we do meet isomorphisms of categories

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

Marco Grandis wrote in part:

>Colin McLarty wrote:

>>Grothendieck gave it a fine nuance in Tohoku (p. 125) saying "Aucune
>>des equivalences de categories qu'on rencontre en pratique n'est un
>>isomorphisme (none of the equivalences one meets in practice are
>>isomorphisms)."  He stressed that we must distinguish isomorphisms
>>from equivalences.  Throughout that and later works he *constructs* a
>>great many categories up to isomorphism, and not just up to
>>equivalence.  We do not meet these isomorphisms, we construct them --
>>and it is quite important that once constructed they are not merely
>>equivalences.

>We do meet isomorphisms of categories. Only, they are so obvious that
>sometimes we do not see them.

>The category of abelian groups is (canonically) isomorphic to the category
>of Z-modules.

[further examples cut]

In all of these examples (although obviously not all examples of isomorphisms),
this is more than just an isomorphism; it's an isomorphism over Set.
That is, it's an isomorphism in the slice category Cat/Set.
It may seem beside the point, but in fact it is also important
that it's an isomorphism in the full subcategory of Cat/Set
whose objects are only the faithful functors to Set;
call this the category Conc of CONCRETE categories.
(So they are all concrete isomorphisms of concrete categories.)

If you take a strictly speak-no-evil approach to category theory
(perhaps even going so far as to found your mathematics on FOLDS),
then it is impossible to state that two categories are isomorphic,
because you must speak of equality of objects (or functors) to do this.
In this approach, Cat and Cat/Set are bicategories but not categories.

But it IS still possible to state that two concrete categories are isomorphic;
the bicategory Conc is (up to equivalence) a locally posetal bicategory
(so if you ignore the non-invertible transformations, it's a category).
So it is possible (and necessary) to say, even when you speak no evil,
that all of Marco's examples are concrete isomorphisms.

So I agree that it is important that these are not mere equivalences,
but I claim (playing the role of an equality-is-evil partisan)
that what is important is not so much that they are isomorphisms
as that they are concrete.


--Toby


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