Question

通过直接将定义和法律翻译成Haskell，我很开心学习类别理论。 Haskell当然不是Coq，但它帮助我获得了类别理论的直觉。我的问题是：以下是合理的＆＃34;翻译＆＃34; Haskell中一个类别的定义？

{-
  The following definition of a category is adapted from "Basic Category Theory" by Jaap van Oosten:

  A category is given by a collection of objects and a collection of morphisms.

  Each morphism has a domain and a codomain which are objects.

  One writes f:X->Y (or X -f-> Y) if X is the domain of the morphism f, and Y its codomain.

  One also writes X = dom(f) and Y = cod(f)

  Given two morphisms f and g such that cod(f) = dom(g), the composition
  of f and g, written (g f), is defined and has domain dom(f) and codomain cod(g):
  (X -f-> Y -g-> Z) = (X -(g f)-> Z)

  Composition is associative, that is: given f : X -> Y , g : Y -> Z and h : Z -> W, h (g f) = (h g) f

  For every object X there is an identity morphism idX : X -> X, satisfying
  (idX g) = g for every g : Y -> X and (f idX) = f for every f : X -> Y.
-}

module Code.CategoryTheory where

---------------------------------------------------------------------

data Category o m =
  Category
  {
  -- A category is given by a collection of objects and a collection of morphisms:
    objects    :: [o],
    morphisms  :: [m],
  -- Each morphism has a domain and a codomain which are objects:
    domain     :: m -> o,
    codomain   :: m -> o,
  -- Given two morphisms f and g such that codomain f = domain g,
  -- the composition of f and g, written (g f), is defined:
    compose    :: m -> m -> Maybe m,
  -- For every object X there is an identity morphism idX : X -> X
    identity   :: o -> m
  }

---------------------------------------------------------------------

-- Check if (Category o m) is truly a category (category laws)

is_category :: (Eq o,Eq m) => Category o m -> Bool

is_category cat =
     domains_are_objects         cat
  && codomains_are_objects       cat
  && composition_is_defined      cat
  && composition_is_associative  cat
  && identity_is_identity        cat

---------------------------------------------------------------------

-- Each morphism has a domain and a codomain which are objects:

domains_are_objects :: Eq o => Category o m -> Bool

domains_are_objects cat =
  all (\m -> elem (domain cat m) (objects cat)) (morphisms cat)

codomains_are_objects :: Eq o => Category o m -> Bool

codomains_are_objects cat =
  all (\m -> elem (codomain cat m) (objects cat)) (morphisms cat)

---------------------------------------------------------------------

-- Given two morphisms f and g such that cod(f) = dom(g),
-- the composition of f and g, written (g f), is defined
-- and has domain dom(f) and codomain cod(g)

composition_is_defined :: Eq o => Category o m -> Bool

composition_is_defined cat =
  go $ morphisms cat
  where
    go []       = True
    go (m : mx) = all (go2 m) mx && go mx

    go2 g f =
      if domain cat g /= codomain cat f then
        True
      else
        case compose cat g f of
          Nothing -> False
          Just gf -> domain cat gf == domain cat f && codomain cat gf == codomain cat g

---------------------------------------------------------------------

-- Composition is associative, that is:
-- given f:X->Y, g:Y->Z and h:Z->W, h (g f) = (h g) f

composition_is_associative :: (Eq o,Eq m) => Category o m -> Bool

composition_is_associative cat =
  go $ morphisms cat
  where
    go []           = True
    go (m : mx)     = go2 m mx && go mx

    go2 _ []        = True
    go2 f (g : mx)  = all (go3 f g) mx && go2 f mx

    go3 f g h =
      if codomain cat f == domain cat g && codomain cat g == domain cat h then
        case (compose cat g f, compose cat h g) of
          (Just gf, Just hg) ->
            case (compose cat h gf, compose cat hg f) of
              (Just hgf0, Just hgf1) -> hgf0 == hgf1
              _                      -> False
          _ -> False
      else
        True

---------------------------------------------------------------------

-- For every object X there is an identity morphism idX : X -> X, satisfying
-- (idX g) = g for every g : Y -> X -- and (f idX) = f for every f : X -> Y.

identity_is_identity :: (Eq m,Eq o) => Category o m -> Bool

identity_is_identity cat =
  go $ objects cat
  where
    go []     = True
    go (o:ox) = all (go2 o) (morphisms cat) && go ox

    go2 o m =
      if domain cat m == o then
        case compose cat m (identity cat o) of
          Nothing -> False
          Just mo -> mo == m
      else if codomain cat m == o then
        case compose cat (identity cat o) m of
          Nothing -> False
          Just im -> im == m
      else
        True

---------------------------------------------------------------------

instance (Show m,Show o) => Show (Category o m) where
  show cat = "Category{objects=" ++ show (objects cat) ++ ",morphisms=" ++ show (morphisms cat) ++ "}"

---------------------------------------------------------------------

testCategory :: Category String (String,String,String)

testCategory =
  Category
  {
    objects   = ["A","B","C","D"],
    morphisms = [("f","A","B"),("g","B","C"),("h","C","D"),("i","A","D")],
    domain    = \(_,a,_) -> a,
    codomain  = \(_,_,b) -> b,
    compose   = \(g,gd,gc) (f,fd,fc) ->
            if fc /= gd then
              Nothing
            else if gd == gc then
              Just (f,fd,fc)
            else if fd == fc then
              Just (g,gd,gc)
            else
              Just (g ++ "." ++ f,fd,gc),
    identity = \o -> ("id" ++ show o, o, o)
  }

---------------------------------------------------------------------

main :: IO ()

main = do
  putStrLn "Category Theory"
  let cat = testCategory
  putStrLn $ show cat
  putStrLn $ "Is category: " ++ show (is_category cat)

Answer 1

这似乎并不是原始结构翻译的糟糕传递。但是你不能利用类型系统为你做任何检查。

数据类别包（https://hackage.haskell.org/package/data-category）使用一个巧妙的技巧来进行构建＆＃34;一级升级＆＃34;并强制执行态射等组成的某些属性......

核心是

class Category k where
  src :: k a b -> Obj k a
  tgt :: k a b -> Obj k b
  (.) :: k b c -> k a b -> k a c

type Obj k a = k a a

在这里，他只代表态射及其构图，然后将对象简单地捕捉到这些物体上的身份态射。我发现这个库非常强大，可以表达它。

在Haskell中定义类别和类别定律

1 个答案: