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Hask

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Current revision (20:35, 13 September 2012) (edit) (undo)
(id is a function :))
 
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'''Hask''' is the name usually given to the [[Category theory|category]] having Haskell types as objects and Haskell functions between them as morphisms.
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'''Hask''' is the [[Category theory|category]] of Haskell types and functions.
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A type-constructor that is an instance of the Functor class is an endofunctor on Hask.
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The objects of '''Hask''' are Haskell types, and the morphisms from objects <hask>A</hask> to <hask>B</hask> are Haskell functions of type <hask>A -> B</hask>. The identity morphism for object <hask>A</hask> is <hask>id :: A -> A</hask>, and the composition of morphisms <hask>f</hask> and <hask>g</hask> is <hask>f . g = \x -> f (g x)</hask>.
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* [http://www.cs.gunma-u.ac.jp/~hamana/Papers/cpo.pdf Makoto Hamana: ''What is the category for Haskell?'']
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== Is '''Hask''' even a category? ==
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== The seq problem ==
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Consider:
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The right identity law fails in '''Hask''' if we distinguish values which can be distinguished by <hask>seq</hask>, since:
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<haskell>
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undef1 = undefined :: a -> b
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undef2 = \_ -> undefined
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</haskell>
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<hask>id . undefined = \x -> id (undefined x) = \x -> undefined x</hask>
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Note that these are not the same value:
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should be equal to <hask>undefined</hask>, but can be distinguished from it using <hask>seq</hask>:
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<haskell>
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seq undef1 () = undefined
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seq undef2 () = ()
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</haskell>
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ghci> (undefined :: Int -> Int) `seq` ()
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This might be a problem, because <hask>undef1 . id = undef2</hask>. In order to make '''Hask''' a category, we define two functions <hask>f</hask> and <hask>g</hask> as the same morphism if <hask>f x = g x</hask> for all <hask>x</hask>. Thus <hask>undef1</hask> and <hask>undef2</hask> are different ''values'', but the same ''morphism'' in '''Hask'''.
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* Exception: Prelude.undefined
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ghci> (id . undefined :: Int -> Int) `seq` ()
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()
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== '''Hask''' is not Cartesian closed ==
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Actual '''Hask''' does not have sums, products, or an initial object, and <hask>()</hask> is not a terminal object. The Monad identities fail for almost all instances of the Monad class.
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{| class="wikitable"
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|+ Why '''Hask''' isn't as nice as you'd thought.
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! scope="col" |
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! scope="col" | Initial Object
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! scope="col" | Terminal Object
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! scope="col" | Sum
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! scope="col" | Product
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! scope="col" | Product
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|-
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! scope="row" | Type
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| <hask>data Empty</hask>
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| <hask>data () = ()</hask>
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| <hask>data Either a b
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= Left a | Right b</hask>
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| <hask>data (a,b) =
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(,) { fst :: a, snd :: b}</hask>
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| <hask>data P a b =
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P {fstP :: !a, sndP :: !b}</hask>
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|-
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! scope="row" | Requirement
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| There is a unique function
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<br /><hask>u :: Empty -> r</hask>
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| There is a unique function
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<br /><hask>u :: r -> ()</hask>
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| For any functions
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<br /><hask>f :: a -> r</hask>
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<br /><hask>g :: b -> r</hask>
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there is a unique function
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<hask>u :: Either a b -> r</hask>
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 +
such that:
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<hask>u . Left = f</hask>
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<br /><hask>u . Right = g</hask>
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| For any functions
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<br /><hask>f :: r -> a</hask>
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<br /><hask>g :: r -> b</hask>
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there is a unique function
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<hask>u :: r -> (a,b)</hask>
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such that:
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<hask>fst . u = f</hask>
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<br /><hask>snd . u = g</hask>
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| For any functions
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<br /><hask>f :: r -> a</hask>
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<br /><hask>g :: r -> b</hask>
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there is a unique function
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<hask>u :: r -> P a b</hask>
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such that:
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<hask>fstP . u = f</hask>
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<br /><hask>sndP . u = g</hask>
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|-
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! scope="row" | Candidate
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| <hask>u1 r = case r of {}</hask>
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| <hask>u1 _ = ()</hask>
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| <hask>u1 (Left a) = f a</hask>
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<br /><hask>u1 (Right b) = g b</hask>
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| <hask>u1 r = (f r,g r)</hask>
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| <hask>u1 r = P (f r) (g r)</hask>
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|-
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! scope="row" | Example failure condition
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| <hask>r ~ ()</hask>
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| <hask>r ~ ()</hask>
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| <hask>r ~ ()</hask>
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<br /><hask>f _ = ()</hask>
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<br /><hask>g _ = ()</hask>
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| <hask>r ~ ()</hask>
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<br /><hask>f _ = undefined</hask>
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<br /><hask>g _ = undefined</hask>
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| <hask>r ~ ()</hask>
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<br /><hask>f _ = ()</hask>
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<br /><hask>g _ = undefined</hask>
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|-
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! scope="row" | Alternative u
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| <hask>u2 _ = ()</hask>
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| <hask>u2 _ = undefined</hask>
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| <hask>u2 _ = ()</hask>
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| <hask>u2 _ = undefined</hask>
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|
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|-
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! scope="row" | Difference
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| <hask>u1 undefined = undefined</hask>
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<br /><hask>u2 undefined = ()</hask>
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| <hask>u1 _ = ()</hask>
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<br /><hask>u2 _ = undefined</hask>
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| <hask>u1 undefined = undefined</hask>
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<br /><hask>u2 undefined = ()</hask>
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| <hask>u1 _ = (undefined,undefined)</hask>
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<br /><hask>u2 _ = undefined</hask>
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| <hask>f _ = ()</hask>
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<br /><hask>(fstP . u1) _ = undefined</hask>
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|- style="background: red;"
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! scope="row" | Result
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! scope="col" | FAIL
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! scope="col" | FAIL
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! scope="col" | FAIL
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! scope="col" | FAIL
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! scope="col" | FAIL
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|}
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== "Platonic" '''Hask''' ==
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 +
Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottom values. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products, and instances of Functor and Monad really are endofunctors and monads.
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== Links ==
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* [http://www.cs.gunma-u.ac.jp/~hamana/Papers/cpo.pdf Makoto Hamana: ''What is the category for Haskell?'']
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* [http://www.cs.nott.ac.uk/~nad/publications/danielsson-popl2006-tr.pdf Nils A. Danielsson, John Hughes, Patrik Jansson, and Jeremy Gibbons. ''Fast and loose reasoning is morally correct.'']
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{{stub}}
 
[[Category:Mathematics]]
[[Category:Mathematics]]
 +
[[Category:Theoretical foundations]]

Current revision

Hask is the category of Haskell types and functions.

The objects of Hask are Haskell types, and the morphisms from objects
A
to
B
are Haskell functions of type
A -> B
. The identity morphism for object
A
is
id :: A -> A
, and the composition of morphisms
f
and
g
is
f . g = \x -> f (g x)
.

Contents

1 Is Hask even a category?

Consider: 

undef1 = undefined :: a -> b
undef2 = \_ -> undefined

Note that these are not the same value: 

seq undef1 () = undefined
seq undef2 () = ()
This might be a problem, because
undef1 . id = undef2
. In order to make Hask a category, we define two functions
f
and
g
as the same morphism if
f x = g x
for all
x
. Thus
undef1
and
undef2
are different values, but the same morphism in Hask.

2 Hask is not Cartesian closed

Actual Hask does not have sums, products, or an initial object, and
()
is not a terminal object. The Monad identities fail for almost all instances of the Monad class.
Why Hask isn't as nice as you'd thought.
Initial Object Terminal Object Sum Product Product
Type
data Empty
data () = ()
data Either a b
 = Left a | Right b
data (a,b) =
 (,) { fst :: a, snd :: b}
data P a b =
 P {fstP :: !a, sndP :: !b}
Requirement There is a unique function
u :: Empty -> r
 There is a unique function
u :: r -> ()
 For any functions
f :: a -> r

g :: b -> r

there is a unique function

u :: Either a b -> r

such that:

u . Left = f

u . Right = g
 For any functions
f :: r -> a

g :: r -> b

there is a unique function

u :: r -> (a,b)

such that:

fst . u = f

snd . u = g
 For any functions
f :: r -> a

g :: r -> b

there is a unique function

u :: r -> P a b

such that:

fstP . u = f

sndP . u = g
Candidate
u1 r = case r of {}
u1 _ = ()
u1 (Left a) = f a

u1 (Right b) = g b
u1 r = (f r,g r)
u1 r = P (f r) (g r)
Example failure condition
r ~ ()
r ~ ()
r ~ ()

f _ = ()

g _ = ()
r ~ ()

f _ = undefined

g _ = undefined
r ~ ()

f _ = ()

g _ = undefined
Alternative u
u2 _ = ()
u2 _ = undefined
u2 _ = ()
u2 _ = undefined
Difference
u1 undefined = undefined

u2 undefined = ()
u1 _ = ()

u2 _ = undefined
u1 undefined = undefined

u2 undefined = ()
u1 _ = (undefined,undefined)

u2 _ = undefined
f _ = ()

(fstP . u1) _ = undefined
Result FAIL FAIL FAIL FAIL FAIL

3 "Platonic" Hask

Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottom values. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products, and instances of Functor and Monad really are endofunctors and monads.

4 Links

Retrieved from "http://www.haskell.org/haskellwiki/Hask"

Categories: Mathematics | Theoretical foundations