Re: [Haskell-cafe] From monads to monoids in a small category

Moreover, `m a` is 'a' plus some terminal element , for example Nothing, [], Left _ etc, So a morphism (a -> m a) contains all the morphisms of (m a -> m a). 2012/9/5 Alberto G. Corona :

Alexander,

In my post (excuses for my dyslexia) I try to demonstrate that the codomain (m a), from the point of view of C. Theory, can be seen as 'a' plus, optionally, some additional element, so a monadic morphism (a -> m a) is part of a endofunctor in (m a -> m a)

When considering the concept of arrow from category theory, (not the concept of function), a point in the domain can "send" more than one arrow to the codomain, while a function do not. I think that this is the most interesting part of the interpretation, if I´m right.

About this, I found this article revealing:

http://cdsmith.wordpress.com/2012/04/18/why-do-monads-matter/

Therefore, codomains of (a -> m a) which are containers with multiple a elements can be considered as multi-arrow morphisms from 'a' to 'a' with the optional addition of some special elements that denote special conditions.

For example

(a -> [a])

May be considered as the general signature of the morphisms from the set 'a' to the set ('a' + [])

So a monadic transformation (a -> [a]) can be considered as a point transformation of a endofunctor within the set (a + []).

Alberto

2012/9/5 Alexander Solla :

...

On Tue, Sep 4, 2012 at 4:21 PM, Alexander Solla wrote:

...

On Tue, Sep 4, 2012 at 3:39 AM, Alberto G. Corona wrote:

...

"Monads are monoids in the category of endofunctors"

This Monoid instance for the endofunctors of the set of all elements of (m a) typematch in Haskell with FlexibleInstances:

instance Monad m => Monoid (a -> m a) where mappend = (>=>) -- kleisly operator mempty = return

The objects of a Kliesli category for a monad m aren't endofunctors. You want something like:

instance Monad m => Monoid (m a -> m (m a)) where ...

/These/ are endofunctors, in virtue of join transforming an m (m a) into an (m a).

Actually, even these aren't endofunctors, for a similar reason that : you "really" want something like

instance Monad m => Monoid (m a -> m a) where mempty = id mappend = undefined -- exercise left to the reader

(i.e., you want to do plumbing through the Eilenberg-Moore category for a monad, instead of the Kliesli category for a monad -- my last message exposes the kind of plumping you want, but not the right types.)