Peter May phrased the fundamental theorem of Galois theory as:

Let G = Gal(E/F) be the Galois group of a finite Galois extension E/F. Define an isomorphism of categories between the category of intermediate fields F\subset K\subset E and field maps K >--> L that fix F pointwise and the category of orbits G/H and G-maps between them.

Thanks, Peter, for that excellent example. For a moment, I was going to object that it is uninteresting because the categories in question are simply posets, until I looked again and saw that they are not what I was expecting. I want to point out that, to those of us who "speak no evil" (I can only really speak for myself, but I expect some others to agree), this example is very different from the concrete isomorphisms, such as that between the categories of boolean rings and boolean algebras. In the latter case, I argued before that what really matters is that we have a concrete equivalence and that the 2-category Conc (whose objects are the faithful functors to the category of sets) and that Conc is a 2-poset: parallel concrete natural transformations are equal (and equality makes sense to us in that context). But in this new example, we have an isomorphism of *strict* categories. That is, there really is a notion of equality of objects in each category, even if our foundation does not provide equality as identity by fiat, because we can define what it means for two objects to be equal: K = L iff, for every element x of F, x in K if and only if x in L (and similarly for the other category). In material set theory, the axiom of extensionality shows that this matches equality as identity; in a structural foundation like ETCS, the two equalities do not match and it is really the defined one that we want, not identity at all. (I should also remark that Peter's categories are small categories, at least if your foundations are sufficiently impredicative to have small power sets, as is usual.) --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]