[PRL] isomorphic fixed points

[William D Clinger]

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

>    Isomorphisms constitute a very undiscriminating equivalence 
> relationship.  In the category of sets, all pairs of sets with the same 
> cardinality are isomorphic.
>
>    We can use cardinality to argue that all fixed points of the Maybe 
> functor (in the category of sets) are isomorphic.

When a CS argument depends upon both the Axiom of Choice and the
Continuum Hypothesis, I reach for my Occam's Razor (TM).

>    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.

That is where the arrows come in.

Will