2010/5/29 Eduardo J. Dubuc <edubuc@dm.uba.ar> Expresses my main point when I quoted Grothendieck on equivalence and isomorphism.
isomorphisms appear not only in examples but are essential also in the theory. For example Grothendieck defines limits and colimits of categories (as universal pseudocones) in SGA4 by means of an isomorphism of categories. Same for toposes.
When AG says "none of the equivalences we meet in practice are isomorphisms" he has in mind lots of examples that I will not even try to survey. (For a really simple one, the category of sheaves defined as espaces etales on on a topological space versus the category of sheaves defined as suitable functors on the site of open subsets.) But when he defines functor categories, or derived categories, and a lot of other things like that, he defines them up to unique isomorphism over the data. A topos E will often be defined only up to equivalence. But, given E, its derived category is defined up to unique isomorphism and one constantly uses the fact that various induced functors are isomorphisms. AG's practice constantly distinguishes isomorphisms from equivalences, and thus distinguishes identity of objects from isomorphism of them. best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]