[PRL] isomorphic fixed points

[Riccardo Pucella]

> I was not at the talk that motivated this.  I am responding only
> to Peter's written remarks.

Ditto for me. 


> >     Once one accepts that isomorphism is a poor notion of
> >  equivalence, one might find this very troubling for category
> >  theory.  If objects are so frequently isomorphic, are not the
> >  results of category theory somewhat vacuous?  We are talk about
> >  lists, and Nats etc as if they were different "objects", but in
> >  the end, they all seem to be isomorphic!  I think the answer to
> >  this is to remember that what are important are the arrows and not
> > the objects.
>
> Category theorists seldom concern themselves with isomorphism of
> objects.  Natural isomorphism of functors is more important, both
> for category theory and for computer science.

Indeed. In fact, most things of interest in category theory are constructed
up-to-isomorphism, meaning that the same result will be obtained if we
start from isomorphic objects. Isomorphism is an equivalence to get rid of
irrelevant details. Most set-theoretic constructions do not depend at all
on the elements of the set. 


If you have a set-theoretic construction that does depend on the elements
of the set, chances are you are using additional structure on the set (if
only at the level of the elements of the set), and this structure should be
reflected by the category (different objects, or different arrows). 


(I recognize that Peter was saying all along that isomorphism is not the
right form of equivalence to consider. All I am saying is that the use of
sets as an example of this point is a bit of a red herring, since people
talk about sets when they often have something with more structure in
mind.)


 Cheers, 
 Riccardo