[Haskell-beginners] Are functors in Haskell always injective?

[Julian Birch]

(Another beginner here)

That's my understanding too.  Functors in Haskell have to operate on the
whole of Hask, which isn't true of functors in general.  Equally, they have
to map to a subset of Haskell (although that mapping could be isomorphic to
Hask e.g. Identity Functor).  It's the first restriction that's the most
problematic, since it prevents set from being a Functor.  This is fixable (
http://okmij.org/ftp/Haskell/RestrictedMonad.lhs) but not part of "standard
Haskell".  The second restriction basically just means you've got to do
some book-keeping to get back to the original type.

Julian


On 5 July 2014 06:51, Dimitri DeFigueiredo <defigueiredo at ucdavis.edu> wrote:

>  Oops! Sorry for the typos! Fixing that below.
>
>
> Hi All,
>
> My understanding is that a functor in Haskell is made of two "maps".
> One that maps objects to objects that in Hask means types into types (i.e.
> a type constructor)
> And one that maps arrows into arrows, i.e. fmap.
>
> My understanding is that a functor F in category theory is required to
> preserve the domain and codomain of arrows, but it doesn't have to be
> injective. In other words, two objects X and Y of category C (i.e. two
> types in Hask) can be mapped into the same object Z (same type) in category
> D. As long as the "homomorphism" law holds:
>
> F(f:X->Y) = F(f):F(X)->F(Y)
>
> However, I don't think there is any way this mapping of types cannot be
> injective in Haskell. It seems that a type constructor, when called with
> two distinct type will always yield another two **distinct** types. (E.g.
> Int and Double yield Maybe Int and Maybe Double) So, it seems that Functors
> in Haskell are actually more restrictive than functors can be in general.
> Is this observation correct or did I misunderstand something?
>
>
> Thanks!
>
>
> Dimitri
>
> Em 05/07/14 02:42, Dimitri DeFigueiredo escreveu:
>
> Hi All,
>
> My understanding is that a functor in Haskell is made of two "maps".
> One that maps objects to objects that in Hask means types into types (i.e.
> a type constructor)
> And one that maps arrows into arrows, i.e. fmap.
>
> My understanding is that a functor F in category theory is required to
> preserve the domain and codomain of arrows, but it doesn't have to be
> injective. In other words, two objects X and Y of category C (i.e. two
> types in Hask) can be mapped into the same object Z (same type) in category
> Z. As long as the "homomorphism" law holds:
>
> F(f:X->Y) = F(f):F(X)->F(Y)
>
> However, I don't think there is any way this mapping of types cannot be
> injective in Haskell. It seems that a type constructor, when called with
> two distinct type will always yield another two *distinct* types. (E.g. Int
> and Fload yield Maybe Int and Maybe) So, it seems that Functors in Haskell
> are actually more restrictive than functors can be in general. Is this
> observation correct or did I misunderstand something?
>
>
> Thanks!
>
>
> Dimitri
>
>
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