I am not sure I follow how the endofunctor gave me the 2nd functor. As I read the transformation there are two catagories C and D and two functors F and G between the same two catagories. My problem is that I only have one functor between the Hask and List catagories. So where does the 2nd functor come into picture that also maps between the same C and D catagories? Thanks Daryoush On Tue, Apr 21, 2009 at 4:01 PM, Dan Weston wrote:

You are on the right track. The usual construction is that Hask is the category (with types as objects and functions as morphisms).

Functor F is then an endofunctor taking Hask to itself:

a -> F a f -> fmap f

So, for F = []:

a -> [a] f -> map f

Natural transformations are then any fully polymorphic (no context) unary function. The polymorphism is what makes them natural, since there is no method to treat one object (type) different from another.

Daryoush Mehrtash wrote:

...

In category theory functors are defined between two category of C and D where every object and morphism from C is mapped to D.

I am trying to make sense of the above definition with functor class in Haskell. Let say I am dealing with List type. When I define List to be a instance of a functor I am saying the source category (C) is Haskell types and the destination category is List (D). In this the "fmap" is implementation of the mapping between every morphism in my Haskell Categroy (C) to morphism in List cataegory (D). With type constructor I also have the mapping of types (objects in Haskell Category, or my source cataegroy C) to List category (D). So my functor in the catarogy sense is actually the fmap and type constructor. Am I remotely correct?

If this is correct.... With this example, then can you then help me understand the transformation between functors and natural transformations? Specifically, what does it means to have two different functors between Haskell cateogry and List category? Thanks,

Daryoush

Daryoush Mehrtash wrote:

I am not sure I follow how the endofunctor gave me the 2nd functor.

As I read the transformation there are two catagories C and D and two functors F and G between the same two catagories. My problem is that I only have one functor between the Hask and List catagories. So where does the 2nd functor come into picture that also maps between the same C and D catagories?

You don't have a category List, you have the image of the List functor (in Hask). With the definition: data List a =... instance Functor List where fmap =... The type constructor "List" provides the object half of an endofunctor on Hask, and the function "fmap @List" provides the morphism half of the endofunctor. Now, if you declare another endofunctor, G: data G a =... instance Functor G where fmap =... then a natural transformation from List to G must provide a mapping (forall a. List a -> G a) for objects and a mapping (forall a b. (List a -> List b) -> (G a -> G b)) for morphisms, satisfying all the necessary laws. -- Live well, ~wren

Here F is the identity functor, and G is the list functor. And yes, C=D= category of (a subset of) Haskell types. Are you saying the function that goes from list functor to singleton funtor is a natural transformation? But aren't they functors to different subset of Haskell Types? The Haskell Wikibooks also sayshttp://en.wikibooks.org/wiki/Haskell/Category_theory#Functors_on_Haskthe same thing:

Functors in Haskell are from *Hask* to *func*, where *func* is the subcategory of *Hask* defined on just that functor's types. E.g. the list functor goes from *Hask* to *Lst*, where *Lst* is the category containing only *list types*, that is, [T] for any type T. The morphisms in *Lst* are functions defined on list types, that is, functions [T] -> [U] for types T, U.

So in your example there is C that is Hask. But there are two D's, D1 that is all List types, and D2 all singleton types. In this example I guess, the Singleton types are subset of List types which are subset of Hask. Is that related to natural transformation or unrelated? Daryoush On Wed, Apr 22, 2009 at 12:18 AM, Kim-Ee Yeoh wrote:

Daryoush Mehrtash-2 wrote:

...

I am not sure I follow how the endofunctor gave me the 2nd functor.

As I read the transformation there are two catagories C and D and two functors F and G between the same two catagories. My problem is that I only have one functor between the Hask and List catagories. So where does the 2nd functor come into picture that also maps between the same C and D catagories?

Consider singleton :: a -> [a] singleton x = [x]

Here F is the identity functor, and G is the list functor. And yes, C=D= category of (a subset of) Haskell types.

-- View this message in context: http://www.nabble.com/Functor-and-Haskell-tp23166441p23170956.html Sent from the Haskell - Haskell-Cafe mailing list archive at Nabble.com.

_______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe

-- Daryoush Weblog: http://perlustration.blogspot.com/

Thanks for this example I get the point now. (at least i think i do :) ) One more question.... This all being on the same category then the functor transformation can also be view as a simple morphism too. In this example the listToMaybe can be viewed as morphism between list and Maybe types that are both in the Hask categroy too. right? If so then what would viewing the morphism as natural transformation by you? Thanks, Daryoush P.S. for others interested in the same topic I found this on Haskell wikis that add some more detail: http://www.haskell.org/haskellwiki/Category_theory/Natural_transformation On Wed, Apr 22, 2009 at 3:40 PM, Ross Paterson wrote:

On Wed, Apr 22, 2009 at 03:14:03PM -0700, Daryoush Mehrtash wrote:

...

The Haskell Wikibooks also says the same thing:

Functors in Haskell are from Hask to func, where func is the subcategory of Hask defined on just that functor's types. E.g. the list functor goes from Hask to Lst, where Lst is the category containing only list types, that is, [T] for any type T. The morphisms in Lst are functions defined on list types, that is, functions [T] -> [U] for types T, U.

That's true, but not a particularly helpful view. Any functor F : C -> D can be viewed as a functor from C to the full subcategory of D on objects of the form F A for A an object of C. But then different functors map to different categories and you can't talk about natural transformations between them. Composing functors also becomes impossible.

The simple view is that [], Maybe and Id are all functors from Hask to Hask. Then listToMaybe :: [a] -> Maybe a is a natural transformation from [] to Maybe, because

fmap f . listToMaybe = listToMaybe . map f _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe

Thanks this was helpful. In many of Conal Elliot's writings I see that he shows that his semantic function is a natural transformation. Is that just basically showing the polymorphic nature of his semantic functions, or are there other benifits you get by showing a particular function is a natural transformation? Daryoush On Thu, Apr 23, 2009 at 12:34 PM, Dan Doel wrote:

...

Thanks for this example I get the point now. (at least i think i do :) )

One more question.... This all being on the same category then the functor transformation can also be view as a simple morphism too. In this example the listToMaybe can be viewed as morphism between list and Maybe types

On Thursday 23 April 2009 2:44:48 pm Daryoush Mehrtash wrote: that

...

are both in the Hask categroy too. right? If so then what would viewing the morphism as natural transformation by you?

listToMaybe in general wouldn't be a morphism in the category, because morphisms would be from concrete types to other concrete types. [1] So, if you'll excuse some notation I just made up (with a little help from GHC core notation :)):

listToMaybe@Int :: [Int] -> Maybe Int listToMaybe@Char :: [Char] -> Maybe Char listToMaybe@String :: [String] -> Maybe String

are all morphisms in the alleged Hask category. Each polymorphic function (similar to the above one, at least) defines a family of morphisms like that. *But*, that's what a natural transformation is: a family of morphisms, one for each object in the category, that commute with functor application in a certain way. Thus, one can look at the fully polymorphic listToMaybe as a natural transformation:

listToMaybe :: [] -> Maybe

-- Dan

[1] Maybe you could make up a category where polymorphic types are objects as well, but that doesn't seem to be the way people typically go about applying category theory to Haskell. _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe

Daryoush Mehrtash-2 wrote:

...

singleton :: a -> [a] singleton x = [x]

Here F is the identity functor, and G is the list functor. And yes, C=D= category of (a subset of) Haskell types.

Are you saying the function that goes from list functor to singleton funtor is a natural transformation?

Here *singleton* is the natural transformation from the identity functor to the list functor. Daryoush Mehrtash-2 wrote:

But aren't they functors to different subset of Haskell Types?

They're usually treated as endofunctors on Hask, for reasons Ross Paterson has given. -- View this message in context: http://www.nabble.com/Functor-and-Haskell-tp23166441p23189784.html Sent from the Haskell - Haskell-Cafe mailing list archive at Nabble.com.