Re: Terminology: Remarks

["Jean Bénabou" <jean.benabou@wanadoo.fr>]

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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