[Haskell-cafe] Join and it's relation to >>= and return

Ron de Bruijn rondebruijn at yahoo.com
Wed Jun 9 17:34:02 EDT 2004

Hello again,

I have thought a while about morphisms and although I
had written down in my paper that a functor and also a
natural transformation are also morphisms, but in a
different category, I now am not sure anymore of this.

If you see everything(objects and morphisms) as dots
and arrows, and some arrows and some dots are just
some more complex than others, then this holds, but is
that legal? Intuitively seen, it is. 

According to some paper, morphisms are not like
functions, in that you can apply some morphism to an
object, but that you can only compose them. But I
would say that if you have the morphism f:a->b, that
is a arrow from dot a to dot b. That there clearly is
a notion of following that arrow, in effect applying a
function. And suppose there is the following path of
morphisms: a---->b---->c---->d, with a..d are dots. 
Then I would say there are three functions(constructed
by composition)(in fact more, because of identity
mapping) from a that when followed give new objects. 
This following of arrows, looks a lot like general
function application, as in f(x) = 2x for example. 

It's btw quite hard to write the essence of monads
down in a clear and precise way. I hope you can give
some feedback on the above. 
Ragards, Ron

P.S. The question about multiplication still stands.
Probably multiplication is a set of laws defined on a
mathematical object that must hold. And for each
mathematical object there is such definition. Is this

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