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