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


-- | Categorical types and classes
--   
--   Categorical types and classes
@package category
@version 0.2.3.0

module Control.Category.Groupoid

-- | <a>Category</a> where every morphism is iso
--   
--   Laws:
--   
--   <pre>
--   <a>id</a> = f <a>.</a> <a>invert</a> f
--   <a>id</a> = <a>invert</a> f <a>.</a> f
--   </pre>
class Category k => Groupoid k
invert :: Groupoid k => k a b -> k b a

module Data.Morphism.Endo
newtype Endo s a
Endo :: s a a -> Endo s a
[endo] :: Endo s a -> s a a
instance forall k (s :: k -> k -> *) (a :: k). Control.Category.Category s => GHC.Base.Semigroup (Data.Morphism.Endo.Endo s a)
instance forall k (s :: k -> k -> *) (a :: k). Control.Category.Category s => GHC.Base.Monoid (Data.Morphism.Endo.Endo s a)
instance forall k (s :: k -> k -> *) (a :: k). Control.Category.Groupoid.Groupoid s => Algebra.Group (Data.Morphism.Endo.Endo s a)

module Control.Category.Dual
newtype Dual k a b
Dual :: k b a -> Dual k a b
[dual] :: Dual k a b -> k b a
instance forall k1 k2 (k3 :: k2 -> k1 -> *) (a :: k1) (b :: k2). Algebra.Group (k3 b a) => Algebra.Group (Control.Category.Dual.Dual k3 a b)
instance forall k1 k2 (k3 :: k2 -> k1 -> *) (a :: k1) (b :: k2). GHC.Base.Monoid (k3 b a) => GHC.Base.Monoid (Control.Category.Dual.Dual k3 a b)
instance forall k1 k2 (k3 :: k2 -> k1 -> *) (a :: k1) (b :: k2). GHC.Base.Semigroup (k3 b a) => GHC.Base.Semigroup (Control.Category.Dual.Dual k3 a b)
instance forall k1 (k2 :: k1 -> k1 -> *). Control.Category.Category k2 => Control.Category.Category (Control.Category.Dual.Dual k2)
instance forall k1 (k2 :: k1 -> k1 -> *). Control.Category.Groupoid.Groupoid k2 => Control.Category.Groupoid.Groupoid (Control.Category.Dual.Dual k2)

module Control.Category.Const2

-- | Notes: <a>Const2</a> '()' is the indiscrete category.
newtype Const2 a b c
Const2 :: a -> Const2 a b c
instance forall a k1 (b :: k1) k2 (c :: k2). Algebra.Group a => Algebra.Group (Control.Category.Const2.Const2 a b c)
instance forall a k1 (b :: k1) k2 (c :: k2). GHC.Base.Monoid a => GHC.Base.Monoid (Control.Category.Const2.Const2 a b c)
instance forall a k1 (b :: k1) k2 (c :: k2). GHC.Base.Semigroup a => GHC.Base.Semigroup (Control.Category.Const2.Const2 a b c)
instance (GHC.Base.Semigroup a, GHC.Base.Monoid a) => Control.Category.Category (Control.Category.Const2.Const2 a)
instance (GHC.Base.Semigroup a, Algebra.Group a) => Control.Category.Groupoid.Groupoid (Control.Category.Const2.Const2 a)

module Control.Categorical.Functor

-- | Laws:
--   
--   <pre>
--   <a>map</a> <a>id</a> = <a>id</a>
--   <a>map</a> (f <a>.</a> g) = <a>map</a> f <a>.</a> <a>map</a> g
--   </pre>
class (Category s, Category t) => Functor (s :: α -> α -> *) (t :: β -> β -> *) (f :: α -> β)
map :: Functor s t f => s a b -> t (f a) (f b)

-- | <i>Deprecated: Use Endofunctor</i>
type EndoFunctor s = Functor s s
type Endofunctor s = Functor s s
(<$>) :: Functor s (->) f => s a b -> f a -> f b
infixl 4 <$>
newtype NT s f g
NT :: (forall a. s (f a) (g a)) -> NT s f g
[nt] :: NT s f g -> forall a. s (f a) (g a)
instance forall k1 k2 (s :: k2 -> k2 -> *). Control.Category.Category s => Control.Category.Category (Control.Categorical.Functor.NT s)
instance forall k1 k2 (s :: k2 -> k2 -> *). Control.Category.Groupoid.Groupoid s => Control.Category.Groupoid.Groupoid (Control.Categorical.Functor.NT s)
instance forall k1 k2 (s :: k2 -> k2 -> *) (f :: k2 -> *). Control.Categorical.Functor.Functor s (->) f => Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT s) (Control.Categorical.Functor.NT (->)) (Data.Functor.Compose.Compose f)
instance Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (->)) (Control.Categorical.Functor.NT (Control.Categorical.Functor.NT (->))) Data.Functor.Compose.Compose
instance forall k (f :: k -> *). Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (->)) (Control.Categorical.Functor.NT (->)) (Data.Functor.Sum.Sum f)
instance Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (->)) (Control.Categorical.Functor.NT (Control.Categorical.Functor.NT (->))) Data.Functor.Sum.Sum
instance forall k (f :: k -> *). Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (->)) (Control.Categorical.Functor.NT (->)) (Data.Functor.Product.Product f)
instance Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (->)) (Control.Categorical.Functor.NT (Control.Categorical.Functor.NT (->))) Data.Functor.Product.Product
instance Control.Categorical.Functor.Functor (->) (Control.Categorical.Functor.NT (->)) Data.Functor.Const.Const
instance Control.Categorical.Functor.Functor (->) (Control.Categorical.Functor.NT (->)) (,)
instance Control.Categorical.Functor.Functor (->) (Control.Categorical.Functor.NT (->)) Data.Either.Either
instance forall k (s :: k -> k -> *). Control.Category.Category s => Control.Categorical.Functor.Functor (Control.Category.Dual.Dual s) (Control.Categorical.Functor.NT (->)) s
instance GHC.Base.Functor f => Control.Categorical.Functor.Functor (->) (->) f
instance forall k (s :: k -> k -> *) (f :: k -> *) (g :: k -> *). (Control.Categorical.Functor.Functor s (->) f, Control.Categorical.Functor.Functor s (->) g) => Control.Categorical.Functor.Functor s (->) (Data.Functor.Sum.Sum f g)
instance forall k (s :: k -> k -> *) (f :: k -> *) (g :: k -> *). (Control.Categorical.Functor.Functor s (->) f, Control.Categorical.Functor.Functor s (->) g) => Control.Categorical.Functor.Functor s (->) (Data.Functor.Product.Product f g)
instance forall α (s :: α -> α -> *) a. Control.Category.Category s => Control.Categorical.Functor.Functor s (->) (Data.Functor.Const.Const a)
instance Control.Categorical.Functor.Functor (->) (->) Data.Functor.Identity.Identity
instance forall α (s :: α -> α -> *). Control.Category.Category s => Control.Categorical.Functor.Functor s (->) Data.Proxy.Proxy
instance Control.Categorical.Functor.Functor (->) (->) ((,) a)
instance Control.Categorical.Functor.Functor (->) (->) (Data.Either.Either a)
instance forall α (s :: α -> α -> *) (a :: α). Control.Category.Category s => Control.Categorical.Functor.Functor s (->) (s a)
instance forall β α (s :: α -> α -> *) (t :: β -> β -> *) (f :: α -> β). Control.Categorical.Functor.Functor s t f => Control.Categorical.Functor.Functor (Control.Category.Dual.Dual s) (Control.Category.Dual.Dual t) f

module Data.Morphism.Iso
data Iso s a b
Iso :: s a b -> s b a -> Iso s a b
instance forall k (s :: k -> k -> *) (a :: k) (b :: k). (GHC.Base.Semigroup (s a b), GHC.Base.Semigroup (s b a)) => GHC.Base.Semigroup (Data.Morphism.Iso.Iso s a b)
instance forall k (s :: k -> k -> *) (a :: k) (b :: k). (GHC.Base.Semigroup (s a b), GHC.Base.Semigroup (s b a), GHC.Base.Monoid (s a b), GHC.Base.Monoid (s b a)) => GHC.Base.Monoid (Data.Morphism.Iso.Iso s a b)
instance forall k (s :: k -> k -> *) (a :: k) (b :: k). (GHC.Base.Semigroup (s a b), GHC.Base.Semigroup (s b a), Algebra.Group (s a b), Algebra.Group (s b a)) => Algebra.Group (Data.Morphism.Iso.Iso s a b)
instance forall k (s :: k -> k -> *). Control.Category.Category s => Control.Category.Category (Data.Morphism.Iso.Iso s)
instance forall k (s :: k -> k -> *). Control.Category.Category s => Control.Category.Groupoid.Groupoid (Data.Morphism.Iso.Iso s)
instance forall β α (s :: α -> α -> *) (t :: β -> β -> *) (f :: α -> β). Control.Categorical.Functor.Functor s t f => Control.Categorical.Functor.Functor (Data.Morphism.Iso.Iso s) t f
instance forall β α (s :: α -> α -> *) (t :: β -> β -> *) (f :: α -> β). Control.Categorical.Functor.Functor s t f => Control.Categorical.Functor.Functor (Data.Morphism.Iso.Iso s) (Control.Category.Dual.Dual t) f

module Control.Categorical.Monad
class Endofunctor s m => Monad s m
unit :: Monad s m => a `s` m a
join :: Monad s m => m (m a) `s` m a
bind :: Monad s m => (a `s` m b) -> m a `s` m b
(<=<) :: Monad s m => (b `s` m c) -> (a `s` m b) -> a `s` m c
infixr 1 <=<
(>=>) :: Monad s m => (a `s` m b) -> (b `s` m c) -> a `s` m c
infixr 1 >=>
newtype Kleisli s m a b
Kleisli :: (a `s` m b) -> Kleisli s m a b
[kleisli] :: Kleisli s m a b -> a `s` m b
class Endofunctor s ɯ => Comonad s ɯ
counit :: Comonad s ɯ => ɯ a `s` a
cut :: Comonad s ɯ => ɯ a `s` ɯ (ɯ a)
cobind :: Comonad s ɯ => (ɯ a `s` b) -> ɯ a `s` ɯ b
(=<=) :: Comonad s ɯ => (ɯ b `s` c) -> (ɯ a `s` b) -> ɯ a `s` c
infixr 1 =<=
(=>=) :: Comonad s ɯ => (ɯ a `s` b) -> (ɯ b `s` c) -> ɯ a `s` c
infixr 1 =>=
newtype Cokleisli s ɯ a b
Cokleisli :: (ɯ a `s` b) -> Cokleisli s ɯ a b
[cokleisli] :: Cokleisli s ɯ a b -> ɯ a `s` b
instance forall k (s :: k -> k -> *) (ɯ :: k -> k). Control.Categorical.Monad.Comonad s ɯ => Control.Category.Category (Control.Categorical.Monad.Cokleisli s ɯ)
instance forall k1 k2 (s :: k2 -> k2 -> *) (ɯ :: k1 -> k2) (a :: k1). Control.Category.Category s => Control.Categorical.Functor.Functor s (->) (Control.Categorical.Monad.Cokleisli s ɯ a)
instance forall k1 k2 (s :: k2 -> k2 -> *) (t :: k1 -> k1 -> *) (ɯ :: k2 -> k1). Control.Categorical.Functor.Functor s t ɯ => Control.Categorical.Functor.Functor (Control.Category.Dual.Dual s) (Control.Categorical.Functor.NT (->)) (Control.Categorical.Monad.Cokleisli t ɯ)
instance forall β (s :: β -> β -> *) (ɯ :: β -> β). Control.Categorical.Monad.Comonad s ɯ => Control.Categorical.Functor.Functor (Control.Categorical.Monad.Cokleisli s ɯ) s ɯ
instance Control.Categorical.Monad.Comonad (->) Data.Functor.Identity.Identity
instance Control.Categorical.Monad.Comonad (->) GHC.Base.NonEmpty
instance GHC.Base.Monoid m => Control.Categorical.Monad.Comonad (->) ((->) m)
instance Control.Categorical.Monad.Comonad (->) ((,) a)
instance Control.Categorical.Monad.Comonad (->) (Data.Semigroup.Arg a)
instance forall k (s :: k -> k -> *) (m :: k -> k). Control.Categorical.Monad.Monad s m => Control.Category.Category (Control.Categorical.Monad.Kleisli s m)
instance forall k1 k2 (s :: k2 -> k2 -> *) (t :: k1 -> k1 -> *) (m :: k2 -> k1) (a :: k1). Control.Categorical.Functor.Functor s t m => Control.Categorical.Functor.Functor s (->) (Control.Categorical.Monad.Kleisli t m a)
instance forall k1 k2 (s :: k2 -> k2 -> *) (m :: k1 -> k2). Control.Category.Category s => Control.Categorical.Functor.Functor (Control.Category.Dual.Dual s) (Control.Categorical.Functor.NT (->)) (Control.Categorical.Monad.Kleisli s m)
instance forall β (s :: β -> β -> *) (m :: β -> β). Control.Categorical.Monad.Monad s m => Control.Categorical.Functor.Functor (Control.Categorical.Monad.Kleisli s m) s m
instance GHC.Base.Monad m => Control.Categorical.Monad.Monad (->) m

module Data.Functor.Trans.Writer
newtype WriterT p w f a
WriterT :: f (p w a) -> WriterT p w f a
[runWriterT] :: WriterT p w f a -> f (p w a)
instance forall k1 k2 (f :: * -> *) (s :: k2 -> k2 -> *) (p :: k1 -> k2 -> *) (w :: k1). (Control.Categorical.Functor.Functor (->) (->) f, Control.Categorical.Functor.Functor s (->) (p w)) => Control.Categorical.Functor.Functor s (->) (Data.Functor.Trans.Writer.WriterT p w f)
instance (GHC.Base.Monoid w, Control.Categorical.Monad.Monad (->) f) => Control.Categorical.Monad.Monad (->) (Data.Functor.Trans.Writer.WriterT (,) w f)
instance forall k (p :: k -> * -> *) (w :: k) (f :: * -> *). (Control.Categorical.Monad.Comonad (->) (p w), Control.Categorical.Monad.Comonad (->) f) => Control.Categorical.Monad.Comonad (->) (Data.Functor.Trans.Writer.WriterT p w f)
instance Control.Categorical.Monad.Monad (->) f => Control.Categorical.Monad.Monad (->) (Data.Functor.Trans.Writer.WriterT Data.Either.Either w f)
instance forall k1 k2 k3 (p :: k1 -> k2 -> k3) (w :: k1). Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (->)) (Control.Categorical.Functor.NT (->)) (Data.Functor.Trans.Writer.WriterT p w)

module Data.Functor.Trans.Reader
newtype ReaderT s r f a
ReaderT :: (r `s` f a) -> ReaderT s r f a
[runReaderT] :: ReaderT s r f a -> r `s` f a
instance forall k1 k2 (s :: k2 -> k2 -> *) (t :: k1 -> k1 -> *) (f :: k2 -> k1) (r :: k1). Control.Categorical.Functor.Functor s t f => Control.Categorical.Functor.Functor s (->) (Data.Functor.Trans.Reader.ReaderT t r f)
instance forall k1 k2 (t :: k2 -> k2 -> *) (f :: k2 -> *) (s :: k1 -> * -> *) (r :: k1). (Control.Categorical.Functor.Functor t (->) f, Control.Categorical.Functor.Functor (->) (->) (s r)) => Control.Categorical.Functor.Functor t (->) (Data.Functor.Trans.Reader.ReaderT s r f)
instance Control.Categorical.Monad.Monad (->) f => Control.Categorical.Monad.Monad (->) (Data.Functor.Trans.Reader.ReaderT (->) r f)
instance Control.Categorical.Monad.Comonad (->) ɯ => Control.Categorical.Monad.Comonad (->) (Data.Functor.Trans.Reader.ReaderT (,) r ɯ)
instance forall k1 k2 k3 (t :: k3 -> k3 -> *) (s :: k2 -> k3 -> *) (r :: k2). Control.Categorical.Functor.Functor t (->) (s r) => Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT t) (Control.Categorical.Functor.NT (->)) (Data.Functor.Trans.Reader.ReaderT s r)
instance forall k1 k2 (s :: k2 -> * -> *) (r :: k2). Control.Categorical.Monad.Monad (->) (s r) => Control.Categorical.Monad.Monad (Control.Categorical.Functor.NT (->)) (Data.Functor.Trans.Reader.ReaderT s r)
instance forall k1 k2 (s :: k2 -> * -> *) (r :: k2). Control.Categorical.Monad.Comonad (->) (s r) => Control.Categorical.Monad.Comonad (Control.Categorical.Functor.NT (->)) (Data.Functor.Trans.Reader.ReaderT s r)
instance forall k1 k2 k3 (t :: k3 -> k3 -> *) (s :: k3 -> k2 -> *). Control.Categorical.Functor.Functor t (Control.Categorical.Functor.NT (->)) s => Control.Categorical.Functor.Functor t (Control.Categorical.Functor.NT (Control.Categorical.Functor.NT (->))) (Data.Functor.Trans.Reader.ReaderT s)

module Data.Functor.Trans.Identity
newtype IdentityT f a
IdentityT :: f a -> IdentityT f a
[runIdentityT] :: IdentityT f a -> f a
instance forall k (s :: k -> k -> *) (f :: k -> *). Control.Categorical.Functor.Functor s (->) f => Control.Categorical.Functor.Functor s (->) (Data.Functor.Trans.Identity.IdentityT f)
instance Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (->)) (Control.Categorical.Functor.NT (->)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Monad (->) m => Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (Control.Categorical.Monad.Kleisli (->) m)) (Control.Categorical.Functor.NT (Control.Categorical.Monad.Kleisli (->) m)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Comonad (->) ɯ => Control.Categorical.Functor.Functor (Control.Categorical.Functor.NT (Control.Categorical.Monad.Cokleisli (->) ɯ)) (Control.Categorical.Functor.NT (Control.Categorical.Monad.Cokleisli (->) ɯ)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Monad (Control.Categorical.Functor.NT (->)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Comonad (Control.Categorical.Functor.NT (->)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Monad (->) m => Control.Categorical.Monad.Monad (Control.Categorical.Functor.NT (Control.Categorical.Monad.Kleisli (->) m)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Comonad (->) ɯ => Control.Categorical.Monad.Monad (Control.Categorical.Functor.NT (Control.Categorical.Monad.Cokleisli (->) ɯ)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Comonad (->) ɯ => Control.Categorical.Monad.Comonad (Control.Categorical.Functor.NT (Control.Categorical.Monad.Cokleisli (->) ɯ)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Monad (->) m => Control.Categorical.Monad.Comonad (Control.Categorical.Functor.NT (Control.Categorical.Monad.Kleisli (->) m)) Data.Functor.Trans.Identity.IdentityT
instance Control.Categorical.Monad.Monad (->) f => Control.Categorical.Monad.Monad (->) (Data.Functor.Trans.Identity.IdentityT f)
instance Control.Categorical.Monad.Comonad (->) f => Control.Categorical.Monad.Comonad (->) (Data.Functor.Trans.Identity.IdentityT f)
instance forall k (s :: k -> k -> *) (m :: * -> *) (f :: k -> *). (Control.Categorical.Functor.Functor s (Control.Categorical.Monad.Kleisli (->) m) f, Control.Categorical.Functor.Endofunctor (->) m) => Control.Categorical.Functor.Functor s (Control.Categorical.Monad.Kleisli (->) m) (Data.Functor.Trans.Identity.IdentityT f)
instance forall k (s :: k -> k -> *) (ɯ :: * -> *) (f :: k -> *). (Control.Categorical.Functor.Functor s (Control.Categorical.Monad.Cokleisli (->) ɯ) f, Control.Categorical.Functor.Endofunctor (->) ɯ) => Control.Categorical.Functor.Functor s (Control.Categorical.Monad.Cokleisli (->) ɯ) (Data.Functor.Trans.Identity.IdentityT f)