Dear Michael, I remind you that it was Benabou who observed (and proved) the equivalence between anafunctors and representable distributors, reproduced in the nLab as "folklore", from which I concluded (without any knowledge of anafunctors) that the stack completion of a category C in S represents anafunctors with target C. I am glad that Makkai is now aware of this fact, which gives a universal flavor to his subject, whatever "morally" means. I know nothing about the "axiom of cardinal selection". As for there being an example of an elementaru topos which does not satisfy the "axiom of stack compleitons", Joyal gave one lomg ago and Lawvere mentioned it in his 1974 Montreal lectures. Take a group G with a proper class of subgroups having a small index in G. The topos [G, Sets] is an example. Also, so far as I know, it is not yet known (Hyland, Robinson, and Rosolini,"The discrete objects in the effective topos", Proc. London Math. Soc. (3) 60 (1990, 1-36)) whether the full internal subcategory Q on the subquotients of N in Eff (the effective topos) has an internal stack completion. The stack completion is identified as Orth(Delta 2), families of discrete objects. Regards,Marta
Date: Mon, 11 Jul 2011 18:20:42 -0700 Subject: Re: categories: RE: stacks (was: size_question_encore) From: mshulman@ucsd.edu To: martabunge@hotmail.com CC: david.roberts@adelaide.edu.au; joyal.andre@uqam.ca; categories@mta.ca
Is the "axiom of stack completions" related to the "axiom of small cardinality selection" used by Makkai to prove that the bicategory of anafunctors is cartesian closed? I think I recall a remark in Makkai's paper to the effect that the stack completion of a category C is at least morally the same as the category Ana(1,C) of "ana-objects" of C.
Are there known examples of elementary toposes which violate the axiom of stack completions?
On Mon, Jul 11, 2011 at 5:32 AM, Marta Bunge <martabunge@hotmail.com> wrote:
Concerning size matters, let me observe that my construction of the stack completion (Bunge, Cahiers 1979) is meaningful regardless of size questions, that is, for any base topos S. The outcome, however, of applying it to an internal category need no longer be internal. For this reason I introduce an "axiom of stack completions" which guarantees that stack completions of internal categories be again internal,and which is satisfied by any S a Grothehdieck topos. The question of stating such an axiom as an additional axiom to the ones for elementary toposes was proposed as a problem by Lawvere in his Montreal lectures in 1974.
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On Tue, Jul 12, 2011 at 5:30 AM, Marta Bunge <martabunge@hotmail.com> wrote:
I am glad that Makkai is now aware of this fact, which gives a universal flavor to his subject, whatever "morally" means.
The paper I was referring to is the one that first introduced anafunctors, so I think he's been aware of it since the beginning (I suspect it was a primary motivation, even). The word "morally" was my own weasel word, to cover the fact that I didn't have time to look up the paper and remind myself what precisely he actually wrote. (-:
As for there being an example of an elementary topos which does not satisfy the "axiom of stack completions", Joyal gave one long ago and Lawvere mentioned it in his 1974 Montreal lectures. Take a group G with a proper class of subgroups having a small index in G. The topos [G, Sets] is an example.
Ah, thanks. That makes sense. The question about the effective topos is also intriguing! Are there any interesting non-Grothendieck elementary toposes which are known to satisfy the axiom of stack completions? (By "interesting" I mean to exclude toposes such as the category of sets smaller than some strong limit cardinal -- not to say that such toposes are not interesting for other purposes.) Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Mike, As for stacks being the primary motivation Makkai had for anafunctors, that is not the impression I got from people attending his course, as they not only expressed interest in this comment of mine, but also someone suggested reducing the theory of stacks to that of anafunctors. In fact, I was told by another that the introduction of anafunctors was quite different from that of stacks, as it had more to do with expressing anafunctors in FOLDS. Since you are interested in this, why don't you try getting it directly from the horse's mouth? The answer to your other question (Are there any interesting non-Grothendieck elementary toposes whichare known to satisfy the axiom of stack completions?) is still open. Bob Pare and I investigated this question a long time ago (more than 30 years ago, actually) and in our attempts to formulate this axiom, all we could say was that every Grothedieck topos satisfied it. I forgot to mention that an elementary formulation of ASC ("axiom of stack completions") is till missing. All I mean by it at the moment is that it holds for an elementary topos S if, for any category C in S, the stack completion \tilde([C]) of the fibration [C] over S which is the externalization of C, which is explicilty constructed, is equivalent to the externalization [\tilde(C)] of a category \tilde(C) in S. Regards,Marta
Date: Tue, 12 Jul 2011 07:33:46 -0700 Subject: Re: categories: RE: stacks (was: size_question_encore) From: mshulman@ucsd.edu To: martabunge@hotmail.com CC: david.roberts@adelaide.edu.au; joyal.andre@uqam.ca; categories@mta.ca
On Tue, Jul 12, 2011 at 5:30 AM, Marta Bunge <martabunge@hotmail.com> wrote:
I am glad that Makkai is now aware of this fact, which gives a universal flavor to his subject, whatever "morally" means.
The paper I was referring to is the one that first introduced anafunctors, so I think he's been aware of it since the beginning (I suspect it was a primary motivation, even). The word "morally" was my own weasel word, to cover the fact that I didn't have time to look up the paper and remind myself what precisely he actually wrote. (-:
As for there being an example of an elementary topos which does not satisfy the "axiom of stack completions", Joyal gave one long ago and Lawvere mentioned it in his 1974 Montreal lectures. Take a group G with a proper class of subgroups having a small index in G. The topos [G, Sets] is an example.
Ah, thanks. That makes sense. The question about the effective topos is also intriguing!
Are there any interesting non-Grothendieck elementary toposes which are known to satisfy the axiom of stack completions? (By "interesting" I mean to exclude toposes such as the category of sets smaller than some strong limit cardinal -- not to say that such toposes are not interesting for other purposes.)
Mike
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Dear Michael, You wrote:
Are there known examples of elementary toposes which violate the axiomof stack completions?
Here is my favorite example. Let C(2) be the cyclic group of order 2. It suffices to construct a topos E for which the cardinality of set of isomorphism classes of C(2)-torsor is larger than the cardinality of the set of global sections of any object of E. Let G=C(2)^I be the product of I copies of C(2), where I is an infinite set. The group G is compact totally disconnected. Let me denote the topos of continuous G-sets by BG. There is then a canonical bijection between the following three sets 1) the set of isomorphism classes of C(2)-torsors in BG 2) the set of isomorphism classes of geometric morphisms BC(2)--->BG 3) the set of continuous homomomorphisms G-->C(2). Each projection G-->C(2) is a continuous homomomorphism. Hence the cardinality of set of isomorphism classes of C(2)-torsors in BG must be as large as the cardinality of I. The topos E=BG is thus an example when I is a proper class. For those who dont like proper classes, we may and take for E the topos of continuous G-sets in a Grothendieck universe and I to be a set larger than this universe. Best, Andre -------- Message d'origine-------- De: viritrilbia@gmail.com de la part de Michael Shulman Date: lun. 11/07/2011 21:20 À: Marta Bunge Cc: david.roberts@adelaide.edu.au; Joyal, André; categories@mta.ca Objet : Re: categories: RE: stacks (was: size_question_encore) Is the "axiom of stack completions" related to the "axiom of small cardinality selection" used by Makkai to prove that the bicategory of anafunctors is cartesian closed? I think I recall a remark in Makkai's paper to the effect that the stack completion of a category C is at least morally the same as the category Ana(1,C) of "ana-objects" of C. Are there known examples of elementary toposes which violate the axiom of stack completions? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear all: If we take the filtered poset of finite subsets of I, then taking the finite products of C(2) gives an strict pro group, and this is the example of a "Faux topos", SGA4 SLN 269 page 322. Now, this example is exhibited as what we some times call Giraud topos (all exactness properties but without generators) My question is (answer probably well known to the experts): Are Giraud topoi elementary topoi (that is, do they have an Omega and exponentials) ? greetings e.d. On 07/12/2011 12:04 PM, André Joyal wrote:
Dear Michael,
You wrote:
Are there known examples of elementary toposes which violate the axiomof stack completions?
Here is my favorite example.
Let C(2) be the cyclic group of order 2. It suffices to construct a topos E for which the cardinality of set of isomorphism classes of C(2)-torsor is larger than the cardinality of the set of global sections of any object of E.
Let G=C(2)^I be the product of I copies of C(2), where I is an infinite set. The group G is compact totally disconnected. Let me denote the topos of continuous G-sets by BG.
There is then a canonical bijection between the following three sets
1) the set of isomorphism classes of C(2)-torsors in BG
2) the set of isomorphism classes of geometric morphisms BC(2)--->BG
3) the set of continuous homomomorphisms G-->C(2).
Each projection G-->C(2) is a continuous homomomorphism. Hence the cardinality of set of isomorphism classes of C(2)-torsors in BG must be as large as the cardinality of I.
The topos E=BG is thus an example when I is a proper class.
For those who dont like proper classes, we may and take for E the topos of continuous G-sets in a Grothendieck universe and I to be a set larger than this universe.
Best, Andre
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Dear Mike,
Does it not work to say that every internal category admits a weak equivalence functor to an internal category which is a stack?
Sure. This is so by Corollary 2.11 in Bunge-Pare. No problem with internal categories or internal weak equivalence functors of course. But how does one internalize the notion of a stack? It comes down to parametrizing all epimorphisms in the topos itself. All the best,Marta ----------------------------------------
Date: Tue, 12 Jul 2011 11:45:41 -0700 Subject: Re: categories: RE: stacks (was: size_question_encore) From: mshulman@ucsd.edu To: marta.bunge@mcgill.ca CC: david.roberts@adelaide.edu.au; joyal.andre@uqam.ca; categories@mta.ca
On Tue, Jul 12, 2011 at 7:56 AM, Marta Bunge <martabunge@hotmail.com> wrote:
As for stacks being the primary motivation Makkai had for anafunctors, that is not the impression I got from people attending his course,
Okay, thanks for the correction.
I forgot to mention that an elementary formulation of ASC ("axiom of stack completions") is till missing.
Does it not work to say that every internal category admits a weak equivalence functor to an internal category which is a stack?
Mike
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On Tue, 12 Jul 2011, Eduardo Dubuc wrote:
Dear all:
If we take the filtered poset of finite subsets of I, then taking the finite products of C(2) gives an strict pro group, and this is the example of a "Faux topos", SGA4 SLN 269 page 322.
Now, this example is exhibited as what we some times call Giraud topos (all exactness properties but without generators)
My question is (answer probably well known to the experts):
Are Giraud topoi elementary topoi (that is, do they have an Omega and exponentials) ?
greetings e.d.
The answer is no. Let M be the free monoid on a proper class of generators (indexed by the ordinals, say), and consider the category of M-sets (that is, sets equipped with a proper class of independent endomorphisms e_\alpha, \alpha \in Ord). This satisfies all the Giraud axioms except for the generating set, but doesn't have a subobject classifier. To see this, let \kappa be a cardinal, and consider the M-set A_\kappa whose underlying set is the set of \kappa-sequences of natural numbers with all but finitely many terms zero, with e_\alpha acting as the successor map on the \alpha-th term for \alpha < \kappa, and as the identity otherwise. Clearly, A_\kappa has \kappa distinct sub-M-sets, so it has \kappa distinct morphisms to \Omega if the latter exists. But it's generated as an M-set by the single element (0,0,0,...), so this must map to \kappa distinct elements of \Omega, and in particular \Omega has cardinality at least \kappa. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
André Joyal -
Eduardo Dubuc -
Marta Bunge -
Michael Shulman -
Prof. Peter Johnstone