Pertinent remark, but anyway, if you define equivalence by "A FUNCTOR such that there exist a quasi inverse", this relation is symmetric, thus an equivalence relation (without using choice because you have no need to define a quasi inverse) on spite that it has a direction. When I said "highly non symmetric" I was using the word "symmetric" with its every day meaning, not the technical mathematical meaning of an equivalence relation. Often in practice equivalence relations are highly non symmetric. e.d. On 03/05/13 20:22, Marta Bunge wrote:

Dear Eduardo,

This is an addendum to what I have written to Jean Benabou (and Toby Bartels), but referring to your comment. It is not always the case that an equivalence F: S^G -> S^K, for S a topos, and G,K small (internal) groupies, is induced by an equivalence f:G->K, not even by a weak equivalence functor f:G->K. However, given that S^G and S^G are equivalent categories, it follows that the stack completions G' and K' are equivalent. In particular, this (the Morita equivalence theorem for internal groupoids in a topos) is an instance where it is necessary to consider, not just wef G->K or K->G, but the equivalence relation "G weakly equivalent to K".

Regards, Marta

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Date: Thu, 2 May 2013 22:41:38 -0300 From: edubuc@dm.uba.ar To: categories@TobyBartels.name CC: categories@mta.ca Subject: categories: Re: Terminology: Remarks

On 02/05/13 03:46, Toby Bartels wrote:

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Jean B?nabou wrote in small part:

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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

Equivalence of categories in practice is highly non symmetric. Usually one direction is defined and canonical, and the other is choice dependent and as such they are many of them. A definition of equivalence should reflect this fact, thus, it is not "a pair of functors such etc etc", but, either "A FUNCTOR full and faithful and essentially surjective", or "A FUNCTOR such that there exist a quasi inverse" if you do not want to use choice.

e.d.

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