Portability  Rank2Types 

Stability  provisional 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Safe Haskell  SafeInferred 
 type Iso a b c d = forall k f. (Isomorphic k, Functor f) => k (c > f d) (a > f b)
 type :<> a b = Iso a a b b
 iso :: (Isomorphic k, Functor f) => (a > b) > (b > a) > k (b > f b) (a > f a)
 isos :: (Isomorphic k, Functor f) => (a > c) > (c > a) > (b > d) > (d > b) > k (c > f d) (a > f b)
 ala :: Simple Iso a b > ((a > b) > c > b) > c > a
 auf :: Simple Iso a b > ((d > b) > c > b) > (d > a) > c > a
 under :: Isomorphism (c > Mutator d) (a > Mutator b) > (a > b) > c > d
 from :: Isomorphic k => Isomorphism a b > k b a
 via :: Isomorphic k => Isomorphism a b > k a b
 data Isomorphism a b = Isomorphism (a > b) (b > a)
 class Category k => Isomorphic k where
 isomorphic :: (a > b) > (b > a) > k a b
 isomap :: ((a > b) > c > d) > ((b > a) > d > c) > k a b > k c d
 _const :: Iso a b (Const a c) (Const b d)
 identity :: Iso a b (Identity a) (Identity b)
 newtype ReifiedIso a b c d = ReifyIso {
 reflectIso :: Iso a b c d
 type SimpleIso a b = Iso a a b b
 type SimpleReifiedIso a b = ReifiedIso a a b b
Isomorphism Lenses
type Iso a b c d = forall k f. (Isomorphic k, Functor f) => k (c > f d) (a > f b)Source
Isomorphim families can be composed with other lenses using either (.
) and id
from the Prelude or from Control.Category. However, if you compose them
with each other using (.
) from the Prelude, they will be dumbed down to a
mere Lens
.
import Control.Category import Prelude hiding ((.
),id
)
typeIso
a b c d = forall k f. (Isomorphic
k,Functor
f) =>Overloaded
k f a b c d
iso :: (Isomorphic k, Functor f) => (a > b) > (b > a) > k (b > f b) (a > f a)Source
isos :: (Isomorphic k, Functor f) => (a > c) > (c > a) > (b > d) > (d > b) > k (c > f d) (a > f b)Source
ala :: Simple Iso a b > ((a > b) > c > b) > c > aSource
Based on ala
from Conor McBride's work on Epigram.
>>>
:m + Data.Monoid.Lens Data.Foldable
>>>
ala _sum foldMap [1,2,3,4]
10
auf :: Simple Iso a b > ((d > b) > c > b) > (d > a) > c > aSource
Based on ala'
from Conor McBride's work on Epigram.
Mnemonically, the German auf plays a similar role to à la, and the combinator
is ala
with an extra function argument.
under :: Isomorphism (c > Mutator d) (a > Mutator b) > (a > b) > c > dSource
Primitive isomorphisms
from :: Isomorphic k => Isomorphism a b > k b aSource
via :: Isomorphic k => Isomorphism a b > k a bSource
Convert from an Isomorphism
back to any Isomorphic
value.
This is useful when you need to store an isomoprhism as a data type inside a container and later reconstitute it as an overloaded function.
data Isomorphism a b Source
A concrete data type for isomorphisms.
This lets you place an isomorphism inside a container without using ImpredicativeTypes
.
Isomorphism (a > b) (b > a) 
class Category k => Isomorphic k whereSource
Used to provide overloading of isomorphism application
This is a Category
with a canonical mapping to it from the
category of isomorphisms over Haskell types.
isomorphic :: (a > b) > (b > a) > k a bSource
Build this morphism out of an isomorphism
The intention is that by using isomorphic
, you can supply both halves of an
isomorphism, but k can be instantiated to (>)
, so you can freely use
the resulting isomorphism as a function.
isomap :: ((a > b) > c > d) > ((b > a) > d > c) > k a b > k c dSource
Map a morphism in the target category using an isomorphism between morphisms in Hask.
Common Isomorphisms
Storing Isomorphisms
newtype ReifiedIso a b c d Source
Useful for storing isomorphisms in containers.
ReifyIso  

Simplicity
type SimpleReifiedIso a b = ReifiedIso a a b bSource
typeSimpleReifiedIso
=Simple
ReifiedIso