On 23 May 2010, at 17:39, Colin McLarty wrote:
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.
We do meet isomorphisms of categories. Only, they are so obvious that sometimes we do not see them. For instance: The category of abelian groups is (canonically) isomorphic to the category of Z-modules. Groups are often defined as semigroups satisfying two conditions; but they can also be defined as sets with a zeroary operation, a unary operation and a binary operation satisfying certain axioms. Again, we have two isomorphic categories. An unbiased definition would give a third isomorphic category (and one can form infinitely many intermediate cases between the second and the third, likely of little interest). Algebras for the free group monad are directly linked with the unbiased version, yet not the same. Lattices (with 0 and 1) can be defined as ordered sets satisfying some conditions; or as sets with two binary operations satisfying other conditions; then one can add two zeroary operations;... Best regards Marco Grandis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]