README.md

Notes on categorical structures in Haskell.

These are a collection of some notes on category theory collected from various sources. Sometimes I learn something new about category theory this way.

things I've learned recently (sorry if this is newbish!)

1. Language pragmas are important for extending GHC. So far I've used PolyKinds, TypeOperators, MultiParamTypeClasses, RankNTypes.

2. Yoneda's lemma could be rephrased as saying the right Kan extension of a functor along the identity is itself.

3. Kan extensions are pretty cool. Given an adjoint pair, we can write each functor as a Kan extension of the other. Precisely, if F:C->D and G:D->C are an adjoint pair (F = left) then G is the left Kan extension of id:C->C along F, and F is the right Kan extension of id:D->D along G,

G = Lan F Identity

and

F = Ran G Identity

This all conspires to leading to the codensity monad. In essence, adjoint pairs give rise to monads. But when we don't have an adjoint pair, but instead just a right Kan extension Ran G Identity, we can still form the codensity monad. This monad agrees with the monad of an adjunction if the right Kan extension is preserved by the left adjoint F.