categories: Re: Time for functors to grow up; three queries

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

Just a couple of comments in response to Tom Leinster's posting.

On Sat, 9 Feb 2002, Tom Leinster wrote:

> EQUIVALENCES.  When I've tried to teach the notion of equivalence of
> categories to newcomers to the subject, I've said something like this.
> In general we don't care whether two objects of a category are equal, only
> whether they're isomorphic; for instance, if a group theorist says that two
> groups are the same, she doesn't usually know or care whether, by some
> miracle of set theory, they're actually equal - isomorphism is all that
> matters.  So the notion of isomorphism of categories is unreasonably
> strict, requiring as it does that certain equalities of objects occur.  We
> therefore replace those equalities by (natural) isomorphisms to obtain the
> notion of equivalence.  
> 
> All well and good and perfectly standard: but there follows an
> uncomfortable moment.  With the build-up above, I've led my student to
> expect that equality of objects and isomorphism of categories are such
> outrageously over-strict notions that they hardly ever occur in nature;
> that if you pluck out of the air a random example of two categories which
> are "essentially the same" then that'll be an equivalence, not an
> isomorphism.  Actually, that's not the case: there are plenty of perfectly
> natural examples of isomorphic categories.  For instance, if you take two
> different definitions of topological space - say, a set equipped with open
> subsets or a set equipped with a closure operator, and the corresponding
> notions of continuous map - then the resulting two categories are
> isomorphic.  The same goes for different definitions of group obtained by
> varying the generators and relations presenting the theory.  And the same
> goes in other categories of sets-with-structure.  

This is really a consequence of the fact that you're thinking of the
category of sets as a unique entity (if not God-given, then at least
handed down to us by Zermelo and Fraenkel). Of course, if you have two
equivalent single-sorted theories, and you consider their models in the
same category, you will get isomorphic categories of models. But that
won't happen if you consider equivalent theories whose underlying sorts
are different; nor will it happen if you consider models of the same
theory in two equivalent categories (for example, the category of sheaves
on a space X and the category of local homeomorphisms over X -- even for
a one-point space X, these aren't isomorphic, though they are both
equivalent to (Sets)).

> MONADICITY.  The situation's even more extreme here.  Right now I can't
> think of *any* natural example of a monadic adjunction in which the
> comparison functor is an equivalence but not an isomorphism (though I'm
> sure someone can put me right). 

I can: the monadic adjunction between any topos and its opposite.

Peter Johnstone