Re: Terminology: Remarks

[Marta Bunge <martabunge@hotmail.com>]

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







> 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:
>> 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
>>
> 
> 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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