[Haskell-cafe] A question about monad based on http://en.wikipedia.org/wiki/Monad_(category_theory)

[wren ng thornton]

On 1/17/11 10:46 PM, C K Kashyap wrote:
> Hi,
> I was going through http://en.wikipedia.org/wiki/Monad_(category_theory) -
> under Formal Definition I notice that monad is a Functor T:C ->  C
>
> My question is - when we think of Maybe as a functor T:C ->  C .... should we
> think that C here refers to Hakell types?

Not quite. A functor is a mapping between categories.

Well, then what's a mapping between categories? It's a pair of maps, one 
mapping objects to objects and the other mapping arrows to arrows, such 
that they preserve the category structure (i.e., obey the functor laws). 
For categories like the ones for lambda calculi, the objects are types 
and the arrows are functions.

Thus, the C there is referring to some category (namely a category 
modeling Haskell's type system). If you wish, you can think of 
categories as being like Either Object Arrow, for a particular class of 
objects and a particular class of arrows. So we'd have,

     T : Either Ob Hom -> Either Ob Hom
     T (Left ob) = ...
     T (Right hom) = ...

Though nobody does that and we prefer our notation to slide over the 
fact that functors and categories have two parts, and so we actually write:

     T : C -> C
     T ob = ...
     T hom = ...

So we might have that T Int = Bool, or T Int = [Int], or whatever. But 
we don't usually give types for the components of a functor, since we 
care more about the functor as a whole (e.g., we don't give types to 
individual lines of a function definition either, we only give a type 
for the whole function).


> As in,
> (Int and Maybe Int are objects in C) and (Int ->  Int and Maybe Int ->  Maybe
> Int are arrows in C) and T is an arrow between them. Is that right? For some
> reason, I was
> imagining them as two different categories C and D.

Generally speaking functors do go between different categories, but 
there's no reason they can't be on the same category (e.g., the identity 
functor :)

Conventional monads are always endofunctors though, since otherwise we 
couldn't talk about the natural transformations like return and join (we 
don't have arrows between objects in different categories). There are 
some generalizations of monads that relax this however.

-- 
Live well,
~wren