-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Category theory
--   
--   Data-category is a collection of categories, and some categorical
--   constructions on them.
--   
--   You can restrict the types of the objects of your category by using a
--   GADT for the arrow type. To be able to proof to the compiler that a
--   type is an object in some category, objects also need to be
--   represented at the value level. The corresponding identity arrow of
--   the object is used for that.
--   
--   See the <a>Boolean</a> and <a>Product</a> categories for some
--   examples.
--   
--   Note: Strictly speaking this package defines Hask-enriched categories,
--   not ordinary categories (which are Set-enriched.) In practice this
--   means we are allowed to ignore <a>undefined</a> (f.e. when talking
--   about uniqueness of morphisms), and we can treat the categories as
--   normal categories.
@package data-category
@version 0.7


module Data.Category

-- | An instance of <tt>Category k</tt> declares the arrow <tt>k</tt> as a
--   category.
class Category k
src :: Category k => k a b -> Obj k a
tgt :: Category k => k a b -> Obj k b
(.) :: Category k => k b c -> k a b -> k a c

-- | Whenever objects are required at value level, they are represented by
--   their identity arrows.
type Obj k a = k a a
data Op k a b
Op :: k b a -> Op k a b
[unOp] :: Op k a b -> k b a
instance Data.Category.Category (->)
instance Data.Category.Category k => Data.Category.Category (Data.Category.Op k)


module Data.Category.Product
data (:**:) :: (* -> * -> *) -> (* -> * -> *) -> * -> * -> *
[:**:] :: c1 a1 b1 -> c2 a2 b2 -> (:**:) c1 c2 (a1, a2) (b1, b2)

-- | The product category of category <tt>c1</tt> and <tt>c2</tt>.
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Category (c1 Data.Category.Product.:**: c2)


module Data.Category.Functor

-- | Functors are arrows in the category Cat.
data Cat :: * -> * -> *
[CatA] :: (Functor ftag, Category (Dom ftag), Category (Cod ftag)) => ftag -> Cat (CatW (Dom ftag)) (CatW (Cod ftag))

-- | We need a wrapper here because objects need to be of kind *, and
--   categories are of kind * -&gt; * -&gt; *.
data CatW :: (* -> * -> *) -> *

-- | Functors map objects and arrows.
class (Category (Dom ftag), Category (Cod ftag)) => Functor ftag where type Dom ftag :: * -> * -> * type Cod ftag :: * -> * -> * type (:%) ftag a :: * where {
    type family Dom ftag :: * -> * -> *;
    type family Cod ftag :: * -> * -> *;
    type family (:%) ftag a :: *;
}

-- | <tt>%</tt> maps arrows.
(%) :: Functor ftag => ftag -> Dom ftag a b -> Cod ftag (ftag :% a) (ftag :% b)
data Id (k :: * -> * -> *)
Id :: Id
data (:.:) g h
[:.:] :: (Functor g, Functor h, Cod h ~ Dom g) => g -> h -> g :.: h
data Const (c1 :: * -> * -> *) (c2 :: * -> * -> *) x
[Const] :: Obj c2 x -> Const c1 c2 x

-- | The constant functor with the same domain and codomain as f.
type ConstF f = Const (Dom f) (Cod f)
data Opposite f
[Opposite] :: Functor f => f -> Opposite f
data OpOp (k :: * -> * -> *)
OpOp :: OpOp
data OpOpInv (k :: * -> * -> *)
OpOpInv :: OpOpInv
data Proj1 (c1 :: * -> * -> *) (c2 :: * -> * -> *)
Proj1 :: Proj1
data Proj2 (c1 :: * -> * -> *) (c2 :: * -> * -> *)
Proj2 :: Proj2
data (:***:) f1 f2
(:***:) :: f1 -> f2 -> (:***:) f1 f2
data DiagProd (k :: * -> * -> *)
DiagProd :: DiagProd
type Tuple1 c1 c2 a = (Const c2 c1 a :***: Id c2) :.: DiagProd c2

-- | <a>Tuple1</a> tuples with a fixed object on the left.
tuple1 :: (Category c1, Category c2) => Obj c1 a -> Tuple1 c1 c2 a
type Tuple2 c1 c2 a = Swap c2 c1 :.: Tuple1 c2 c1 a

-- | <a>Tuple2</a> tuples with a fixed object on the right.
tuple2 :: (Category c1, Category c2) => Obj c2 a -> Tuple2 c1 c2 a
type Swap (c1 :: * -> * -> *) (c2 :: * -> * -> *) = (Proj2 c1 c2 :***: Proj1 c1 c2) :.: DiagProd (c1 :**: c2)

-- | <a>swap</a> swaps the 2 categories of the product of categories.
swap :: (Category c1, Category c2) => Swap c1 c2
data Hom (k :: * -> * -> *)
Hom :: Hom
type (:*-:) x k = Hom k :.: Tuple1 (Op k) k x

-- | The covariant functor Hom(X,--)
homX_ :: Category k => Obj k x -> x :*-: k
type (:-*:) k x = Hom k :.: Tuple2 (Op k) k x

-- | The contravariant functor Hom(--,X)
hom_X :: Category k => Obj k x -> k :-*: x
type HomF f g = Hom (Cod f) :.: (Opposite f :***: g)
homF :: (Functor f, Functor g, Cod f ~ Cod g) => f -> g -> HomF f g
type Star f = HomF (Id (Cod f)) f
star :: Functor f => f -> Star f
type Costar f = HomF f (Id (Cod f))
costar :: Functor f => f -> Costar f
instance Data.Category.Category Data.Category.Functor.Cat
instance Data.Category.Category k => Data.Category.Functor.Functor (Data.Category.Functor.Id k)
instance (Data.Category.Category (Data.Category.Functor.Cod g), Data.Category.Category (Data.Category.Functor.Dom h)) => Data.Category.Functor.Functor (g Data.Category.Functor.:.: h)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.Functor.Const c1 c2 x)
instance (Data.Category.Category (Data.Category.Functor.Dom f), Data.Category.Category (Data.Category.Functor.Cod f)) => Data.Category.Functor.Functor (Data.Category.Functor.Opposite f)
instance Data.Category.Category k => Data.Category.Functor.Functor (Data.Category.Functor.OpOp k)
instance Data.Category.Category k => Data.Category.Functor.Functor (Data.Category.Functor.OpOpInv k)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.Functor.Proj1 c1 c2)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.Functor.Proj2 c1 c2)
instance (Data.Category.Functor.Functor f1, Data.Category.Functor.Functor f2) => Data.Category.Functor.Functor (f1 Data.Category.Functor.:***: f2)
instance Data.Category.Category k => Data.Category.Functor.Functor (Data.Category.Functor.DiagProd k)
instance Data.Category.Category k => Data.Category.Functor.Functor (Data.Category.Functor.Hom k)


module Data.Category.NaturalTransformation

-- | <tt>f :~&gt; g</tt> is a natural transformation from functor f to
--   functor g.
type (:~>) f g = forall c d. (c ~ Dom f, c ~ Dom g, d ~ Cod f, d ~ Cod g) => Nat c d f g

-- | A component for an object <tt>z</tt> is an arrow from <tt>F z</tt> to
--   <tt>G z</tt>.
type Component f g z = Cod f (f :% z) (g :% z)

-- | 'n ! a' returns the component for the object <tt>a</tt> of a natural
--   transformation <tt>n</tt>. This can be generalized to any arrow
--   (instead of just identity arrows).
(!) :: (Category c, Category d) => Nat c d f g -> c a b -> d (f :% a) (g :% b)
infixl 9 !

-- | Horizontal composition of natural transformations.
o :: (Category c, Category d, Category e) => Nat d e j k -> Nat c d f g -> Nat c e (j :.: f) (k :.: g)

-- | The identity natural transformation of a functor.
natId :: Functor f => f -> Nat (Dom f) (Cod f) f f
srcF :: Nat c d f g -> f
tgtF :: Nat c d f g -> g

-- | Natural transformations are built up of components, one for each
--   object <tt>z</tt> in the domain category of <tt>f</tt> and <tt>g</tt>.
data Nat :: (* -> * -> *) -> (* -> * -> *) -> * -> * -> *
[Nat] :: (Functor f, Functor g, c ~ Dom f, c ~ Dom g, d ~ Cod f, d ~ Cod g) => f -> g -> (forall z. Obj c z -> Component f g z) -> Nat c d f g

-- | The category of endofunctors.
type Endo k = Nat k k
type Presheaves k = Nat (Op k) (->)
type Profunctors c d = Nat (Op d :**: c) (->)
compAssoc :: (Functor f, Functor g, Functor h, Dom f ~ Cod g, Dom g ~ Cod h) => f -> g -> h -> Nat (Dom h) (Cod f) ((f :.: g) :.: h) (f :.: (g :.: h))
compAssocInv :: (Functor f, Functor g, Functor h, Dom f ~ Cod g, Dom g ~ Cod h) => f -> g -> h -> Nat (Dom h) (Cod f) (f :.: (g :.: h)) ((f :.: g) :.: h)
idPrecomp :: Functor f => f -> Nat (Dom f) (Cod f) (f :.: Id (Dom f)) f
idPrecompInv :: Functor f => f -> Nat (Dom f) (Cod f) f (f :.: Id (Dom f))
idPostcomp :: Functor f => f -> Nat (Dom f) (Cod f) (Id (Cod f) :.: f) f
idPostcompInv :: Functor f => f -> Nat (Dom f) (Cod f) f (Id (Cod f) :.: f)
constPrecompIn :: Nat j d (f :.: Const j c x) g -> Nat j d (Const j d (f :% x)) g
constPrecompOut :: Nat j d f (g :.: Const j c x) -> Nat j d f (Const j d (g :% x))
constPostcompIn :: Nat j d (Const k d x :.: f) g -> Nat j d (Const j d x) g
constPostcompOut :: Nat j d f (Const k d x :.: g) -> Nat j d f (Const j d x)
data FunctorCompose (c :: * -> * -> *) (d :: * -> * -> *) (e :: * -> * -> *)
FunctorCompose :: FunctorCompose

-- | Composition of endofunctors is a functor.
type EndoFunctorCompose k = FunctorCompose k k k

-- | <tt>Precompose f e</tt> is the functor such that <tt>Precompose f e :%
--   g = g :.: f</tt>, for functors <tt>g</tt> that compose with <tt>f</tt>
--   and with codomain <tt>e</tt>.
type Precompose f e = FunctorCompose (Dom f) (Cod f) e :.: Tuple2 (Nat (Cod f) e) (Nat (Dom f) (Cod f)) f
precompose :: (Category e, Functor f) => f -> Precompose f e

-- | <tt>Postcompose f c</tt> is the functor such that <tt>Postcompose f c
--   :% g = f :.: g</tt>, for functors <tt>g</tt> that compose with
--   <tt>f</tt> and with domain <tt>c</tt>.
type Postcompose f c = FunctorCompose c (Dom f) (Cod f) :.: Tuple1 (Nat (Dom f) (Cod f)) (Nat c (Dom f)) f
postcompose :: (Category e, Functor f) => f -> Postcompose f e
data Wrap f h
Wrap :: f -> h -> Wrap f h
data Apply (c1 :: * -> * -> *) (c2 :: * -> * -> *)
Apply :: Apply
data Tuple (c1 :: * -> * -> *) (c2 :: * -> * -> *)
Tuple :: Tuple
instance Data.Category.Category d => Data.Category.Category (Data.Category.NaturalTransformation.Nat c d)
instance (Data.Category.Category c, Data.Category.Category d, Data.Category.Category e) => Data.Category.Functor.Functor (Data.Category.NaturalTransformation.FunctorCompose c d e)
instance (Data.Category.Functor.Functor f, Data.Category.Functor.Functor h) => Data.Category.Functor.Functor (Data.Category.NaturalTransformation.Wrap f h)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.NaturalTransformation.Apply c1 c2)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.NaturalTransformation.Tuple c1 c2)


module Data.Category.RepresentableFunctor
data Representable f repObj
Representable :: f -> Obj (Dom f) repObj -> (forall k z. (Dom f ~ k, Cod f ~ (->)) => Obj k z -> f :% z -> k repObj z) -> (forall k. (Dom f ~ k, Cod f ~ (->)) => f :% repObj) -> Representable f repObj
[representedFunctor] :: Representable f repObj -> f
[representingObject] :: Representable f repObj -> Obj (Dom f) repObj
[represent] :: Representable f repObj -> forall k z. (Dom f ~ k, Cod f ~ (->)) => Obj k z -> f :% z -> k repObj z
[universalElement] :: Representable f repObj -> forall k. (Dom f ~ k, Cod f ~ (->)) => f :% repObj
unrepresent :: (Functor f, Dom f ~ k, Cod f ~ (->)) => Representable f repObj -> k repObj z -> f :% z
covariantHomRepr :: Category k => Obj k x -> Representable (x :*-: k) x
contravariantHomRepr :: Category k => Obj k x -> Representable (k :-*: x) x
type InitialUniversal x u a = Representable ((x :*-: Cod u) :.: u) a

-- | An initial universal property, a universal morphism from x to u.
initialUniversal :: Functor u => u -> Obj (Dom u) a -> Cod u x (u :% a) -> (forall y. Obj (Dom u) y -> Cod u x (u :% y) -> Dom u a y) -> InitialUniversal x u a
type TerminalUniversal x u a = Representable ((Cod u :-*: x) :.: Opposite u) a

-- | A terminal universal property, a universal morphism from u to x.
terminalUniversal :: Functor u => u -> Obj (Dom u) a -> Cod u (u :% a) x -> (forall y. Obj (Dom u) y -> Cod u (u :% y) x -> Dom u y a) -> TerminalUniversal x u a


module Data.Category.Adjunction
data Adjunction c d f g
Adjunction :: f -> g -> Profunctors c d (Costar f) (Star g) -> Profunctors c d (Star g) (Costar f) -> Adjunction c d f g
[leftAdjoint] :: Adjunction c d f g -> f
[rightAdjoint] :: Adjunction c d f g -> g
[leftAdjunctN] :: Adjunction c d f g -> Profunctors c d (Costar f) (Star g)
[rightAdjunctN] :: Adjunction c d f g -> Profunctors c d (Star g) (Costar f)
mkAdjunction :: (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall a b. Obj d a -> c (f :% a) b -> d a (g :% b)) -> (forall a b. Obj c b -> d a (g :% b) -> c (f :% a) b) -> Adjunction c d f g
mkAdjunctionUnits :: (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall a. Obj d a -> Component (Id d) (g :.: f) a) -> (forall a. Obj c a -> Component (f :.: g) (Id c) a) -> Adjunction c d f g
leftAdjunct :: Adjunction c d f g -> Obj d a -> c (f :% a) b -> d a (g :% b)
rightAdjunct :: Adjunction c d f g -> Obj c b -> d a (g :% b) -> c (f :% a) b
adjunctionUnit :: Adjunction c d f g -> Nat d d (Id d) (g :.: f)
adjunctionCounit :: Adjunction c d f g -> Nat c c (f :.: g) (Id c)
idAdj :: Category k => Adjunction k k (Id k) (Id k)
composeAdj :: Adjunction d e f g -> Adjunction c d f' g' -> Adjunction c e (f' :.: f) (g :.: g')
data AdjArrow c d
[AdjArrow] :: (Category c, Category d) => Adjunction c d f g -> AdjArrow (CatW c) (CatW d)
initialPropAdjunction :: forall f g c d. (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall y. Obj d y -> InitialUniversal y g (f :% y)) -> Adjunction c d f g
terminalPropAdjunction :: forall f g c d. (Functor f, Functor g, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => f -> g -> (forall x. Obj c x -> TerminalUniversal x f (g :% x)) -> Adjunction c d f g
adjunctionInitialProp :: Adjunction c d f g -> Obj d y -> InitialUniversal y g (f :% y)
adjunctionTerminalProp :: Adjunction c d f g -> Obj c x -> TerminalUniversal x f (g :% x)
precomposeAdj :: Category e => Adjunction c d f g -> Adjunction (Nat c e) (Nat d e) (Precompose g e) (Precompose f e)
postcomposeAdj :: Category e => Adjunction c d f g -> Adjunction (Nat e c) (Nat e d) (Postcompose f e) (Postcompose g e)
contAdj :: Adjunction (Op (->)) (->) (Opposite ((->) :-*: r) :.: OpOpInv (->)) ((->) :-*: r)
instance Data.Category.Category Data.Category.Adjunction.AdjArrow


module Data.Category.Unit
data Unit a b
[Unit] :: Unit () ()

-- | <a>Unit</a> is the category with one object.
instance Data.Category.Category Data.Category.Unit.Unit


module Data.Category.Coproduct
data I1 a
data I2 a
data (:++:) :: (* -> * -> *) -> (* -> * -> *) -> * -> * -> *
[I1] :: c1 a1 b1 -> (:++:) c1 c2 (I1 a1) (I1 b1)
[I2] :: c2 a2 b2 -> (:++:) c1 c2 (I2 a2) (I2 b2)

-- | The coproduct category of categories <tt>c1</tt> and <tt>c2</tt>.
data Inj1 (c1 :: * -> * -> *) (c2 :: * -> * -> *)
Inj1 :: Inj1

-- | <a>Inj1</a> is a functor which injects into the left category.
data Inj2 (c1 :: * -> * -> *) (c2 :: * -> * -> *)
Inj2 :: Inj2

-- | <a>Inj2</a> is a functor which injects into the right category.
data (:+++:) f1 f2
(:+++:) :: f1 -> f2 -> (:+++:) f1 f2

-- | <tt>f1 :+++: f2</tt> is the coproduct of the functors <tt>f1</tt> and
--   <tt>f2</tt>.
data CodiagCoprod (k :: * -> * -> *)
CodiagCoprod :: CodiagCoprod

-- | <a>CodiagCoprod</a> is the codiagonal functor for coproducts.
data Cotuple1 (c1 :: * -> * -> *) (c2 :: * -> * -> *) a
Cotuple1 :: (Obj c1 a) -> Cotuple1 a

-- | <a>Cotuple1</a> projects out to the left category, replacing a value
--   from the right category with a fixed object.
data Cotuple2 (c1 :: * -> * -> *) (c2 :: * -> * -> *) a
Cotuple2 :: (Obj c2 a) -> Cotuple2 a

-- | <a>Cotuple2</a> projects out to the right category, replacing a value
--   from the left category with a fixed object.
data (:>>:) :: (* -> * -> *) -> (* -> * -> *) -> * -> * -> *
[I1A] :: c1 a1 b1 -> (:>>:) c1 c2 (I1 a1) (I1 b1)
[I12] :: Obj c1 a -> Obj c2 b -> (:>>:) c1 c2 (I1 a) (I2 b)
[I2A] :: c2 a2 b2 -> (:>>:) c1 c2 (I2 a2) (I2 b2)

-- | The directed coproduct category of categories <tt>c1</tt> and
--   <tt>c2</tt>.
data NatAsFunctor f g
NatAsFunctor :: (Nat (Dom f) (Cod f) f g) -> NatAsFunctor f g

-- | A natural transformation <tt>Nat c d</tt> is isomorphic to a functor
--   from <tt>c :**: 2</tt> to <tt>d</tt>.
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Category (c1 Data.Category.Coproduct.:++: c2)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.Coproduct.Inj1 c1 c2)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.Coproduct.Inj2 c1 c2)
instance (Data.Category.Functor.Functor f1, Data.Category.Functor.Functor f2) => Data.Category.Functor.Functor (f1 Data.Category.Coproduct.:+++: f2)
instance Data.Category.Category k => Data.Category.Functor.Functor (Data.Category.Coproduct.CodiagCoprod k)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.Coproduct.Cotuple1 c1 c2 a1)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Functor.Functor (Data.Category.Coproduct.Cotuple2 c1 c2 a2)
instance (Data.Category.Category c1, Data.Category.Category c2) => Data.Category.Category (c1 Data.Category.Coproduct.:>>: c2)
instance (Data.Category.Functor.Functor f, Data.Category.Functor.Functor g, Data.Category.Functor.Dom f ~ Data.Category.Functor.Dom g, Data.Category.Functor.Cod f ~ Data.Category.Functor.Cod g) => Data.Category.Functor.Functor (Data.Category.Coproduct.NatAsFunctor f g)


module Data.Category.Void
data Void a b
magic :: Void a b -> x

-- | <a>Void</a> is the category with no objects.
voidNat :: (Functor f, Functor g, Dom f ~ Void, Dom g ~ Void, Cod f ~ d, Cod g ~ d) => f -> g -> Nat Void d f g
data Magic (k :: * -> * -> *)
Magic :: Magic

-- | Since there is nothing to map in <a>Void</a>, there's a functor from
--   it to any other category.
instance Data.Category.Category Data.Category.Void.Void
instance Data.Category.Category k => Data.Category.Functor.Functor (Data.Category.Void.Magic k)


module Data.Category.Limit
data Diag :: (* -> * -> *) -> (* -> * -> *) -> *
[Diag] :: Diag j k

-- | The diagonal functor with the same domain and codomain as <tt>f</tt>.
type DiagF f = Diag (Dom f) (Cod f)

-- | A cone from N to F is a natural transformation from the constant
--   functor to N to F.
type Cone f n = Nat (Dom f) (Cod f) (ConstF f n) f

-- | A co-cone from F to N is a natural transformation from F to the
--   constant functor to N.
type Cocone f n = Nat (Dom f) (Cod f) f (ConstF f n)

-- | The vertex (or apex) of a cone.
coneVertex :: Cone f n -> Obj (Cod f) n

-- | The vertex (or apex) of a co-cone.
coconeVertex :: Cocone f n -> Obj (Cod f) n

-- | Limits in a category <tt>k</tt> by means of a diagram of type
--   <tt>j</tt>, which is a functor from <tt>j</tt> to <tt>k</tt>.
type Limit f = LimitFam (Dom f) (Cod f) f

-- | An instance of <tt>HasLimits j k</tt> says that <tt>k</tt> has all
--   limits of type <tt>j</tt>.
class (Category j, Category k) => HasLimits j k

-- | <a>limit</a> returns the limiting cone for a functor <tt>f</tt>.
limit :: HasLimits j k => Obj (Nat j k) f -> Cone f (Limit f)

-- | <a>limitFactorizer</a> shows that the limiting cone is universal –
--   i.e. any other cone of <tt>f</tt> factors through it by returning the
--   morphism between the vertices of the cones.
limitFactorizer :: HasLimits j k => Obj (Nat j k) f -> (forall n. Cone f n -> k n (Limit f))
data LimitFunctor (j :: * -> * -> *) (k :: * -> * -> *)
LimitFunctor :: LimitFunctor

-- | The limit functor is right adjoint to the diagonal functor.
limitAdj :: forall j k. HasLimits j k => Adjunction (Nat j k) k (Diag j k) (LimitFunctor j k)
rightAdjointPreservesLimits :: (HasLimits j c, HasLimits j d) => Adjunction c d f g -> Obj (Nat j c) t -> d (Limit (g :.: t)) (g :% Limit t)
rightAdjointPreservesLimitsInv :: (HasLimits j c, HasLimits j d) => Obj (Nat c d) g -> Obj (Nat j c) t -> d (g :% Limit t) (Limit (g :.: t))

-- | Colimits in a category <tt>k</tt> by means of a diagram of type
--   <tt>j</tt>, which is a functor from <tt>j</tt> to <tt>k</tt>.
type Colimit f = ColimitFam (Dom f) (Cod f) f

-- | An instance of <tt>HasColimits j k</tt> says that <tt>k</tt> has all
--   colimits of type <tt>j</tt>.
class (Category j, Category k) => HasColimits j k

-- | <a>colimit</a> returns the limiting co-cone for a functor <tt>f</tt>.
colimit :: HasColimits j k => Obj (Nat j k) f -> Cocone f (Colimit f)

-- | <a>colimitFactorizer</a> shows that the limiting co-cone is universal
--   – i.e. any other co-cone of <tt>f</tt> factors through it by returning
--   the morphism between the vertices of the cones.
colimitFactorizer :: HasColimits j k => Obj (Nat j k) f -> (forall n. Cocone f n -> k (Colimit f) n)
data ColimitFunctor (j :: * -> * -> *) (k :: * -> * -> *)
ColimitFunctor :: ColimitFunctor

-- | The colimit functor is left adjoint to the diagonal functor.
colimitAdj :: forall j k. HasColimits j k => Adjunction k (Nat j k) (ColimitFunctor j k) (Diag j k)
leftAdjointPreservesColimits :: (HasColimits j c, HasColimits j d) => Adjunction c d f g -> Obj (Nat j d) t -> c (f :% Colimit t) (Colimit (f :.: t))
leftAdjointPreservesColimitsInv :: (HasColimits j c, HasColimits j d) => Obj (Nat d c) f -> Obj (Nat j d) t -> c (Colimit (f :.: t)) (f :% Colimit t)
class Category k => HasTerminalObject k where type TerminalObject k :: * where {
    type family TerminalObject k :: *;
}
terminalObject :: HasTerminalObject k => Obj k (TerminalObject k)
terminate :: HasTerminalObject k => Obj k a -> k a (TerminalObject k)
class Category k => HasInitialObject k where type InitialObject k :: * where {
    type family InitialObject k :: *;
}
initialObject :: HasInitialObject k => Obj k (InitialObject k)
initialize :: HasInitialObject k => Obj k a -> k (InitialObject k) a
data Zero
class Category k => HasBinaryProducts k where type BinaryProduct (k :: * -> * -> *) x y :: * l *** r = (l . proj1 (src l) (src r)) &&& (r . proj2 (src l) (src r)) where {
    type family BinaryProduct (k :: * -> * -> *) x y :: *;
}
proj1 :: HasBinaryProducts k => Obj k x -> Obj k y -> k (BinaryProduct k x y) x
proj2 :: HasBinaryProducts k => Obj k x -> Obj k y -> k (BinaryProduct k x y) y
(&&&) :: HasBinaryProducts k => k a x -> k a y -> k a (BinaryProduct k x y)
(***) :: HasBinaryProducts k => k a1 b1 -> k a2 b2 -> k (BinaryProduct k a1 a2) (BinaryProduct k b1 b2)
data ProductFunctor (k :: * -> * -> *)
ProductFunctor :: ProductFunctor
data (:*:) p q
[:*:] :: (Functor p, Functor q, Dom p ~ Dom q, Cod p ~ k, Cod q ~ k, HasBinaryProducts k) => p -> q -> p :*: q

-- | A specialisation of the limit adjunction to products.
prodAdj :: HasBinaryProducts k => Adjunction (k :**: k) k (DiagProd k) (ProductFunctor k)
class Category k => HasBinaryCoproducts k where type BinaryCoproduct (k :: * -> * -> *) x y :: * l +++ r = (inj1 (tgt l) (tgt r) . l) ||| (inj2 (tgt l) (tgt r) . r) where {
    type family BinaryCoproduct (k :: * -> * -> *) x y :: *;
}
inj1 :: HasBinaryCoproducts k => Obj k x -> Obj k y -> k x (BinaryCoproduct k x y)
inj2 :: HasBinaryCoproducts k => Obj k x -> Obj k y -> k y (BinaryCoproduct k x y)
(|||) :: HasBinaryCoproducts k => k x a -> k y a -> k (BinaryCoproduct k x y) a
(+++) :: HasBinaryCoproducts k => k a1 b1 -> k a2 b2 -> k (BinaryCoproduct k a1 a2) (BinaryCoproduct k b1 b2)
data CoproductFunctor (k :: * -> * -> *)
CoproductFunctor :: CoproductFunctor
data (:+:) p q
[:+:] :: (Functor p, Functor q, Dom p ~ Dom q, Cod p ~ k, Cod q ~ k, HasBinaryCoproducts k) => p -> q -> p :+: q

-- | A specialisation of the colimit adjunction to coproducts.
coprodAdj :: HasBinaryCoproducts k => Adjunction k (k :**: k) (CoproductFunctor k) (DiagProd k)
instance (Data.Category.Category j, Data.Category.Category k) => Data.Category.Functor.Functor (Data.Category.Limit.Diag j k)
instance Data.Category.Limit.HasLimits j k => Data.Category.Functor.Functor (Data.Category.Limit.LimitFunctor j k)
instance Data.Category.Limit.HasColimits j k => Data.Category.Functor.Functor (Data.Category.Limit.ColimitFunctor j k)
instance Data.Category.Limit.HasTerminalObject k => Data.Category.Limit.HasLimits Data.Category.Void.Void k
instance Data.Category.Limit.HasTerminalObject (->)
instance Data.Category.Limit.HasTerminalObject Data.Category.Functor.Cat
instance (Data.Category.Category c, Data.Category.Limit.HasTerminalObject d) => Data.Category.Limit.HasTerminalObject (Data.Category.NaturalTransformation.Nat c d)
instance Data.Category.Limit.HasTerminalObject Data.Category.Unit.Unit
instance (Data.Category.Limit.HasTerminalObject c1, Data.Category.Limit.HasTerminalObject c2) => Data.Category.Limit.HasTerminalObject (c1 Data.Category.Product.:**: c2)
instance (Data.Category.Category c1, Data.Category.Limit.HasTerminalObject c2) => Data.Category.Limit.HasTerminalObject (c1 Data.Category.Coproduct.:>>: c2)
instance Data.Category.Limit.HasInitialObject k => Data.Category.Limit.HasColimits Data.Category.Void.Void k
instance Data.Category.Limit.HasInitialObject (->)
instance Data.Category.Limit.HasInitialObject Data.Category.Functor.Cat
instance (Data.Category.Category c, Data.Category.Limit.HasInitialObject d) => Data.Category.Limit.HasInitialObject (Data.Category.NaturalTransformation.Nat c d)
instance (Data.Category.Limit.HasInitialObject c1, Data.Category.Limit.HasInitialObject c2) => Data.Category.Limit.HasInitialObject (c1 Data.Category.Product.:**: c2)
instance Data.Category.Limit.HasInitialObject Data.Category.Unit.Unit
instance (Data.Category.Limit.HasInitialObject c1, Data.Category.Category c2) => Data.Category.Limit.HasInitialObject (c1 Data.Category.Coproduct.:>>: c2)
instance (Data.Category.Limit.HasLimits i k, Data.Category.Limit.HasLimits j k, Data.Category.Limit.HasBinaryProducts k) => Data.Category.Limit.HasLimits (i Data.Category.Coproduct.:++: j) k
instance Data.Category.Limit.HasBinaryProducts (->)
instance Data.Category.Limit.HasBinaryProducts Data.Category.Functor.Cat
instance Data.Category.Limit.HasBinaryProducts Data.Category.Unit.Unit
instance (Data.Category.Limit.HasBinaryProducts c1, Data.Category.Limit.HasBinaryProducts c2) => Data.Category.Limit.HasBinaryProducts (c1 Data.Category.Product.:**: c2)
instance (Data.Category.Limit.HasBinaryProducts c1, Data.Category.Limit.HasBinaryProducts c2) => Data.Category.Limit.HasBinaryProducts (c1 Data.Category.Coproduct.:>>: c2)
instance Data.Category.Limit.HasBinaryProducts k => Data.Category.Functor.Functor (Data.Category.Limit.ProductFunctor k)
instance (Data.Category.Category (Data.Category.Functor.Dom p), Data.Category.Category (Data.Category.Functor.Cod p)) => Data.Category.Functor.Functor (p Data.Category.Limit.:*: q)
instance (Data.Category.Category c, Data.Category.Limit.HasBinaryProducts d) => Data.Category.Limit.HasBinaryProducts (Data.Category.NaturalTransformation.Nat c d)
instance (Data.Category.Limit.HasColimits i k, Data.Category.Limit.HasColimits j k, Data.Category.Limit.HasBinaryCoproducts k) => Data.Category.Limit.HasColimits (i Data.Category.Coproduct.:++: j) k
instance Data.Category.Limit.HasBinaryCoproducts Data.Category.Functor.Cat
instance Data.Category.Limit.HasBinaryCoproducts Data.Category.Unit.Unit
instance (Data.Category.Limit.HasBinaryCoproducts c1, Data.Category.Limit.HasBinaryCoproducts c2) => Data.Category.Limit.HasBinaryCoproducts (c1 Data.Category.Product.:**: c2)
instance (Data.Category.Limit.HasBinaryCoproducts c1, Data.Category.Limit.HasBinaryCoproducts c2) => Data.Category.Limit.HasBinaryCoproducts (c1 Data.Category.Coproduct.:>>: c2)
instance Data.Category.Limit.HasBinaryCoproducts k => Data.Category.Functor.Functor (Data.Category.Limit.CoproductFunctor k)
instance (Data.Category.Category (Data.Category.Functor.Dom p), Data.Category.Category (Data.Category.Functor.Cod p)) => Data.Category.Functor.Functor (p Data.Category.Limit.:+: q)
instance (Data.Category.Category c, Data.Category.Limit.HasBinaryCoproducts d) => Data.Category.Limit.HasBinaryCoproducts (Data.Category.NaturalTransformation.Nat c d)
instance Data.Category.Limit.HasInitialObject k => Data.Category.Limit.HasTerminalObject (Data.Category.Op k)
instance Data.Category.Limit.HasTerminalObject k => Data.Category.Limit.HasInitialObject (Data.Category.Op k)
instance Data.Category.Limit.HasBinaryCoproducts k => Data.Category.Limit.HasBinaryProducts (Data.Category.Op k)
instance Data.Category.Limit.HasBinaryProducts k => Data.Category.Limit.HasBinaryCoproducts (Data.Category.Op k)
instance Data.Category.Category k => Data.Category.Limit.HasLimits Data.Category.Unit.Unit k
instance (Data.Category.Limit.HasInitialObject (i Data.Category.Coproduct.:>>: j), Data.Category.Category k) => Data.Category.Limit.HasLimits (i Data.Category.Coproduct.:>>: j) k
instance Data.Category.Category k => Data.Category.Limit.HasColimits Data.Category.Unit.Unit k
instance (Data.Category.Limit.HasTerminalObject (i Data.Category.Coproduct.:>>: j), Data.Category.Category k) => Data.Category.Limit.HasColimits (i Data.Category.Coproduct.:>>: j) k
instance Data.Category.Limit.HasLimits (->) (->)
instance Data.Category.Limit.HasColimits (->) (->)


-- | Comma categories.
module Data.Category.Comma
data CommaO :: * -> * -> * -> *
[CommaO] :: (Cod t ~ k, Cod s ~ k) => Obj (Dom t) a -> k (t :% a) (s :% b) -> Obj (Dom s) b -> CommaO t s (a, b)
data (:/\:) :: * -> * -> * -> * -> *
[CommaA] :: CommaO t s (a, b) -> Dom t a a' -> Dom s b b' -> CommaO t s (a', b') -> (t :/\: s) (a, b) (a', b')
commaId :: CommaO t s (a, b) -> Obj (t :/\: s) (a, b)

-- | The comma category T \downarrow S
type ObjectsFUnder f a = ConstF f a :/\: f
type ObjectsFOver f a = f :/\: ConstF f a
type ObjectsUnder c a = Id c `ObjectsFUnder` a
type ObjectsOver c a = Id c `ObjectsFOver` a
initialUniversalComma :: forall u x c a a_. (Functor u, c ~ (u `ObjectsFUnder` x), HasInitialObject c, (a_, a) ~ InitialObject c) => u -> InitialUniversal x u a
terminalUniversalComma :: forall u x c a a_. (Functor u, c ~ (u `ObjectsFOver` x), HasTerminalObject c, (a, a_) ~ TerminalObject c) => u -> TerminalUniversal x u a
instance (Data.Category.Category (Data.Category.Functor.Dom t), Data.Category.Category (Data.Category.Functor.Dom s)) => Data.Category.Category (t Data.Category.Comma.:/\: s)


module Data.Category.Monoidal

-- | A monoidal category is a category with some kind of tensor product. A
--   tensor product is a bifunctor, with a unit object.
class (Functor f, Dom f ~ (Cod f :**: Cod f)) => TensorProduct f where type Unit f :: * where {
    type family Unit f :: *;
}
unitObject :: TensorProduct f => f -> Obj (Cod f) (Unit f)
leftUnitor :: (TensorProduct f, Cod f ~ k) => f -> Obj k a -> k (f :% (Unit f, a)) a
leftUnitorInv :: (TensorProduct f, Cod f ~ k) => f -> Obj k a -> k a (f :% (Unit f, a))
rightUnitor :: (TensorProduct f, Cod f ~ k) => f -> Obj k a -> k (f :% (a, Unit f)) a
rightUnitorInv :: (TensorProduct f, Cod f ~ k) => f -> Obj k a -> k a (f :% (a, Unit f))
associator :: (TensorProduct f, Cod f ~ k) => f -> Obj k a -> Obj k b -> Obj k c -> k (f :% (f :% (a, b), c)) (f :% (a, f :% (b, c)))
associatorInv :: (TensorProduct f, Cod f ~ k) => f -> Obj k a -> Obj k b -> Obj k c -> k (f :% (a, f :% (b, c))) (f :% (f :% (a, b), c))

-- | If a category has all products, then the product functor makes it a
--   monoidal category, with the terminal object as unit.

-- | If a category has all coproducts, then the coproduct functor makes it
--   a monoidal category, with the initial object as unit.

-- | Functor composition makes the category of endofunctors monoidal, with
--   the identity functor as unit.

-- | <tt>MonoidObject f a</tt> defines a monoid <tt>a</tt> in a monoidal
--   category with tensor product <tt>f</tt>.
data MonoidObject f a
MonoidObject :: Cod f (Unit f) a -> Cod f ((f :% (a, a))) a -> MonoidObject f a
[unit] :: MonoidObject f a -> Cod f (Unit f) a
[multiply] :: MonoidObject f a -> Cod f ((f :% (a, a))) a
trivialMonoid :: TensorProduct f => f -> MonoidObject f (Unit f)
coproductMonoid :: (HasInitialObject k, HasBinaryCoproducts k) => Obj k a -> MonoidObject (CoproductFunctor k) a

-- | <tt>ComonoidObject f a</tt> defines a comonoid <tt>a</tt> in a
--   comonoidal category with tensor product <tt>f</tt>.
data ComonoidObject f a
ComonoidObject :: Cod f a (Unit f) -> Cod f a (f :% (a, a)) -> ComonoidObject f a
[counit] :: ComonoidObject f a -> Cod f a (Unit f)
[comultiply] :: ComonoidObject f a -> Cod f a (f :% (a, a))
trivialComonoid :: TensorProduct f => f -> ComonoidObject f (Unit f)
productComonoid :: (HasTerminalObject k, HasBinaryProducts k) => Obj k a -> ComonoidObject (ProductFunctor k) a
data MonoidAsCategory f m a b
[MonoidValue] :: (TensorProduct f, Dom f ~ (k :**: k), Cod f ~ k) => f -> MonoidObject f m -> k (Unit f) m -> MonoidAsCategory f m m m

-- | A monoid as a category with one object.

-- | A monad is a monoid in the category of endofunctors.
type Monad f = MonoidObject (EndoFunctorCompose (Dom f)) f
mkMonad :: (Functor f, Dom f ~ k, Cod f ~ k) => f -> (forall a. Obj k a -> Component (Id k) f a) -> (forall a. Obj k a -> Component (f :.: f) f a) -> Monad f
monadFunctor :: Monad f -> f
idMonad :: Category k => Monad (Id k)

-- | A comonad is a comonoid in the category of endofunctors.
type Comonad f = ComonoidObject (EndoFunctorCompose (Dom f)) f
mkComonad :: (Functor f, Dom f ~ k, Cod f ~ k) => f -> (forall a. Obj k a -> Component f (Id k) a) -> (forall a. Obj k a -> Component f (f :.: f) a) -> Comonad f
idComonad :: Category k => Comonad (Id k)

-- | Every adjunction gives rise to an associated monad.
adjunctionMonad :: Adjunction c d f g -> Monad (g :.: f)

-- | Every adjunction gives rise to an associated monad transformer.
adjunctionMonadT :: (Dom m ~ c) => Adjunction c d f g -> Monad m -> Monad ((g :.: m) :.: f)

-- | Every adjunction gives rise to an associated comonad.
adjunctionComonad :: Adjunction c d f g -> Comonad (f :.: g)

-- | Every adjunction gives rise to an associated comonad transformer.
adjunctionComonadT :: (Dom w ~ d) => Adjunction c d f g -> Comonad w -> Comonad ((f :.: w) :.: g)
instance (Data.Category.Limit.HasTerminalObject k, Data.Category.Limit.HasBinaryProducts k) => Data.Category.Monoidal.TensorProduct (Data.Category.Limit.ProductFunctor k)
instance (Data.Category.Limit.HasInitialObject k, Data.Category.Limit.HasBinaryCoproducts k) => Data.Category.Monoidal.TensorProduct (Data.Category.Limit.CoproductFunctor k)
instance Data.Category.Category k => Data.Category.Monoidal.TensorProduct (Data.Category.NaturalTransformation.EndoFunctorCompose k)
instance Data.Category.Category (Data.Category.Monoidal.MonoidAsCategory f m)


-- | The cube category.
module Data.Category.Cube
data Z
data S n
data Sign
M :: Sign
P :: Sign
data Cube :: * -> * -> *
[Z] :: Cube Z Z
[S] :: Cube x y -> Cube (S x) (S y)
[Y] :: Sign -> Cube x y -> Cube x (S y)
[X] :: Cube x y -> Cube (S x) y
data Sign0
SM :: Sign0
S0 :: Sign0
SP :: Sign0
data ACube :: * -> *
[Nil] :: ACube Z
[Cons] :: Sign0 -> ACube n -> ACube (S n)
data Forget
Forget :: Forget

-- | Turn <tt>Cube x y</tt> arrows into <tt>ACube x -&gt; ACube y</tt>
--   functions.
data Add
Add :: Add

-- | Ordinal addition is a bifuntor, it concattenates the maps as it were.
instance Data.Category.Category Data.Category.Cube.Cube
instance Data.Category.Limit.HasTerminalObject Data.Category.Cube.Cube
instance Data.Category.Functor.Functor Data.Category.Cube.Forget
instance Data.Category.Functor.Functor Data.Category.Cube.Add
instance Data.Category.Monoidal.TensorProduct Data.Category.Cube.Add


-- | Dialg(F,G), the category of (F,G)-dialgebras and (F,G)-homomorphisms.
module Data.Category.Dialg

-- | Objects of Dialg(F,G) are (F,G)-dialgebras.
data Dialgebra f g a
[Dialgebra] :: (Category c, Category d, Dom f ~ c, Dom g ~ c, Cod f ~ d, Cod g ~ d, Functor f, Functor g) => Obj c a -> d (f :% a) (g :% a) -> Dialgebra f g a

-- | Arrows of Dialg(F,G) are (F,G)-homomorphisms.
data Dialg f g a b
[DialgA] :: (Category c, Category d, Dom f ~ c, Dom g ~ c, Cod f ~ d, Cod g ~ d, Functor f, Functor g) => Dialgebra f g a -> Dialgebra f g b -> c a b -> Dialg f g a b
dialgId :: Dialgebra f g a -> Obj (Dialg f g) a
dialgebra :: Obj (Dialg f g) a -> Dialgebra f g a

-- | The category of (F,G)-dialgebras.
type Alg f = Dialg f (Id (Dom f))
type Algebra f a = Dialgebra f (Id (Dom f)) a
type Coalg f = Dialg (Id (Dom f)) f
type Coalgebra f a = Dialgebra (Id (Dom f)) f a

-- | The initial F-algebra is the initial object in the category of
--   F-algebras.
type InitialFAlgebra f = InitialObject (Alg f)

-- | The terminal F-coalgebra is the terminal object in the category of
--   F-coalgebras.
type TerminalFAlgebra f = TerminalObject (Coalg f)

-- | A catamorphism of an F-algebra is the arrow to it from the initial
--   F-algebra.
type Cata f a = Algebra f a -> Alg f (InitialFAlgebra f) a

-- | A anamorphism of an F-coalgebra is the arrow from it to the terminal
--   F-coalgebra.
type Ana f a = Coalgebra f a -> Coalg f a (TerminalFAlgebra f)
data NatNum
Z :: () -> NatNum
S :: NatNum -> NatNum
primRec :: (() -> t) -> (t -> t) -> NatNum -> t

-- | The category for defining the natural numbers and primitive recursion
--   can be described as <tt>Dialg(F,G)</tt>, with
--   <tt>F(A)=&lt;1,A&gt;</tt> and <tt>G(A)=&lt;A,A&gt;</tt>.
data FreeAlg m
FreeAlg :: (Monad m) -> FreeAlg m

-- | <tt>FreeAlg</tt> M takes <tt>x</tt> to the free algebra <tt>(M x,
--   mu_x)</tt> of the monad <tt>M</tt>.
data ForgetAlg m
ForgetAlg :: ForgetAlg m

-- | <tt>ForgetAlg m</tt> is the forgetful functor for <tt>Alg m</tt>.
eilenbergMooreAdj :: (Functor m, Dom m ~ k, Cod m ~ k) => Monad m -> Adjunction (Alg m) k (FreeAlg m) (ForgetAlg m)
instance Data.Category.Category (Data.Category.Dialg.Dialg f g)
instance Data.Category.Limit.HasInitialObject (Data.Category.Dialg.Dialg (Data.Category.Functor.Tuple1 (->) (->) ()) (Data.Category.Functor.DiagProd (->)))
instance (Data.Category.Functor.Functor m, Data.Category.Functor.Dom m ~ k, Data.Category.Functor.Cod m ~ k) => Data.Category.Functor.Functor (Data.Category.Dialg.FreeAlg m)
instance (Data.Category.Functor.Functor m, Data.Category.Functor.Dom m ~ k, Data.Category.Functor.Cod m ~ k) => Data.Category.Functor.Functor (Data.Category.Dialg.ForgetAlg m)


-- | This is an attempt at the Kleisli category, and the construction of an
--   adjunction for each monad.
module Data.Category.Kleisli
data Kleisli m a b
[Kleisli] :: (Functor m, Dom m ~ k, Cod m ~ k) => Monad m -> Obj k b -> k a (m :% b) -> Kleisli m a b
kleisliId :: (Functor m, Dom m ~ k, Cod m ~ k) => Monad m -> Obj k a -> Kleisli m a a

-- | The category of Kleisli arrows.
data KleisliAdjF m
KleisliAdjF :: (Monad m) -> KleisliAdjF m
data KleisliAdjG m
KleisliAdjG :: (Monad m) -> KleisliAdjG m
kleisliAdj :: (Functor m, Dom m ~ k, Cod m ~ k) => Monad m -> Adjunction (Kleisli m) k (KleisliAdjF m) (KleisliAdjG m)
instance Data.Category.Category (Data.Category.Kleisli.Kleisli m)
instance (Data.Category.Functor.Functor m, Data.Category.Functor.Dom m ~ k, Data.Category.Functor.Cod m ~ k) => Data.Category.Functor.Functor (Data.Category.Kleisli.KleisliAdjF m)
instance (Data.Category.Functor.Functor m, Data.Category.Functor.Dom m ~ k, Data.Category.Functor.Cod m ~ k) => Data.Category.Functor.Functor (Data.Category.Kleisli.KleisliAdjG m)


-- | The (augmented) simplex category.
module Data.Category.Simplex
data Simplex :: * -> * -> *
[Z] :: Simplex Z Z
[Y] :: Simplex x y -> Simplex x (S y)
[X] :: Simplex x (S y) -> Simplex (S x) (S y)
data Z
data S n
suc :: Obj Simplex n -> Obj Simplex (S n)
data Forget
Forget :: Forget
data Fin :: * -> *
[Fz] :: Fin (S n)
[Fs] :: Fin n -> Fin (S n)
data Add
Add :: Add

-- | The maps <tt>0 -&gt; 1</tt> and <tt>2 -&gt; 1</tt> form a monoid,
--   which is universal, c.f. <a>Replicate</a>.
universalMonoid :: MonoidObject Add (S Z)
data Replicate f a
Replicate :: f -> (MonoidObject f a) -> Replicate f a
instance Data.Category.Category Data.Category.Simplex.Simplex
instance Data.Category.Limit.HasInitialObject Data.Category.Simplex.Simplex
instance Data.Category.Limit.HasTerminalObject Data.Category.Simplex.Simplex
instance Data.Category.Functor.Functor Data.Category.Simplex.Forget
instance Data.Category.Functor.Functor Data.Category.Simplex.Add
instance Data.Category.Monoidal.TensorProduct Data.Category.Simplex.Add
instance Data.Category.Monoidal.TensorProduct f => Data.Category.Functor.Functor (Data.Category.Simplex.Replicate f a)


module Data.Category.Yoneda
type YonedaEmbedding k = Postcompose (Hom k) (Op k) :.: (Postcompose (Swap k (Op k)) (Op k) :.: Tuple k (Op k))

-- | The Yoneda embedding functor, <tt>C -&gt; Set^(C^op)</tt>.
yonedaEmbedding :: Category k => YonedaEmbedding k
data Yoneda (k :: * -> * -> *) f
Yoneda :: Yoneda f

-- | <a>Yoneda</a> converts a functor <tt>f</tt> into a natural
--   transformation from the hom functor to f.

-- | <a>fromYoneda</a> and <a>toYoneda</a> are together the isomophism from
--   the Yoneda lemma.
fromYoneda :: (Category k, Functor f, Dom f ~ Op k, Cod f ~ (->)) => f -> Nat (Op k) (->) (Yoneda k f) f
toYoneda :: (Category k, Functor f, Dom f ~ Op k, Cod f ~ (->)) => f -> Nat (Op k) (->) f (Yoneda k f)
instance (Data.Category.Category k, Data.Category.Functor.Functor f, Data.Category.Functor.Dom f ~ Data.Category.Op k, Data.Category.Functor.Cod f ~ (->)) => Data.Category.Functor.Functor (Data.Category.Yoneda.Yoneda k f)


module Data.Category.CartesianClosed

-- | A category is cartesian closed if it has all products and exponentials
--   for all objects.
class (HasTerminalObject k, HasBinaryProducts k) => CartesianClosed k where type Exponential k y z :: * where {
    type family Exponential k y z :: *;
}
apply :: CartesianClosed k => Obj k y -> Obj k z -> k (BinaryProduct k (Exponential k y z) y) z
tuple :: CartesianClosed k => Obj k y -> Obj k z -> k z (Exponential k y (BinaryProduct k z y))
(^^^) :: CartesianClosed k => k z1 z2 -> k y2 y1 -> k (Exponential k y1 z1) (Exponential k y2 z2)
data ExpFunctor (k :: * -> * -> *)
ExpFunctor :: ExpFunctor

-- | The exponential as a bifunctor.

-- | Exponentials in <tt>Hask</tt> are functions.

-- | Exponentials in <tt>Cat</tt> are the functor categories.
type PShExponential k y z = (Presheaves k :-*: z) :.: Opposite ((ProductFunctor (Presheaves k) :.: Tuple2 (Presheaves k) (Presheaves k) y) :.: YonedaEmbedding k)
pshExponential :: Category k => Obj (Presheaves k) y -> Obj (Presheaves k) z -> PShExponential k y z

-- | The category of presheaves on a category <tt>C</tt> is cartesian
--   closed for any <tt>C</tt>.

-- | The product functor is left adjoint the the exponential functor.
curryAdj :: CartesianClosed k => Obj k y -> Adjunction k k (ProductFunctor k :.: Tuple2 k k y) (ExpFunctor k :.: Tuple1 (Op k) k y)

-- | From the adjunction between the product functor and the exponential
--   functor we get the curry and uncurry functions, generalized to any
--   cartesian closed category.
curry :: CartesianClosed k => Obj k x -> Obj k y -> Obj k z -> k (BinaryProduct k x y) z -> k x (Exponential k y z)
uncurry :: CartesianClosed k => Obj k x -> Obj k y -> Obj k z -> k x (Exponential k y z) -> k (BinaryProduct k x y) z

-- | From every adjunction we get a monad, in this case the State monad.
type State k s a = Exponential k s (BinaryProduct k a s)
stateMonadReturn :: CartesianClosed k => Obj k s -> Obj k a -> k a (State k s a)
stateMonadJoin :: CartesianClosed k => Obj k s -> Obj k a -> k (State k s (State k s a)) (State k s a)
type Context k s a = BinaryProduct k (Exponential k s a) s
contextComonadExtract :: CartesianClosed k => Obj k s -> Obj k a -> k (Context k s a) a
contextComonadDuplicate :: CartesianClosed k => Obj k s -> Obj k a -> k (Context k s a) (Context k s (Context k s a))
instance Data.Category.CartesianClosed.CartesianClosed k => Data.Category.Functor.Functor (Data.Category.CartesianClosed.ExpFunctor k)
instance Data.Category.CartesianClosed.CartesianClosed (->)
instance Data.Category.CartesianClosed.CartesianClosed Data.Category.Functor.Cat
instance Data.Category.Category k => Data.Category.CartesianClosed.CartesianClosed (Data.Category.NaturalTransformation.Presheaves k)


-- | <i>2</i>, or the Boolean category. It contains 2 objects, one for true
--   and one for false. It contains 3 arrows, 2 identity arrows and one
--   from false to true.
module Data.Category.Boolean
data Fls
data Tru
data Boolean a b
[Fls] :: Boolean Fls Fls
[F2T] :: Boolean Fls Tru
[Tru] :: Boolean Tru Tru

-- | <tt>Boolean</tt> is the category with true and false as objects, and
--   an arrow from false to true.

-- | False is the initial object in the Boolean category.

-- | True is the terminal object in the Boolean category.

-- | Conjunction is the binary product in the Boolean category.

-- | Disjunction is the binary coproduct in the Boolean category.

-- | Implication makes the Boolean category cartesian closed.
trueProductMonoid :: MonoidObject (ProductFunctor Boolean) Tru
falseCoproductComonoid :: ComonoidObject (CoproductFunctor Boolean) Fls
trueProductComonoid :: ComonoidObject (ProductFunctor Boolean) Tru
falseCoproductMonoid :: MonoidObject (CoproductFunctor Boolean) Fls
trueCoproductMonoid :: MonoidObject (CoproductFunctor Boolean) Tru
falseProductComonoid :: ComonoidObject (ProductFunctor Boolean) Fls
instance Data.Category.Category Data.Category.Boolean.Boolean
instance Data.Category.Limit.HasInitialObject Data.Category.Boolean.Boolean
instance Data.Category.Limit.HasTerminalObject Data.Category.Boolean.Boolean
instance Data.Category.Limit.HasBinaryProducts Data.Category.Boolean.Boolean
instance Data.Category.Limit.HasBinaryCoproducts Data.Category.Boolean.Boolean
instance Data.Category.CartesianClosed.CartesianClosed Data.Category.Boolean.Boolean


module Data.Category.Fix
newtype Fix f a b
Fix :: (f (Fix f) a b) -> Fix f a b

-- | <tt><a>Fix</a> f</tt> is the fixed point category for a category
--   combinator <tt>f</tt>.

-- | <tt>Fix f</tt> inherits its (co)limits from <tt>f (Fix f)</tt>.

-- | <tt>Fix f</tt> inherits its (co)limits from <tt>f (Fix f)</tt>.

-- | <tt>Fix f</tt> inherits its (co)limits from <tt>f (Fix f)</tt>.

-- | <tt>Fix f</tt> inherits its (co)limits from <tt>f (Fix f)</tt>.

-- | <tt>Fix f</tt> inherits its exponentials from <tt>f (Fix f)</tt>.
data Wrap (f :: (* -> * -> *) -> * -> * -> *)
Wrap :: Wrap

-- | The <a>Wrap</a> functor wraps <a>Fix</a> around <tt>f (Fix f)</tt>.

-- | Take the <a>Omega</a> category, add a new disctinct object, and an
--   arrow from that object to every object in <a>Omega</a>, and you get
--   <a>Omega</a> again.
type Omega = Fix ((:>>:) Unit)
instance Data.Category.Category (f (Data.Category.Fix.Fix f)) => Data.Category.Category (Data.Category.Fix.Fix f)
instance Data.Category.Limit.HasInitialObject (f (Data.Category.Fix.Fix f)) => Data.Category.Limit.HasInitialObject (Data.Category.Fix.Fix f)
instance Data.Category.Limit.HasTerminalObject (f (Data.Category.Fix.Fix f)) => Data.Category.Limit.HasTerminalObject (Data.Category.Fix.Fix f)
instance Data.Category.Limit.HasBinaryProducts (f (Data.Category.Fix.Fix f)) => Data.Category.Limit.HasBinaryProducts (Data.Category.Fix.Fix f)
instance Data.Category.Limit.HasBinaryCoproducts (f (Data.Category.Fix.Fix f)) => Data.Category.Limit.HasBinaryCoproducts (Data.Category.Fix.Fix f)
instance Data.Category.CartesianClosed.CartesianClosed (f (Data.Category.Fix.Fix f)) => Data.Category.CartesianClosed.CartesianClosed (Data.Category.Fix.Fix f)
instance Data.Category.Category (f (Data.Category.Fix.Fix f)) => Data.Category.Functor.Functor (Data.Category.Fix.Wrap f)


module Data.Category.NNO
class HasTerminalObject k => HasNaturalNumberObject k where type NaturalNumberObject k :: * where {
    type family NaturalNumberObject k :: *;
}
zero :: HasNaturalNumberObject k => k (TerminalObject k) (NaturalNumberObject k)
succ :: HasNaturalNumberObject k => k (NaturalNumberObject k) (NaturalNumberObject k)
primRec :: HasNaturalNumberObject k => k (TerminalObject k) a -> k a a -> k (NaturalNumberObject k) a
data NatNum
Z :: NatNum
S :: NatNum -> NatNum
type Nat = Fix ((:++:) Unit)
data PrimRec z s
PrimRec :: z -> s -> PrimRec z s
instance Data.Category.NNO.HasNaturalNumberObject (->)
instance Data.Category.NNO.HasNaturalNumberObject Data.Category.Functor.Cat
instance (Data.Category.Functor.Functor z, Data.Category.Functor.Functor s, Data.Category.Functor.Dom z ~ Data.Category.Unit.Unit, Data.Category.Functor.Cod z ~ Data.Category.Functor.Dom s, Data.Category.Functor.Dom s ~ Data.Category.Functor.Cod s) => Data.Category.Functor.Functor (Data.Category.NNO.PrimRec z s)