[Haskell-cafe] Re: Why monoids will abide...
Mauro J. Jaskelioff
mjj at Cs.Nott.AC.UK
Wed Jan 21 11:15:23 EST 2009
Andrzej Jaworski wrote:
> Monads are monoids in categories of functors C -> C Arrows are monoids
> in subcategories of bifunctors (C^op) x C -> C Trees are a playing
> ground for functors in general:-)
This is the nice thing about category theory! plenty of reuse of concepts :)
The situation for Arrows is a bit more complex. Monoids (C^op) x C -> C
are equivalent to Freyd categories (Heunen and Jacobs, MFPS 2006) , but
Arrows in Haskell are actually indexed-Freyd categories, as explained by
Bob Atkey in "What is a Categorical Model of Arrows?"
(MSFP 2008, http://homepages.inf.ed.ac.uk/ratkey/arrows.pdf)
The realization that monads are monoids would be far more useful in a
language with kind polymorphism.
However, this shouldn't stop us from dreaming...
All the best,
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