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POMonad theory and Haskell
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copied!<p>Most tutorials seem to give a lot of examples of monads (IO, state, list and so on) and then expect the reader to be able to abstract the overall principle and then they mention category theory. I don't tend to learn very well by trying generalise from examples and I would like to understand from a theoretical point of view why this pattern is so important.</p> <p>Judging from this thread: <a href="https://stackoverflow.com/questions/2366/can-anyone-explain-monads">Can anyone explain Monads?</a> this is a common problem, and I've tried looking at most of the tutorials suggested (except the Brian Beck videos which won't play on my linux machine):</p> <p>Does anyone know of a tutorial that starts from category theory and explains IO, state, list monads in those terms? the following is my unsuccessful attempt to do so:</p> <p>As I understand it a monad consists of a triple: an endo-functor and two natural transformations.</p> <p>The functor is usually shown with the type: (a -> b) -> (m a -> m b) I included the second bracket just to emphasise the symmetry.</p> <p>But, this is an endofunctor, so shouldn't the domain and codomain be the same like this?:</p> <p>(a -> b) -> (a -> b)</p> <p>I think the answer is that the domain and codomain both have a type of:</p> <p>(a -> b) | (m a -> m b) | (m m a -> m m b) and so on ...</p> <p>But I'm not really sure if that works or fits in with the definition of the functor given?</p> <p>When we move on to the natural transformation it gets even worse. If I understand correctly a natural transformation is a second order functor (with certain rules) that is a functor from one functor to another one. So since we have defined the functor above the general type of the natural transformations would be: ((a -> b) -> (m a -> m b)) -> ((a -> b) -> (m a -> m b))</p> <p>But the actual natural transformations we are using have type:</p> <p>a -> m a</p> <p>m a -> (a ->m b) -> m b</p> <p>Are these subsets of the general form above? and why are they natural transformations?</p> <p>Martin</p>

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