Re: Terminology: Remarks

[Marta Bunge <martabunge@hotmail.com>]

Dear Jean (and Toby), 


The notion of equivalence of categories suggested by Toby is precisely the one introduced in the paper Marta Bunge and Robert Pare, "Stacks and Equivalence of Indexed Categories", Cahiers de Top. et Geo. Diff. , Vol XX-4 (1979) 373-399.  S is a topos. 

Definition (1.3)  We say that two S-indexed categories A and B are weakly  equivalent if there exists an S-indexed category E and a span A <-E->B of weak equivalence functors (fully faithful and essentially surjective) E->A and E->B.

Proposition (1.4) Weak equivalence is an equivalence relation. Proof. The relation is trivially reflexive and symmetric. Composition is defined by means of (what we call) a 2-pullback, from which it follows that the relation is transitive. 

This answers your question (i).


Proposition (1.6) Let C and D be small categories (internal to S). If  their externalizations [C] and [D] are weakly equivalent, then one can choose D small with [D] ->[C],[D] -> [B] weak equivalences. In this case one can simply resort to entirely internal definition of weak equivalence so that  C <-D->B is a span in S. Proof. See pages 377-378. 

Remark (page 384). Equivalence implies weak equivalence. Weak equivalence implies equivalence iff (AC) holds. Weak equivalence implies local equivalence iff (IAC) holds Local equivalence does not imply weak equivalence. Equivalence implies local equivalence. Local equivalence implies equivalence iff  every object of S with full support has a global section. 

Corollary (2.12) (to Definition 2.10). Let F:A->B be a weak equivalence functor between S-indexed categories where B is a stack. Then F:A->B is "the" (that is, up to equivalence) stack completion of A.

Corollary (2.12). Let A and B be S-indexed categories and let F: A->A', G:B->B' be weak equivalence functors with A' and B' stacks. If A and B are weakly equivalent, then A' and B' are equivalent, and conversely. 

Rekark (at the end of Section 2). The associated stack of any locally small  S-indexed category is constructed in (Marta Bunge, Stack completions and  Morita equivalence for categories in a topos, Cahiers de Top.Gem. Diff  XX-4 (1979) 401-436). The stack completion of a locally small category A  is (can be taken to be) a locally small category A'. However, it is not in genera the case that the stack completion of a small category C  be small. Say that S satisfies the Axiom of Stack Completions (ASC) if the stack  completion of a category in S can be taken to be small. Any Grothendieck topos S satisfies (ASC).


Let me point out that nothing special about S-indexed categories is employed here that could not have been done with fibrations over S. We could have defined weak equivalence of small categories (in S) directly but the general theory of stacks demanded that we dealt with arbitrary S-indexed catgeories, or with arbitrary fibrations over S. 


As for an answer to your question (ii), any small category X with X->1 a weak equivalence gives a span 1<-X->1 as desired. One of them is of course 1<-1->1. Such categories X have 1 as their stack completion, so they are all themselves weakly equivalent. In other words, up to weak equivalence there is only one such span 1<-X->1. 


Best regards,
Marta




> Subject: categories: Re: Terminology: Remarks
> From: jean.benabou@wanadoo.fr
> Date: Fri, 3 May 2013 06:53:12 +0200
> CC: categories@mta.ca
> To: categories@TobyBartels.name
> 
> 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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