ajb-2 wrote:
In Haskell, natural transformations are functions that respect the structure of functors. Since you can't avoid respecting the structure of functors (the language won't let you do otherwise), you get natural transformations for free. (Free as in theorems, not free as in beer.)
It's worth noting that polymorphism is one of those unavoidable, albeit hidden, functors. Polymorphizing a function "forall a" can be thought of as lifting it via the ((->) T_a) functor, where T_a is the type variable of "a". E.g. reverse really has the signature T_a -> [a] -> [a]. But ((->) T_a) is left adjoint to ((,) T_a), which just happens to be the analogous type-explicit way of representing existentials. Adjunctions are a useful tool to describe and analyze such dualities precisely. ajb-2 wrote:
Adjunctions, on the other hand, you have to make yourself. As such, they're more like monads.
Constructing adjunctions, comprising as they do of a pair of functors, does seem double the work of a single monad. Duality OTOH is a powerful guiding principle and may well be easier than working with the monad laws directly. And besides, you get a comonad for free. ajb-2 wrote:
One thing that springs to mind is that an adjunction could connect monads and their associated comonads. Is that a good picture?
Definitely. -- View this message in context: http://www.nabble.com/Functional-programmer%27s-intuition-for-adjunctions--t... Sent from the Haskell - Haskell-Cafe mailing list archive at Nabble.com.