Re: FW: RE [Haskell-cafe] Monad Description For Imperative Programmer

I knew someone was going to catch me wandering into the deep end of the pool! Having read large parts of your blog, I would never presume to tell you anything about Haskell or category theory, but what the hell...

I mostly sympathise with your rant, but I think you need to be clearer about what exactly is concatenated. In general you can't concatenate Monads. What you *can* concatenate are Kleisli arrows (ie. things of type Monad m => a -> m b). You can also apply Kleisli arrows to Monads, and that's what >>= does.

I feel that talking about Monads without Kleisli arrows is like talking about category theory without arrows, or at least sets without functions. In each case, without the latter, the former is more or less useless.

OK, I'll be clearer. I did actually mean Kleisli arrows, though I disagree about your statement about concatenating monad instances and claims of "useless": Prelude> print "Hello" >> return 3 "Hello" 3 Granted, >> is not as "general" as >>=, so combining one monad instance with another is not as "general" as with a Kleisli arrow: concatenating degenerates to simple sequencing. Sequencing print statements is more rather than less useless to many people, but I see your point. Actually, you have made my point! :) The forgetful action of Kleisli arrows acting on a monad (or conversely the free algebra of a monad as a subspace of Kleisli arrows) is to my understanding intimately connected with the specialness of the IO monad. It is the continuous nature of Haskell monads that gives non-IO monads value. So I guess my rant really was about Kleisli arrows not all being forgetful functors, used only for their sequencing effect. It just sounded too hard to pull that argument off without reinforcing the myth that you need to know category theory to have a rant about Haskell tutorials.

Also, I'm having a terminological difficulty that maybe someone can help with:

'Monad' is a type class.

Actually I thought it was a type class constructor. The monad Monad m => m a is continuous in its instance type a, which is important in establishing the relationship between >>= and >>. The Haskell type 'IO ()' is a monad instance that is also isomorphic to the discrete trivial monad, but that is not a Haskell Monad capital-M. I used the term "instance" because the type IO () is an instance of the typeclass IO, not for any more profound reason. Forgive the display of wanton ignorance above. After all, isn't that what ranting is all about? Dan Weston Dan Piponi wrote:

On 8/1/07, Dan Weston wrote:

...

The moral of the story is that monads are less than meets the eye. You can create them and concatenate them

I mostly sympathise with your rant, but I think you need to be clearer about what exactly is concatenated. In general you can't concatenate Monads. What you *can* concatenate are Kleisli arrows (ie. things of type Monad m => a -> m b). You can also apply Kleisli arrows to Monads, and that's what >>= does.

I feel that talking about Monads without Kleisli arrows is like talking about category theory without arrows, or at least sets without functions. In each case, without the latter, the former is more or less useless.

Also, I'm having a terminological difficulty that maybe someone can help with:

'Monad' is a type class.

So what's 'IO'? Is the correct terminology 'instance' as in 'IO is an instance of Monad'. I consider 'IO' to be 'a monad' as that fits with mathematical terminology. But what about an actual object of type 'IO Int', say? Some people have been loosely calling such an object 'a monad'. That doesn't seem quite right. Maybe it's 'an instance of IO Int', though that's stretching the word 'instance' to meaning two different things. And if an object of type IO Int is in instance of IO Int, is it reasonable to also call it an 'instance of IO', or even 'an instance of Monad'? I'm sure there are proper words for all these things if someone fills me in. -- Dan _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe