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
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