[Haskell-cafe] Category Theory woes

[Daniel Fischer]

Am Donnerstag 18 Februar 2010 19:55:31 schrieb Nick Rudnick:
> Gregg Reynolds wrote:
> > On Thu, Feb 18, 2010 at 7:48 AM, Nick Rudnick
> > <joerg.rudnick at t-online.de <mailto:joerg.rudnick at t-online.de>> wrote:
> >
> >     IM(H??)O, a really introductive book on category theory still is
> >     to be written -- if category theory is really that fundamental
> >     (what I believe, due to its lifting of restrictions usually
> >     implicit at 'orthodox maths'), than it should find a reflection in
> >     our every day's common sense, shouldn't it?
> >
> >
> > Goldblatt works for me.
>
> Accidentially, I have Goldblatt here, although I didn't read it before
> -- you agree with me it's far away from every day's common sense, even
> for a hobby coder?? I mean, this is not «Head first categories», is it?
> ;-)) With «every day's common sense» I did not mean «a mathematician's
> every day's common sense», but that of, e.g., a housewife or a child...

Doesn't work. You need a lot of training in abstraction to learn very 
abstract concepts. Joe Sixpack's common sense isn't prepared for that.

>
> But I have became curious now for Goldblatt...
>
> >     * the definition of open/closed sets in topology with the boundary
> >     elements of a closed set to considerable extent regardable as
> >     facing to an «outside» (so that reversing these terms could even
> >     appear more intuitive, or «bordered» instead of closed and
> >     «unbordered» instead of open),
> >
> >
> > Both have a border, just in different places.
>
> Which elements form the border of an open set??

The boundary of an open set is the boundary of its complement.
The boundary may be empty (happens if and only if the set is simultaneously 
open and closed, "clopen", as some say).

>
> >     As an example, let's play a little:
> >
> >     Arrows: Arrows are more fundamental than objects, in fact,
> >     categories may be defined with arrows only. Although I like the
> >     term arrow (more than 'morphism'), I intuitively would find the
> >     term «reference» less contradictive with the actual intention, as
> >     this term
> >
> > Arrows don't refer.
>
> A *referrer* (object) refers to a *referee* (object) by a *reference*
> (arrow).
>

Doesn't work for me. Not in Ens (sets, maps), Grp (groups, homomorphisms), 
Top (topological spaces, continuous mappings), Diff (differential 
manifolds, smooth mappings), ... .