Terminology: Remarks

Terminology: Remarks

[Jean Bénabou]

Dear all,

Let me first thank the persons who have answered my questions on terminology.
Since the answers were different, it seems that the terminology is not standard and that my questions made sense.

Preliminary remarks. 
By now, everybody understands what kind of categories I was talking about. they are very simple, one might be tempted to say trivial. But in many highly non trivial questions they appear either as "building bricks" of more complex constructions (see e.g. fibrations such that all the fibers are of that kind), or as special cases, unavoidable, of more general situations (e.g. Freyd's notion of "equivalence kernels").
Of course such categories can be "internalized ", say in a topos, (this is much too strong), and it would be nice that the terminology should fit also the internal case, and in particular any reference, explicit or implicit, to AC should be avoided.

1- Discrete versus indiscrete, or coarse, etc. The categories  0 and 1 are both discrete and indiscrete So each name night pose a problem. The terminology "discrete" is by now well established. Thus I think one should avoid "indiscrete" and use "coarse" instead. Thus 0 and 1 are discrete and coarse and that is admissible. In the internal case every sub object of 1 is both discrete and indiscrete. So "subterminal" is a nice name to cover both cases.

2- "Essentially". One has to be careful about this word. It seems to mean "up to equivalence". But that depends on what you call "equivalence of categories". There is a very strong 2-categorical notion, namely a pair of functors  f: A --> B and g: B --> A  with fg and gf isomorphic to identities (with or without the adjunction axioms). But it is useless for our purpose. The one 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. 
Thus with AC we might adopt the suggestions of Thomas and use essentially discrete and essential sub terminator. But do we really need AC or even any notion of equivalence at all?

3- Elementary remarks.
Let S be a category with finite limits. I want to "internalize" the two notions for which I'd like a name, suitable not only when S=Set, but in general.
(i) if X is an internal category I denote by Ob(X) and Map(X) the objects of objects and maps of X and by
d: Map(X) --> Ob(X)xOb(X)  with projections Dom and Codom. When S=Set if f: x -->y is a map of X, df is the "direction" of f.
X is a preordered object iff d is a mono. 
It is easy, using the multiplication of X, to express in terms of finite limits, the fact that X is a groupoid. Hence equivalence relations are definable in any category with finite limits, although composition of arbitrary relations is possible only when S is regular. Moreover if F: S --> S' is a functor which preserves finite limits and   X is an equivalence relation so is F(X)
No essentiality no AC is needed
(ii) the functor  X --> 1 is full and faithful iff the direction map d is an iso. This again is preserved by finite limits preserving functors F.
(iii) If F preserves finite limits and is faithful it reflects equivalence relations and property (ii). In particular the Yoneda embedding preserves and reflects the previous properties. this enables us to work with them as if S=Set. No AC is involved .

For all these reasons I'm not too keen about using "essentially"

I have many more remarks about Fred Linton's and David Robert's postings but this mail is already a bit long. So I shall wait a few days before i send another mail. By that time I hope to have some reactions to the present posting and I shall do my best to answer them.

Best regards, Jean













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Re: Terminology: Remarks

[Toby Bartels]

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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Re: Terminology: Remarks

[Tom Leinster]

Toby Bartels wrote:

> 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.

It's interesting, I think, that two categories A and B are equivalent if
and only if there exists a span

    A <-- X --> B

of functors that are full and faithful and *genuinely* surjective on
objects.

What's interesting is that "full and faithful and genuinely surjective on
objects" is a purely graph-theoretic condition: it doesn't refer to the
category structures.  This suggests a pain-free way of defining
equivalence of n-categories, an idea I learned from Carlos Simpson.

For example, if I remember correctly, two bicategories A and B are
biequivalent if and only if there exists a span

    A <-- X --> B

of strict functors that are surjective at every level, i.e. locally
faithful, locally full, locally surjective on objects, and surjective on
objects.

Tom

-- 
The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.



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Re: Terminology: Remarks

[Eduardo J. Dubuc]

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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Re: Terminology: Remarks

[Jean Bénabou]

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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Re: Terminology: Remarks

[Marta Bunge]

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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Re: Terminology: Remarks

[Marta Bunge]

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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Re: Terminology: Remarks

[Toby Bartels]

Jean B?nabou wrote in part:

>(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?

Good question!  In this case, we can compose by pullback.
But when I wrote "the obvious thing to do",
maybe I should have written "one obvious thing to try".
After all, it might not work; in this case, it does.

If it didn't work, another obvious thing to try would be zigzags.
In this case, this gives an equivalent 2-groupoid.
So in principle, one could do either, but spans are simpler.

Generalising from the 2-groupoid of categories to the 2-category of them,
we can use spans A <- X -> B where only A <- X needs to be ff eso.
This is what Michael Makkai did, and it allows one to avoid AC
while retaining the usual results about the 2-category of categories.

(Pace Tom Leinster's recent comment under this thread,
  Makkai actually required A <- X to be strictly surjective on objects,
  but again this does not matter; the resulting 2-category is equivalent.)

Beyond this, let me just say that I agree with Marta's answers.


--Toby


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