On Fri, Apr 13, 2012 at 3:15 PM, Raul Miller <rauldmil...@gmail.com> wrote:
> On Fri, Apr 13, 2012 at 2:38 PM, Marshall Lochbaum <mwlochb...@gmail.com>
> wrote:
> > You're making a mistake in your treatment of functions. A function is
> not a
> > formula. It is an association of a value in the codomain to each value in
> > the domain, and if two functions have all of the same associations (which
> > forces them to have the same domain), then they are identical. Therefore
> >:
> > and 1: with domain {0} are the same function.
>
> The reference I was reading
> http://en.wikibooks.org/wiki/Haskell/Category_theory begins with "a
> directed multigraph with loops".
>
> That means that we can have multiple distinct arrows leading from 0 and 1.
>
> You are telling me that we can only have one arrow leading from 0 to
> 1, which makes their explicit statement pointless. But I see no
> statements in their exposition to support this constraint.
Two objects in an arbitrary category may or may not have many arrows
between them. Categories in general are not about sets and functions. If
you restrict yourself to two objects that represent singleton sets and you
want arrows that represent functions between them, then certainly there is
only one (well, one in each direction one for each identity).
Numbers are used to label vertices (objects) of directed graph examples in
the link Robert sent out. Examples there might call objects 0 and 1, but
these are just names. It does not mean that they represent the sets {0} and
{1} nor that arrows between them represent functions on those sets. Be very
careful not to confuse categories in general with categories that represent
sets and functions.
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