I am not sure I follow how the endofunctor gave me the 2nd functor.<br><br>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?<br>
<br>Thanks<br><br>Daryoush<br><br><br><br><div class="gmail_quote">On Tue, Apr 21, 2009 at 4:01 PM, Dan Weston <span dir="ltr"><<a href="mailto:westondan@imageworks.com">westondan@imageworks.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">You are on the right track. The usual construction is that Hask is the category (with types as objects and functions as morphisms).<br>
<br>
Functor F is then an endofunctor taking Hask to itself:<br>
<br>
a -> F a<br>
f -> fmap f<br>
<br>
So, for F = []:<br>
<br>
a -> [a]<br>
f -> map f<br>
<br>
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.<div><div></div>
<div class="h5"><br>
<br>
Daryoush Mehrtash wrote:<br>
<blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
In category theory functors are defined between two category of C and D where every object and morphism from C is mapped to D. <br>
<br>
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?<br>
<br>
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? <br>
Thanks,<br>
<br>
Daryoush<br>
<br>
<br>
<br>
</blockquote>
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