Re: [Haskell-cafe] N and R are categories, no?
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 ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] N and R are categories, no?
On 3/15/07, Steve Downey [EMAIL PROTECTED] wrote: EOk, i'm trying to write down, not another monad tutorial, because I don't know that much yet, but an explication of my current understanding of monads. But before I write down something that is just flat worng, I thought I'd get a cross check. (and I can't get to #haskell) Monads are Functors. Functors are projections from one category to another such that structure is preserved. One example I have in mind is the embedding of the natural numbers into the real numbers. The mapping is so good, that we don't flinch at saying 1 == 1.0. Monads are endofunctors (functors from one category to itself). This is easy to see from the type of join: join : m (m a) - m a For Haskell monads the category is the category of Haskell types and Haskell functions. In this category N and R are objects, so you'll get the wrong idea trying to see them as categories. / Ulf ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] N and R are categories, no?
That said, N and R are indeed categories; however, considering them as categories should be carefully interlaced with your intuitions about them as types. I haven't formally checked it, but I would bet that this endofunctor over N, called Sign, is a monad: Sign x = x + x Pos = injectLeft Neg = injectRight unit = Pos join (Pos (Pos n)) = Pos n join (Pos (Neg n)) = Neg n join (Neg (Pos n)) = Neg n join (Neg (Neg n)) = Pos n Pos and Neg are just labels for sign. I'm assuming N is the naturals, not the integers; thus this monad might actually be useful :). Also note that this means there is not necessarily a mapping from F x - x. Neg 3 should not necessarily map to 3. Also, this structure is probably satisfies many more laws than just the monad laws--e.g. monoids or monoidals. So while it might not always make sense to consider N and R as categories when learning about category theory and Haskell, it might be helpful to learn about monads (and other notions) in categories simpler than the Fun category of functional types and partial functions--N and R are could be good categories for such learning. Have fun! On 3/15/07, Ulf Norell [EMAIL PROTECTED] wrote: On 3/15/07, Steve Downey [EMAIL PROTECTED] wrote: EOk, i'm trying to write down, not another monad tutorial, because I don't know that much yet, but an explication of my current understanding of monads. But before I write down something that is just flat worng, I thought I'd get a cross check. (and I can't get to #haskell) Monads are Functors. Functors are projections from one category to another such that structure is preserved. One example I have in mind is the embedding of the natural numbers into the real numbers. The mapping is so good, that we don't flinch at saying 1 == 1.0. Monads are endofunctors (functors from one category to itself). This is easy to see from the type of join: join : m (m a) - m a For Haskell monads the category is the category of Haskell types and Haskell functions. In this category N and R are objects, so you'll get the wrong idea trying to see them as categories. / Ulf ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] N and R are categories, no?
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. 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. A functor is then an order preserving map between 2 ordered sets and monad is then a closure (http://en.wikipedia.org/wiki/Closure_operator) - I didn't know this latter fact until I just looked it up. Dominic. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] N and R are categories, no?
Thanks for keeping me honest ;) On 3/15/07, Dominic Steinitz [EMAIL PROTECTED] 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. 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. A functor is then an order preserving map between 2 ordered sets and monad is then a closure (http://en.wikipedia.org/wiki/Closure_operator) - I didn't know this latter fact until I just looked it up. Dominic. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] N and R are categories, no?
EOk, i'm trying to write down, not another monad tutorial, because I don't know that much yet, but an explication of my current understanding of monads. But before I write down something that is just flat worng, I thought I'd get a cross check. (and I can't get to #haskell) Monads are Functors. Functors are projections from one category to another such that structure is preserved. One example I have in mind is the embedding of the natural numbers into the real numbers. The mapping is so good, that we don't flinch at saying 1 == 1.0. The functor that takes us from N to R is probably a Monad, that is, if N and R are categories. The real hard part is tying together how unit, join and bind produce a spacesuit that can protect apples from nuclear waste. I'm still getting that clear in my head, although my recent blinding flash of obviousness that M a is a type, and that of course types can do interesting things, I think gets me further along. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
