N_P1Ch07b_Functors_AcrossCats

Markdown of literate Haskell program. Program source: /src/CTNotes/P1Ch07b_Functors_AcrossCats.lhs

Note about CTFP Part 1 Chapter 7. Functors on non-Hask categories

Ref: https://hackage.haskell.org/package/category-extras-0.53.5/docs/Control-Functor-Categorical.html

Book ref: CTFP Ch 7

 {-# LANGUAGE PolyKinds #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
 {-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE KindSignatures #-}
 {-# LANGUAGE DataKinds #-}

 module CTNotes.P1Ch07b_Functors_AcrossCats where
 import Data.Functor.Const (Const(..))
 import Control.Category 
 import Prelude(undefined, ($))
 import qualified CTNotes.P1Ch03b_FiniteCats as Finite

category-extras defines CFunctor (quoted code):

class (Category r, Category s) => CFunctor f r s | f r -> s, f s -> r where
  cmap :: r a b -> s (f a) (f b)

Ignoring functional dependencies I define non-endofunctors as:

 class (Category r, Category s) => CFunctor f r s  where
    cmap :: r a b -> s (f a) (f b)

With PolyKinds in place this assumes most generic kinds for r and s, r :: k1 -> k1 -> * r :: k2 -> k2 -> *. That makes the definition usable with non-Hask categories like the categories I created in N_P1Ch03b_FiniteCats or N_P1Ch03a_NatsCat

Here is one polymorphic instance:

 instance (Category r) => CFunctor (Const a) r (->) where
   cmap _ (Const v) = Const v

Const is defined (imported from Data.Functor.Const base package) as

newtype Const a b = Const { getConst :: a }

λ> :k Const
Const :: * -> k -> *

PolyKinds keeps kind of b polymorphic but the destination category has to have objects of kind * (it does not need to be (->)).

I was able to define instance of CFunctor polymorphically (in r) because Const does not care about source category. I am using (->) for destination, because this is what the implementation \Const v -> Const v uses.

To show that this works, I use category defined in the imported notes N_P1Ch03b_FiniteCats.

 test :: Const a (x :: Finite.Object) -> Const a (x :: Finite.Object)
 test = cmap Finite.MorphId

But this would not compile:

-- Bad code
test :: Const a (x :: Finite.Object) -> Const a (x :: Finite.Object)
test = cmap Finite.MorphAB

TODO: more examples?