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'.
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.