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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
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Eduardo J. Dubuc -
Jean Bénabou -
Marta Bunge -
Toby Bartels -
Tom Leinster