Re: isomorphisms

[Michael Shulman <shulman@uchicago.edu>]

It seems to me that there is an important distinction here that is not
being emphasized.  Isomorphisms of categories can be *technically*
quite useful.  Knowing that a given equivalence of categories is an
isomorphism, rather than merely an equivalence, can certainly make
things much simpler, or even make things possible that we didn't know
how to do before.  Many examples of this sort have been mentioned.
Another that should be added to the list is the theory of strict
2-categories, 2-limits, 2-adjoints, and so on, all of which is defined
using ordinary enriched category theory over Cat, and hence involves
many isomorphisms of hom-categories.

However, I find that in most or all of these examples, one is not
actually interested in the fact of an isomorphism of categories for
its own sake.  There is no "real meaning" in the fact that two
categories are isomorphic, rather than equivalent; generally it's a
technical accident of how we chose to define them.  It may be a very
*convenient* technical accident, but it is an accident nonetheless.
If we chose our definitions differently, or worked in a different
foundational system (such as one where the notion of "isomorphism of
categories" cannot even be defined), some or all of our isomorphisms
of categories might change to become only equivalences, but I don't
believe that any of the real content of category theory would go away;
it would just become harder to prove.

Mike


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