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

Dominic Steinitz dominic.steinitz at blueyonder.co.uk
Fri Mar 16 11:00:26 EDT 2007

```Jules Bean <jules <at> jellybean.co.uk> writes:

>
> 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,
>

I do enjoy Mornington Crescent (you probably need to listen to BBC Radio 4 to
understand this) so I'll respond. An adjunction gives rise to at least two
monads (Kleisli and Eilenberg-Moore) so I think it is important to state what
the natural transformations are. I believe this thread was started by someone
trying to understand monads so I thought I would clarify that it's important to
know what the natural transformations are. You probably know this but monads
were at one time referred to as triples.

> > 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

That was my whole point. Most examples of categories do have some internal
structure but of course objects don't and this structure is irrelevant. A very
good example to keep in your head when being introduced to category theory is a
partially ordered set and that was the point I was trying to make.

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

Agreed. Mornington Crescent?

Dominic.

```