BM Category Theory 8 : Function objects (Exponentials), Curry / unCurry


[Revision] Product (or Sum = co-Product) is a Bifunctor

  • takes 2 objects to produce 3rd object;
  • takes 2 morphisms to produce 3rd morphism. 

Functions are “morphisms between Category objects”  = a Hom-Set, which itself is an object : ‘a -> b’.

Using the same method of Universal Construction of Product objects in category :  

  1. Define a Pattern ,
  2. Ranking which object is better, 
  3. Pick the best object up to an UNIQUE Isomorphism.

we can similarly construct the Function Objects ‘a => b‘ (in Haskell as ‘a -> b’) :

Intuitively, the above diagram is interpreted as: 

  • evalthe Function object(a=>b) x ato return result b
  • g is up to Unique isomorphism (hSAME as ‘eval: \boxed {g = eval \circ (h \text { x } id )}

Curry / Non-curry : Associative (yellow brackets below are optional).

\boxed {z \to (a \to b ) \iff z \to a \to b }

\boxed { (f a ) \: b \iff f a \: b }

Note: function (f a) returns as a new  function which applies on argument b. Equivalently, also read as “f takes 2 arguments a, b”.


f :: Bool -> Int

Bool = {0, 1}
{ f = (Int \text { x }  Int) = Int^{2} = Int ^ {Bool}}

The number of possibilities of applying f from a to b is \boxed {b^{a}} , hence we call:  
\boxed {\text {Function object f as EXPONENTIAL}}

8.2  Algebraic Data Types


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