[Haskell-cafe] From monads to monoids in a small category
alex.solla at gmail.com
Wed Sep 5 01:46:13 CEST 2012
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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