I have very much appreciated André's subtlety on this issue in conversation 2010/5/22 Joyal, André <joyal.andre@uqam.ca>:
I wrote ------------------------------------------------------------ One lies in the fact that equivalent categories are considered to be the "same", ------------------------------------------------------------- I was careful not to write ------------------------------------------------------------ One lies in the fact that equivalent categories are considered to be the same, -------------------------------------------------------------
John Baez has written carefully on this point too. But not everyone is so careful and Ronnie has good reason to be concerned about a tendency to sweep away distinctions that do need to be made. Isomorphic categories too must be distinguished from one another, some times and for some purposes notably including all currently articulated versions of categorical foundations. 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. It is an interesting impulse in higher category theory to avoid identity in favor of isomorphism on the level of objects, and to avoid isomorphism in favor of equivalence on the level of categories. But so far as I know no one has yet articulated a way to avoid ever using identity of objects and identity of categories. Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]