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1. • CommentRowNumber1.
   • CommentAuthorbblfish
   • CommentTimeAug 23rd 2014
   • (edited Aug 23rd 2014)
   • PermaLink
   Author: bblfish
   Format: MarkdownItexWhile studying [scalaz](https://github.com/scalaz/scalaz) using the help of the nice mapping from Haskell to Scalaz [learning scalaz](http://eed3si9n.com/learning-scalaz/Combined+Pages.html#Reader) I came across the description of Reader. It shows that a function can be considered as a Functor - just as a List or Set can. Since Functors are a map between categories I was wondering what a function functor was a map from and to. So it seems to be able to be from any category to something, which for a while I thought could be the [Arrow Category](http://ncatlab.org/nlab/show/arrow%20category). So that would mean that a function functor would be a map from C to Arr(C). But then I'd need to find * a map $F_0 : Obj(C) \to Obj(Arr(C))$ * a map $F_1 : Mor(C) \to Mor(Arr(C))$ In ScalaZ a functor of functions is [hardwired to an initial type](https://github.com/scalaz/scalaz/blob/series/7.2.x/core/src/main/scala/scalaz/std/Function.scala#L92). So that one has a functor of all function Int => X for example. ( presumably because Arr(C) is somehow a 2 Category, but functors are not ) So I suppose $F_0$ needs to be a map from Obj(c) to the functions of type Int=>X ( taking Int as fixed ) in Arr(c). But then that means that Arr(c) can't be the object of the function functor, as there is no such object in Arr(c).... Any hints?

   While studying scalaz using the help of the nice mapping from Haskell to Scalaz learning scalaz I came across the description of Reader. It shows that a function can be considered as a Functor - just as a List or Set can. Since Functors are a map between categories I was wondering what a function functor was a map from and to. So it seems to be able to be from any category to something, which for a while I thought could be the Arrow Category. So that would mean that a function functor would be a map from C to Arr(C). But then I’d need to find

   • a map $F_0 : Obj(C) \to Obj(Arr(C))$
   • a map $F_1 : Mor(C) \to Mor(Arr(C))$

   In ScalaZ a functor of functions is hardwired to an initial type. So that one has a functor of all function Int => X for example. ( presumably because Arr(C) is somehow a 2 Category, but functors are not ) So I suppose $F_0$ needs to be a map from Obj(c) to the functions of type Int=>X ( taking Int as fixed ) in Arr(c). But then that means that Arr(c) can’t be the object of the function functor, as there is no such object in Arr(c)….

   Any hints?

2. • CommentRowNumber2.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeAug 23rd 2014
   • PermaLink
   Author: IngoBlechschmidt
   Format: MarkdownItexJust a short remark: In Haskell, for a fixed type r, the Reader r monad is a functor Hask -> Hask, where Hask is the category of Haskell types and functions between those. More specifically, it is the functor Hom(r, __) (internal Hom in Hask). This is all modulo the usual caveats about the category Hask. In general category theory, there is a functor Hom(r, __) : C -> Set for any object r of any category C.

   Just a short remark: In Haskell, for a fixed type r, the Reader r monad is a functor Hask -> Hask, where Hask is the category of Haskell types and functions between those. More specifically, it is the functor Hom(r, __) (internal Hom in Hask). This is all modulo the usual caveats about the category Hask.

   In general category theory, there is a functor Hom(r, __) : C -> Set for any object r of any category C.