero m k) => MonadPlus m k where
+ fmplus :: (ObjectPair k (m a) (m a)) => k (m a, m a) (m a)
+
+mplus :: (MonadPlus m (->)) => m a -> m a -> m a
+mplus = curry fmplus
+
+instance (Hask.MonadPlus m, Hask.Applicative m) => MonadZero m (->) where
+ fmzero = const Hask.mzero
+instance (Hask.MonadPlus m, Hask.Applicative m) => MonadPlus m (->) where
+ fmplus = uncurry Hask.mplus
+
+
+class (MonadPlus m k) => MonadFail m k where
+ fail :: (Object k (m a)) => k String (m a)
+
+instance (Hask.MonadPlus m, Hask.Applicative m) => MonadFail m (->) where
+ fail = Hask.fail
+
+
+infixr 1 >=>, <=<
+
+(>=>) :: ( Monad m k, Object k a, Object k b, Object k c
+ , Object k (m b), Object k (m c), Object k (m (m c)))
+ => a `k` m b -> b `k` m c -> a `k` m c
+f >=> g = join . fmap g . f
+(<=<) :: ( Monad m k, Object k a, Object k b, Object k c
+ , Object k (m b), Object k (m c), Object k (m (m c)))
+ => b `k` m c -> a `k` m b -> a `k` m c
+f <=< g = join . fmap f . g
+
+newtype Kleisli m k a b = Kleisli { runKleisli :: k a (m b) }
+
+instance (Monad m k) => Category (Kleisli m k) where
+ type Object (Kleisli m k) o = (Object k o, Object k (m o), Object k (m (m o)))
+ id = Kleisli pure
+ Kleisli a . Kleisli b = Kleisli $ join . fmap a . b
+
+instance ( Monad m a, Cartesian a ) => Cartesian (Kleisli m a) where
+ type PairObjects (Kleisli m a) b c
+ = ( ObjectPair a b c
+ , ObjectPair a (m b) c, ObjectPair a b (m c), ObjectPair a (m b) (m c) )
+ type UnitObject (Kleisli m a) = UnitObject a
+ swap = Kleisli $ pure . swap
+ attachUnit = Kleisli $ pure . attachUnit
+ detachUnit = Kleisli $ pure . detachUnit
+ regroup = Kleisli $ pure . regroup
+ regroup' = Kleisli $ pure . regroup'
+
+instance ( Monad m k, CoCartesian k
+ , Object k (m (ZeroObject k)), Object k (m (m (ZeroObject k)))
+ ) => CoCartesian (Kleisli m k) where
+ type SumObjects (Kleisli m k) b c
+ = ( ObjectSum k b c
+ , ObjectSum k (m b) c, ObjectSum k b (m c), ObjectSum k (m b) (m c) )
+ type ZeroObject (Kleisli m k) = ZeroObject k
+ coSwap = Kleisli $ pure . coSwap
+ attachZero = Kleisli $ pure . attachZero
+ detachZero = Kleisli $ pure . detachZero
+ coRegroup = Kleisli $ pure . coRegroup
+ coRegroup' = Kleisli $ pure . coRegroup'
+
+ maybeAsSum = Kleisli $ pure . maybeAsSum
+ maybeFromSum = Kleisli $ pure . maybeFromSum
+ boolAsSum = Kleisli $ pure . boolAsSum
+ boolFromSum = Kleisli $ pure . boolFromSum
+
+instance ( Monad m a, Arrow a (->), Function a ) => Curry (Kleisli m a) where
+ type MorphObjects (Kleisli m a) c d
+ = ( Object a (Kleisli m a c d), Object a (m (Kleisli m a c d))
+ , Object a (a c (m d))
+ , ObjectMorphism a c d, ObjectMorphism a c (m d), ObjectMorphism a c (m (m d)) )
+ curry (Kleisli fUnc) = Kleisli $ pure . arr Kleisli . curry fUnc
+ uncurry (Kleisli fCur) = Kleisli . arr $
+ \(b,c) -> join . fmap (arr $ ($c) . runKleisli) . fCur $ b
+
+
+
+
+instance (Monad m a, Arrow a q, Cartesian a) => EnhancedCat (Kleisli m a) q where
+ arr f = Kleisli $ pure . arr f
+instance (Monad m a, Morphism a, Curry a) => Morphism (Kleisli m a) where
+ first (Kleisli f) = Kleisli $ fzip . (f *** pure)
+ second (Kleisli f) = Kleisli $ fzip . (pure *** f)
+ Kleisli f *** Kleisli g = Kleisli $ fzip . (f *** g)
+instance (Monad m a, PreArrow a, Curry a) => PreArrow (Kleisli m a) where
+ Kleisli f &&& Kleisli g = Kleisli $ fzip . (f &&& g)
+ terminal = Kleisli $ pure . terminal
+ fst = Kleisli $ pure . fst
+ snd = Kleisli $ pure . snd
+instance (SPDistribute k, Monad m k, PreArrow (Kleisli m k), PreArrChoice (Kleisli m k))
+ => SPDistribute (Kleisli m k) where
+ distribute = Kleisli $ pure . distribute
+ unDistribute = Kleisli $ pure . unDistribute
+ boolAsSwitch = Kleisli $ pure . boolAsSwitch
+ boolFromSwitch = Kleisli $ pure . boolFromSwitch
+instance (Monad m a, WellPointed a, ObjectPoint a (m (UnitObject a)))
+ => WellPointed (Kleisli m a) where
+ type PointObject (Kleisli m a) b = (PointObject a b, PointObject a (m b))
+ globalElement x = Kleisli $ fmap (globalElement x) . pureUnit
+ unit = kleisliUnit
+
+
+-- | /Hask/-Kleislis inherit more or less trivially 'Hask.ArrowChoice'; however this
+-- does not generalise greatly well to non-function categories.
+instance ( Monad m k, Arrow k (->), Function k, PreArrChoice k
+ , Object k (m (ZeroObject k)), Object k (m (m (ZeroObject k)))
+ ) => MorphChoice (Kleisli m k) where
+ left (Kleisli f) = Kleisli . arr $ \case { Left x -> fmap coFst . f $ x
+ ; Right y-> (pure . coSnd)`inCategoryOf`f $ y }
+ right(Kleisli f) = Kleisli . arr $ \case { Left x -> (pure . coFst)`inCategoryOf`f $ x
+ ; Right y-> fmap coSnd . f $ y }
+ Kleisli f +++ Kleisli g = Kleisli . arr $ \case
+ Left x -> fmap coFst . f $ x
+ Right y -> fmap coSnd . g $ y
+instance ( Monad m k, Arrow k (->), Function k, PreArrChoice k
+ , Object k (m (ZeroObject k)), Object k (m (m (ZeroObject k)))
+ ) => PreArrChoice (Kleisli m k) where
+ Kleisli f ||| Kleisli g = Kleisli $ f ||| g
+ initial = Kleisli $ pure . initial
+ coFst = Kleisli $ pure . coFst
+ coSnd = Kleisli $ pure . coSnd
+
+
+kleisliUnit :: forall m a . (Monad m a, WellPointed a)
+ => CatTagged (Kleisli m a) (UnitObject a)
+kleisliUnit = retag (unit :: CatTagged a (UnitObject a))
+
+
+guard ::( MonadPlus m k, Arrow k (->), Function k
+ , UnitObject k ~ (), Object k Bool
+ ) => Bool `k` m ()
+guard = i . choose fmzero pure
+ where i = id
+
+
+when :: ( Monad m k, PreArrow k, u ~ UnitObject k
+ , ObjectPair k (m u) u
+ ) => Bool -> m u `k` m u
+when True = id
+when False = pure . terminal
+unless :: ( Monad m k, PreArrow k, u ~ UnitObject k
+ , ObjectPair k (m u) u
+ ) => Bool -> m u `k` m u
+unless False = id
+unless True = pure . terminal
+
+
+
+forever :: ( Monad m k, Function k, Arrow k (->), Object k a, Object k b
+ , Object k (m a), Object k (m (m a)), ObjectPoint k (m b), Object k (m (m b))
+ ) => m a `k` m b
+forever = i . arr loop
+ where loop a = (join . fmap (const $ loop a)) `inCategoryOf` i $ a
+ i = id
+
+void :: ( Monad m k, PreArrow k
+ , Object k a, Object k (m a), ObjectPair k a u, u ~ UnitObject k
+ ) => m a `k` m (UnitObject k)
+void = fmap terminal
+
+
diff --git a/Data/Foldable/Constrained.hs b/Data/Foldable/Constrained.hs
new file mode 100644
--- /dev/null
+++ b/Data/Foldable/Constrained.hs
@@ -0,0 +1,166 @@
+-- |
+-- Module : Data.Foldable.Constrained
+-- Copyright : (c) 2014 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE KindSignatures #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TupleSections #-}
+
+
+module Data.Foldable.Constrained
+ ( module Control.Category.Constrained
+ , Foldable(..)
+ , fold
+ , traverse_, mapM_, forM_, sequence_
+ , concatMap
+ ) where
+
+
+import Control.Category.Constrained
+import Control.Functor.Constrained
+import Control.Applicative.Constrained
+
+import Prelude hiding (
+ id, (.), ($)
+ , Functor(..)
+ , uncurry, curry
+ , mapM_, sequence_, concatMap
+ )
+import Data.Monoid
+
+import qualified Control.Category.Hask as Hask
+import qualified Control.Arrow as A
+
+import Control.Arrow.Constrained
+
+
+
+
+class (Functor t k l) => Foldable t k l where
+ ffoldl :: ( ObjectPair k a b, ObjectPair l a (t b)
+ ) => k (a,b) a -> l (a,t b) a
+ foldMap :: ( Object k a, Object l (t a), Monoid m, Object k m, Object l m )
+ => (a `k` m) -> t a `l` m
+
+fold :: (Foldable t k k, Monoid m, Object k m, Object k (t m)) => t m `k` m
+fold = foldMap id
+
+newtype Endo' k a = Endo' { runEndo' :: k a a }
+instance (Category k, Object k a) => Monoid (Endo' k a) where
+ mempty = Endo' id
+ mappend (Endo' f) (Endo' g) = Endo' $ f . g
+
+newtype Monoidal_ (r :: * -> * -> *) (s :: * -> * -> *) (f :: * -> *) (u :: *)
+ = Monoidal { runMonoidal :: f u }
+instance ( Monoidal f k k, Function k
+ , u ~ UnitObject k, Monoid u
+ , ObjectPair k u u, ObjectPair k (f u) (f u), Object k (f u,f u)
+ ) => Monoid (Monoidal_ k k f u) where
+ mempty = memptyMdl
+ mappend = mappendMdl
+
+memptyMdl :: forall r s f u v . ( Monoidal f r s, Function s
+ , ObjectPair s u u, Monoid v
+ , u~UnitObject r, v~UnitObject s )
+ => Monoidal_ r s f u
+memptyMdl = Monoidal ((pureUnit :: s v (f u)) $ mempty)
+mappendMdl :: forall r s f u v . ( Monoidal f r s, Function s
+ , ObjectPair r u u, ObjectPair s (f u) (f u)
+ , Object s (f u, f u), Monoid v
+ , u~UnitObject r, v~UnitObject s )
+ => Monoidal_ r s f u -> Monoidal_ r s f u -> Monoidal_ r s f u
+mappendMdl (Monoidal x) (Monoidal y)
+ = Monoidal (combine $ (x, y))
+ where combine :: s (f u, f u) (f u)
+ combine = fzipWith detachUnit
+
+
+
+instance Foldable [] (->) (->) where
+ foldMap _ [] = mempty
+ foldMap f (x:xs) = f x <> foldMap f xs
+ ffoldl f = uncurry $ foldl (curry f)
+
+instance Foldable Maybe (->) (->) where
+ foldMap f Nothing = mempty
+ foldMap f (Just x) = f x
+ ffoldl _ (i,Nothing) = i
+ ffoldl f (i,Just a) = f(i,a)
+
+
+instance ( Foldable f s t, WellPointed s, WellPointed t
+ , Functor f (ConstrainedCategory s o) (ConstrainedCategory t o)
+ ) => Foldable f (ConstrainedCategory s o) (ConstrainedCategory t o) where
+ foldMap (ConstrainedMorphism f) = ConstrainedMorphism $ foldMap f
+ ffoldl (ConstrainedMorphism f) = ConstrainedMorphism $ ffoldl f
+
+-- | Despite the ridiculous-looking signature, this is in fact equivalent
+-- to 'Data.Foldable.traverse_' within Hask.
+traverse_ :: forall t k l o f a b uk ul .
+ ( Foldable t k l, PreArrow k, PreArrow l
+ , Monoidal f l l, Monoidal f k k
+ , ObjectPair l (f ul) (t a), ObjectPair k (f ul) a
+ , ObjectPair l ul (t a), ObjectPair l (t a) ul
+ , ObjectPair k b ul, Object k (f b)
+ , ObjectPair k (f ul) (f ul), ObjectPair k ul ul
+ , uk ~ UnitObject k, ul ~ UnitObject l, uk ~ ul
+ ) => a `k` f b -> t a `l` f ul
+traverse_ f = ffoldl q . first pureUnit . swap . attachUnit
+ where q :: k (f uk, a) (f uk)
+ q = fzipWith detachUnit . second (fmap terminal . f)
+
+-- | The distinction between 'mapM_' and 'traverse_' doesn't really make sense
+-- on grounds of 'Monoidal' / 'Applicative' vs 'Monad', but it has in fact some
+-- benefits to restrict this to endofunctors, to make the constraint list
+-- at least somewhat shorter.
+mapM_ :: forall t k o f a b u .
+ ( Foldable t k k, WellPointed k, Monoidal f k k
+ , u ~ UnitObject k
+ , ObjectPair k (f u) (t a), ObjectPair k (f u) a
+ , ObjectPair k u (t a), ObjectPair k (t a) u
+ , ObjectPair k (f u) (f u), ObjectPair k u u
+ , ObjectPair k b u, Object k (f b)
+ ) => a `k` f b -> t a `k` f u
+mapM_ = traverse_
+
+
+
+forM_ :: forall t k l f a b uk ul .
+ ( Foldable t k l, Monoidal f l l, Monoidal f k k
+ , Function l, Arrow k (->), Arrow l (->), ul ~ UnitObject l
+ , uk ~ UnitObject k, uk ~ ul
+ , ObjectPair l ul ul, ObjectPair l (f ul) (f ul)
+ , ObjectPair l (f ul) (t a), ObjectPair l ul (t a)
+ , ObjectPair l (t a) ul, ObjectPair l (f ul) a
+ , ObjectPair k b (f b), ObjectPair k b ul
+ , ObjectPair k uk uk, ObjectPair k (f uk) a, ObjectPair k (f uk) (f uk)
+ ) => t a -> a `k` f b -> f uk
+forM_ v f = traverse_ f $ v
+
+
+sequence_ :: forall t k l m a b uk ul .
+ ( Foldable t k l, Arrow k (->), Arrow l (->)
+ , uk ~ UnitObject k, ul ~ UnitObject l, uk ~ ul
+ , Monoidal m k k, Monoidal m l l
+ , ObjectPair k a uk, ObjectPair k (t (m a)) uk
+ , ObjectPair k uk uk, ObjectPair k (m uk) (m uk), ObjectPair k (t (m a)) ul
+ , ObjectPair l (m ul) (t (m a)), ObjectPair l ul (t (m a))
+ , ObjectPair l (m uk) (t (m a)), ObjectPair l (t (m a)) ul
+ , ObjectPair k (m uk) (m a)
+ ) => t (m a) `l` m uk
+sequence_ = traverse_ id
+
+
+
+concatMap :: (Foldable f k l, Object k a, Object k [b], Object l (f a), Object l [b])
+ => a `k` [b] -> f a `l` [b]
+concatMap = foldMap
+
diff --git a/Data/Traversable/Constrained.hs b/Data/Traversable/Constrained.hs
new file mode 100644
--- /dev/null
+++ b/Data/Traversable/Constrained.hs
@@ -0,0 +1,91 @@
+-- |
+-- Module : Data.Traversable.Constrained
+-- Copyright : (c) 2014 Justus Sagemüller
+-- License : GPL v3 (see COPYING)
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+--
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TupleSections #-}
+
+
+module Data.Traversable.Constrained
+ ( module Control.Applicative.Constrained
+ , Traversable(..)
+ , sequence, mapM, forM
+ ) where
+
+
+import Control.Category.Constrained
+import Control.Applicative.Constrained
+
+import Prelude hiding (
+ id, const, (.), ($)
+ , Functor(..)
+ , uncurry, curry
+ , mapM, mapM_, sequence
+ )
+import qualified Control.Category.Hask as Hask
+import qualified Control.Arrow as A
+
+import Control.Arrow.Constrained
+
+import Data.Monoid
+
+
+
+
+class (Category k, Category l, Functor s l l, Functor t k k)
+ => Traversable s t k l | s k l -> t, t k l -> s, s t k -> l, s t l -> k where
+ traverse :: ( Monoidal f k l, Object l a, Object l (s a)
+ , ObjectPair k b (t b), ObjectPair l (f b) (f (t b))
+ , ObjectPoint k (t b)
+ ) => a `l` f b -> s a `l` f (t b)
+
+sequence :: ( Traversable t t k k, Monoidal f k k
+ , ObjectPair k a (t a), ObjectPair k (f a) (f (t a))
+ , Object k (t (f a))
+ , ObjectPoint k (t a)
+ ) => t (f a) `k` f (t a)
+sequence = traverse id
+
+instance (Arrow k (->), WellPointed k, Function k, Functor [] k k)
+ => Traversable [] [] k k where
+ traverse f = arr mM
+ where mM [] = constPure [] `inCategoryOf` f $ mempty
+ mM (x:xs) = fzipWith (arr $ uncurry(:)) `inCategoryOf` f
+ $ (f $ x, mM xs)
+
+instance (Arrow k (->), WellPointed k, Function k, Functor Maybe k k)
+ => Traversable Maybe Maybe k k where
+ traverse f = arr mM
+ where mM Nothing = constPure Nothing `inCategoryOf` f $ mempty
+ mM (Just x) = fmap (arr Just) . f $ x
+
+-- data Stupid a = Stupid a
+-- instance Functor Stupid (ConstrainedCategory (->) Num) (->) where
+-- fmap (Stupid (ConstrainedMorphism f)) (Stupid a) = Stupid (f a)
+--
+
+-- | 'traverse', restricted to endofunctors.
+mapM :: ( Traversable t t k k, Monoidal m k k
+ , Object k a, Object k (t a), ObjectPair k b (t b), ObjectPair k (m b) (m (t b))
+ , ObjectPoint k (t b)
+ ) => a `k` m b -> t a `k` m (t b)
+mapM = traverse
+
+-- | Flipped version of 'traverse' / 'mapM'.
+forM :: forall s t k m a b l .
+ ( Traversable s t k l, Monoidal m k l, Function l
+ , Object k b, Object k (t b), ObjectPair k b (t b)
+ , Object l a, Object l (s a), ObjectPair l (m b) (m (t b))
+ , ObjectPoint k (t b)
+ ) => s a -> (a `l` m b) -> m (t b)
+forM v f = traverse f $ v
+
+
diff --git a/Setup.hs b/Setup.hs
new file mode 100644
--- /dev/null
+++ b/Setup.hs
@@ -0,0 +1,6 @@
+module Main (main) where
+
+import Distribution.Simple
+
+main :: IO ()
+main = defaultMain
diff --git a/constrained-categories.cabal b/constrained-categories.cabal
new file mode 100644
--- /dev/null
+++ b/constrained-categories.cabal
@@ -0,0 +1,58 @@
+Name: constrained-categories
+Version: 0.1.0.0
+Category: control
+Synopsis: Constrained clones of the category-theory type classes, using ConstraintKinds.
+Description: Haskell has, and makes great use of, powerful facilities from category
+ theory – basically various variants of functors.
+ .
+ However, all those are just endofunctors in Hask, the category of
+ all Haskell types with functions as morphisms. Which is sufficient
+ for container / control structures that you want to be able to handle
+ any type of data, but otherwise it's a bit limiting, seeing as
+ there are (in maths, science etc.) many categories that cannot properly
+ be represented this way. Commonly used libraries such as
+ thus make
+ little notion of the fact that the objects they deal with actually
+ form a category, instead defining just specialised versions of
+ the operations.
+ .
+ This library generalises functors etc. to a much wider class of
+ categories, by allowing for constraints on objects (so these can have
+ extra properties required). At the same time, we try to keep as close
+ as possible to the well-known Haskell type class hierarchies rather
+ than exactly adopting the mathematicians' notions.
+ .
+ Consider the README file, the examples, and/or the documentation to
+ "Control.Category.Constrained" for how to make use of this.
+License: GPL-3
+License-file: COPYING
+Author: Justus Sagemüller
+Maintainer: (@) sagemuej $ smail.uni-koeln.de
+Homepage: https://github.com/leftaroundabout/constrained-categories
+Build-Type: Simple
+Cabal-Version: >=1.10
+
+source-repository head
+ type: git
+ location: git://github.com/leftaroundabout/constrained-categories.git
+
+Library
+ Default-Language: Haskell2010
+ Build-Depends: base>=4.6 && <5
+ , tagged
+ , void
+ Default-Extensions: ConstraintKinds
+ TypeFamilies
+ FlexibleInstances
+ UndecidableInstances
+ Trustworthy
+ Exposed-modules: Control.Category.Constrained
+ Control.Functor.Constrained
+ Control.Applicative.Constrained
+ Control.Arrow.Constrained
+ Control.Monad.Constrained
+ Control.Category.Hask
+ Control.Category.Constrained.Prelude
+ Data.Foldable.Constrained
+ Data.Traversable.Constrained
+