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

[Alexander Solla]

On Tue, Sep 4, 2012 at 4:21 PM, Alexander Solla <alex.solla at gmail.com>wrote:

>
>
> On Tue, Sep 4, 2012 at 3:39 AM, Alberto G. Corona <agocorona at gmail.com>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.)
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