[Haskell-cafe] Categories (cont.)

[Gershom Bazerman]

On 12/20/12 10:05 PM, Jay Sulzberger wrote:
> What does the code for going backwards looks like?  That is,
> suppose we have an instance of Category with only one object.
> What is the Haskell code for the function which takes the
> category instance and produces a monoid thing, like your integers
> with 1 and usual integer multiplication?  Could we use a
> "constraint" at the level of types, or at some other level, to
> write the code?  Here by "constraint" I mean something like a
> declaration that is a piece of Haskell source code, and not
> something the human author of the code uses to write the code.
instance C.Category k => Monoid (k a a) where
     mempty = C.id
     mappend = (C..)

The above gives witness to the fact that, if I'm using the language 
correctly, if we choose any object (our "a") in any given category, this 
induces a monoid with the identity morphism as unit and composition of 
endomorphisms as append.

The standard libraries in fact provide this instance for the function 
arrow category (under a newtype wrapper):

newtype Endo a = Endo { appEndo :: a -> a }

instance Monoid (Endo a) where
         mempty = Endo id
         Endo f `mappend` Endo g = Endo (f . g)

--Gershom
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