Calculating Functors

• To: types@cis.upenn.edu
• Subject: Calculating Functors
• From: "Samuel E. Moelius III" <usmoeliu@mcs.drexel.edu>
• Date: Thu, 25 Apr 2002 18:31:03 -0400

Let + range over all covariant type operators.  Let - range over all 
contravariant type operators.  Now imagine one has a term, fun+, which 
calculates for any covariant type operator its associated covariant 
functor.  The type of fun+ would be:

	all F:+. all A:*. all B:*. (A -> B) -> F[A] -> F[B]

Now imagine, similarly, a term, fun-, which calculates for any 
contravariant type operator its associated contravariant functor.  The 
type of fun- would be:

	all G:-. all A:*. all B:*. (B -> A) -> G[A] -> G[B]

Question 1.  If one had fun+ only, could one construct fun-?  Similarly, 
could one construct fun+ from fun-?

Question 2.  Assuming one has fun+ and fun-, can one construct a term 
which calculates a functor for any type operator?  That is, a term of 
type

	all H:* -> *. all A:*. all B:*. (A -> B) -> (B -> A) -> H[A] -> H[B]

Sam Moelius

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