[Haskell-cafe] N and R are categories, no?

[Jules Bean]

Dominic Steinitz wrote:
>> I haven't formally checked it, but I would bet that this endofunctor
>> over N, called Sign, is a monad:
>>     
>
> Just to be picky a functor isn't a monad. A monad is a triple consisting of a 
> functor and 2 natural transformations which make certain diagrams commute.
>
>   

Whilst that's true, the statement 'T is a monad' has a perfectly 
sensible meaning. It means "there exist two natural transformations 
which make T a monad". This is often expressed as 'T is monadic' which, 
in turn, is sometimes more concretely defined as 'T has a left adjoint, 
such that the adjunction is monadic'.

> If you are looking for examples, I always think that a partially ordered set 
> is a good because the objects don't have any elements. 
Since we're playing 'pedantry' games, objects in categories don't have 
elements :P However if you take 'element' to mean 'morphism from the 
terminal object' then neither R nor N have terminal objects.

Certainly I'd agree that partial orders probably aren't very interesting 
categories to look for monads in.

Jules