[Haskell-beginners] Category question

[Manfred Lotz]

On Mon, 28 May 2012 13:43:33 -0400 (EDT)
Jay Sulzberger <jays at panix.com> wrote:


> 
> No.  The point is that, by definition, a category, call it C, is
> a struct with two sets, Obj(C) and Mor(C), and further operations:
> 
> 1. head: Mor(C) -> Obj(C)
> 
> 2. tail: Mor(C) -> Obj(C)
> 
> 3. id: Obj(C) -> Mor(C)
> 
> 4. *: Mor(C) x Mor(C) -> Mor(C)
> 
> where head and tail and id are everywhere defined single valued
> maps.  They are all maps of sets.  *, read "composition of
> morphisms" is a map of sets, with signature as displayed, but is
> not usually everywhere defined.  We have then several
> "equational" axioms, which C is required to satisfy to be a
> category.
> 
> (set theoretical note: We have, partly implicitly, ruled out
> categories which are not "small".  See standard texts for this
> locus of difficulty.)
> 
> By the axioms, any object b of C must have defined its associated
> identity morphism id[b].  For many categories, b will always be
> an actual set, and id[b] will be the unique map of sets defined
> by
> 
>    (id[b])(x) = x , for all x in b
> 
> where (id[b])(x) is read "the result of applying id[b] to the element
> x of b".
> 
> But, as explained, many categories have objects which are not
> sets.  Indeed, often, no object is a set.
> 
> The definition of category never mentions whether or not the
> objects are sets.  And, as we have seen, there are many
> categories whose objects are not sets.  (Perhaps categorically
> better: many categories are not directly presented as having
> objects which are sets.)
> 
> to repeat: The concept "category" is larger in extension than the
> concept "category whose objects are sets and whose morphisms are
> maps of sets".
> 
> ad representations of categories:
> 
>    http://en.wikipedia.org/wiki/Yoneda_Lemma
>    [page was last modified on 1 April 2012 at 05:17]
> 
> >
> >
> > I guess that this made me think of idA as idA(x) = x for each x of
> > A. Later when I saw other (more general) definitions I did not read
> > carefully to realize the difference.
> >
> >
> > Thanks a lot for making this clear to me.
> >
> >
> > -- 
> > Manfred
> 
> I will let stand my restatement of what you already know ;)
> 
> oo--JS.
> 

Thanks a lot for the detailed example and explanations. I will study
your post thoroughly.



-- 
Manfred