Copyright  (C) 20142015 Edward Kmett 

License  BSDstyle (see the file LICENSE) 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Stability  provisional 
Portability  portable 
Safe Haskell  Safe 
Language  Haskell98 
This module supplies contravariant analogues to the Applicative
and Alternative
classes.
Contravariant Applicative
class Contravariant f => Divisible f where #
A Divisible
contravariant functor is the contravariant analogue of Applicative
.
Continuing the intuition that Contravariant
functors consume input, a Divisible
contravariant functor also has the ability to be composed "beside" another contravariant
functor.
Serializers provide a good example of Divisible
contravariant functors. To begin
let's start with the type of serializers for specific types:
newtype Serializer a = Serializer { runSerializer :: a > ByteString }
This is a contravariant functor:
instance Contravariant Serializer where contramap f s = Serializer (runSerializer s . f)
That is, given a serializer for a
(s :: Serializer a
), and a way to turn
b
s into a
s (a mapping f :: b > a
), we have a serializer for b
:
contramap f s :: Serializer b
.
Divisible gives us a way to combine two serializers that focus on different
parts of a structure. If we postulate the existance of two primitive
serializers  string :: Serializer String
and int :: Serializer Int
, we
would like to be able to combine these into a serializer for pairs of
String
s and Int
s. How can we do this? Simply run both serializer and
combine their output!
data StringAndInt = StringAndInt String Int stringAndInt :: Serializer StringAndInt stringAndInt = Serializer $ \(StringAndInt s i) > let sBytes = runSerializer string s iBytes = runSerializer int i in sBytes <> iBytes
divide
is a generalization by also taking a contramap
like function to
split any a
into a pair. This conveniently allows you to target fields of
a record, for instance, by extracting the values under two fields and
combining them into a tuple.
To complete the example, here is how to write stringAndInt
using a
Divisible
instance:
instance Divisible Serializer where conquer = Serializer (const mempty) divide toBC bSerializer cSerializer = Serializer $ \a > case toBC a of (b, c) > let bBytes = runSerializer bSerializer b cBytes = runSerializer cSerializer c in bBytes <> cBytes stringAndInt :: Serializer StringAndInt stringAndInt = divide (\(StringAndInt s i) > (s, i)) string int
divide :: (a > (b, c)) > f b > f c > f a #
Conquer acts as an identity for combining Divisible
functors.
Instances
Contravariant Alternative
class Divisible f => Decidable f where #
A Decidable
contravariant functor is the contravariant analogue of Alternative
.
Noting the superclass constraint that f
must also be Divisible
, a Decidable
functor has the ability to "fan out" input, under the intuition that contravariant
functors consume input.
In the dicussion for Divisible
, an example was demonstrated with Serializer
s,
that turn a
s into ByteString
s. Divisible
allowed us to serialize the product
of multiple values by concatenation. By making our Serializer
also Decidable

we now have the ability to serialize the sum of multiple values  for example
different constructors in an ADT.
Consider serializing arbitrary identifiers that can be either String
s or Int
s:
data Identifier = StringId String  IntId Int
We know we have serializers for String
s and Int
s, but how do we combine them
into a Serializer
for Identifier
? Essentially, our Serializer
needs to
scrutinise the incoming value and choose how to serialize it:
identifier :: Serializer Identifier identifier = Serializer $ \identifier > case identifier of StringId s > runSerializer string s IntId i > runSerializer int i
It is exactly this notion of choice that Decidable
encodes. Hence if we add
an instance of Decidable
for Serializer
...
instance Decidable Serializer where lose f = Serializer $ \a > absurd (f a) choose split l r = Serializer $ \a > either (runSerializer l) (runSerializer r) (split a)
Then our identifier
Serializer
is
identifier :: Serializer Identifier identifier = choose toEither string int where toEither (StringId s) = Left s toEither (IntId i) = Right i
Acts as identity to choose
.
Instances
Mathematical definitions
Divisible
In denser jargon, a Divisible
contravariant functor is a monoid object in the category
of presheaves from Hask to Hask, equipped with Day convolution mapping the Cartesian
product of the source to the Cartesian product of the target.
By way of contrast, an Applicative
functor can be viewed as a monoid object in the
category of copresheaves from Hask to Hask, equipped with Day convolution mapping the
Cartesian product of the source to the Cartesian product of the target.
Given the canonical diagonal morphism:
delta a = (a,a)
should be associative with divide
delta
conquer
as a unit
divide
delta
mconquer
= mdivide
delta
conquer
m = mdivide
delta
(divide
delta
m n) o =divide
delta
m (divide
delta
n o)
With more general arguments you'll need to reassociate and project using the monoidal structure of the source category. (Here fst and snd are used in lieu of the more restricted lambda and rho, but this construction works with just a monoidal category.)
divide
f mconquer
=contramap
(fst
. f) mdivide
fconquer
m =contramap
(snd
. f) mdivide
f (divide
g m n) o =divide
f' m (divide
id
n o) where f' a = let (bc, d) = f a; (b, c) = g bc in (b, (c, d))
A note on
conquer
The underlying theory would suggest that this should be:
conquer :: (a > ()) > f a
However, as we are working over a Cartesian category (Hask) and the Cartesian product, such an input
morphism is uniquely determined to be
, so we elide it.const
mempty
Decidable
A Divisible
contravariant functor is a monoid object in the category of presheaves
from Hask to Hask, equipped with Day convolution mapping the cartesian product of the
source to the Cartesian product of the target.
choose
Left
m (lose
f) = mchoose
Right
(lose
f) m = mchoose
f (choose
g m n) o =choose
f' m (choose
id
n o) where f' =either
(either
id
Left
. g) (Right
.Right
) . f
In addition, we expect the same kind of distributive law as is satisfied by the usual
covariant Alternative
, w.r.t Applicative
, which should be fully formulated and
added here at some point!