Dear Toby, I'm preparing an answer to all the mails I received about equivalence of categories. In order to answer to yours, I need the following precisions about your definition: (i) If F: A -> B and G: B -> C are full and faithful essentially surjective functors, so is GF. How do you compose your equivalences? (ii) Let 1 denote the final category. The unique functor 1 -> 1 is the unique equivalence between 1 and 1. How many spans 1 <- X -> 1 are equivalences in your sense? Best regards, Jean Le 2 mai 2013 à 08:46, Toby Bartels a écrit :
Jean B?nabou wrote in small part:
The one [notion of equivalence of categories] which might serve here is f full and faithful and essentially surjective. But unless we have AC it is not symmetric, even for A and B small.
Then the obvious thing to try is to symmetrise it: An equivalence between A and B is a span A <- X -> B of fully faithful and essentially surjective functors.
--Toby
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