RE: stacks (was: size_question_encore)

RE: stacks (was: size_question_encore)

[Marta Bunge]

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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RE: stacks (was: size_question_encore)

[Michael Shulman]

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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RE: stacks (was: size_question_encore)

[Michael Shulman]

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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RE: stacks (was: size_question_encore)

[Michael Shulman]

I guess I misunderstood what you meant by "elementary".  You wanted a
single statement that can be expressed in the internal logic of the
topos?  Many other properties that people refer to as "elementary",
such as the existence of finite limits or power objects, are defined
by quantifying over all objects and morphisms of the category in
question.  Is AC "elementary"?

On Tue, Jul 12, 2011 at 12:56 PM, Marta Bunge <martabunge@hotmail.com> wrote:
>
>
> 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
>

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RE: stacks (was: size_question_encore)

[Marta Bunge]

Dear Mike,

Yes, AC is elementary. Of course, so is ASC by the same token. Quantifying over all epis is not a problem. Thanks for questioning me on that point. I remembered incorrectly what the problem was. Bob Pare and I wanted to posit something like the existence of a "representative (or universal) cover" as an axiom (RC), such that GT => RC => ASC, and also (ET + AC) => RC. We hoped to show that RC had good properties, and more importantly, that Eff satisfied it but this did not work. What is still open is  whether Eff satisfies ASC. 
All the best, 
Marta 





 





> Date: Tue, 12 Jul 2011 19:24:25 -0700
> Subject: Re: categories: RE: stacks (was: size_question_encore)
> From: mshulman@ucsd.edu
> To: marta.bunge@mcgill.ca
> CC: categories@mta.ca
> 
> I guess I misunderstood what you meant by "elementary".  You wanted a
> single statement that can be expressed in the internal logic of the
> topos?  Many other properties that people refer to as "elementary",
> such as the existence of finite limits or power objects, are defined
> by quantifying over all objects and morphisms of the category in
> question.  Is AC "elementary"?
> 

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RE: stacks (was: size_question_encore)

[Michael Shulman]

On Wed, Jul 13, 2011 at 2:16 AM, Marta Bunge <martabunge@hotmail.com> wrote:
> Bob Pare and I wanted to posit something like the existence of a "representative (or universal) cover" as an axiom (RC), such that GT => RC => ASC, and also (ET + AC) => RC. We hoped to show that RC had good properties, and more importantly, that Eff satisfied it but this did not work.

That's a very interesting question!  There seem to be a lot of axioms
of this flavor, which say in various different ways that AC fails "in
only a small way".  Some that I am aware of include:

* Small Violations of Choice (Blass):
   http://nlab.mathforge.org/nlab/show/small+violations+of+choice
* Small Cardinality Selection (Makkai):
   http://nlab.mathforge.org/nlab/show/small+cardinality+selection+axiom
* Axiom of Multiple Choice (Moerdijk & Palmgren):
   http://nlab.mathforge.org/nlab/show/axiom+of+multiple+choice
* Weakly Initial Sets of Covers (Roberts):
   http://nlab.mathforge.org/nlab/show/WISC
* The ex/lex completion of Set (or the topos in question) is well-powered
* The ex/lex completion of Set is a topos, i.e. Set has a generic proof

Some of these hold in any Grothendieck topos, but others apparently
need not, and I have no idea which of them might hold in Eff.  It
would be interesting to know if any of them imply ASC (or André's
proposed strengthening thereof).

Mike


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RE: stacks (was: size_question_encore)

[Eduardo Dubuc]

Is interesting to compare Joyal and Johnstone examples of Giraud faux
topos, which are very similar and very different at the same time.

Take a proper class (large set) I, and let

   A = N (natural numbers)   or  A = Z/2Z (cyclic group of order 2)

Let  M = A^(I) = {f | f(i) = 0 except for finitely many i,  A = N }.

or   M = A^I = { all f, A = Z/2Z }.

For any (small) subset K c I, let M_K = A^(K) or A^K respectively. We
have a continuous surjective morphism M ---> M_K.

M_K is a (small) set with a continuous action of M.

Let E = \beta M.  E is a Giraud topos. M_K \in E.

(A = N is Johnstone example,  A = Z/2Z is Joyal's)

In Johnstone case M_K has at least K different subobjects, used to
disprove that E is an elementary topos (can not have a subobject
classifier).

In Joyal case, all these subobjects dissappear, M_K is connected, and
does not have any non trivial subobjects.

Joyal knows that in this case (i have not tried to prove it) E has a
subobject classifier, and it is an elementary topos.

However we still have all the M ---> M_K, in particular one for each
singleton {i} c I, used to disprove the axiom of stalk completion.

****************

For any K, M_K is a monoid (group in Joyal case), and for K c J, a
continuous surjective morphism M_J ---> M_K, thus a large strict
promonoid (or progroup), and we have a large pro-object of Grothendieck
topoi E_K = \beta M_K and connected transition morphisms. It is known
(Kennison, Tierney, Moerdijk) that in this case \beta commutes with the
inverse limits, denoted lM, lE respectively,

\beta lM = lE (provided the KTM result holds for large limits).


We have a cones M ---> M_K   and  E ---> E_K, which determine arrows

M ---> lM,  E ---> lE. In Joyal case both arrows are isomorphisms (seems
easy for M, then by KTM it follows for E), which may be behind the fact
that E = lE is an elementary topos.

The category lE may not even have small hom sets in Johnstone case.

****************

Let now MA, EA be Joyal's case, and MP, EP be Johnstone case.

We have dense continous morphisms MP ---> MA  and  MP_K ---> MA_K which
determine arrows:

             EP ---> EA      (as categories EA Full ---> EP)

            EP_K ---> EA_K   (as categories EA_K Full ---> EP_K)

              lEP ---> lEA   (as categories lEA Full ---> lEP)

The indexed by K are Kosher, the others are False, some more false than
others.

***************








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RE: stacks (was: size_question_encore)

[Marta Bunge]

Dear Mike,

Thank you for the information. It would take me a while to check them all.   "Weakly initial covers" sounds a bit like "RC" or like one of its variations. 
All the best,
Marta

> Date: Thu, 14 Jul 2011 23:51:46 -0700
> Subject: Re: categories: RE: stacks (was: size_question_encore)
> From: mshulman@ucsd.edu
> To: marta.bunge@mcgill.ca
> CC: categories@mta.ca
> 
> On Wed, Jul 13, 2011 at 2:16 AM, Marta Bunge <martabunge@hotmail.com> wrote:
>> Bob Pare and I wanted to posit something like the existence of a "representative (or universal) cover" as an axiom (RC), such that GT => RC => ASC, and also (ET + AC) => RC. We hoped to show that RC had good properties, and more importantly, that Eff satisfied it but this did not  work.
> 
> That's a very interesting question!  There seem to be a lot of axioms
> of this flavor, which say in various different ways that AC fails "in
> only a small way".  Some that I am aware of include:
> 
> * Small Violations of Choice (Blass):
>   http://nlab.mathforge.org/nlab/show/small+violations+of+choice
> * Small Cardinality Selection (Makkai):
>   http://nlab.mathforge.org/nlab/show/small+cardinality+selection+axiom
> * Axiom of Multiple Choice (Moerdijk & Palmgren):
>   http://nlab.mathforge.org/nlab/show/axiom+of+multiple+choice
> * Weakly Initial Sets of Covers (Roberts):
>   http://nlab.mathforge.org/nlab/show/WISC
> * The ex/lex completion of Set (or the topos in question) is well-powered
> * The ex/lex completion of Set is a topos, i.e. Set has a generic proof
> 
> Some of these hold in any Grothendieck topos, but others apparently
> need not, and I have no idea which of them might hold in Eff.  It
> would be interesting to know if any of them imply ASC (or André's
> proposed strengthening thereof).
> 
> Mike


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RE: stacks (was: size_question_encore)

[Marta Bunge]

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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RE: stacks (was: size_question_encore)

[Eduardo Dubuc]

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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RE: stacks (was: size_question_encore)

[Marta Bunge]

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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RE: stacks (was: size_question_encore)

[Michael Shulman]

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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RE: stacks (was: size_question_encore)

[Marta Bunge]

Dear David,

 

 

 

Whatever is published in categories is
of public domain so anyone can intervene. You have nothing to apologize for.

 

 

 

I am not acquainted with recent work of
Dorette Pronk, but I read her 1995 Utrecht thesis in detail since I was asked
to do so by her advisor. In it, she refers to my paper (Marta Bunge, "An
application of descent to a classification theorem for toposes" , Math.
Proc. Camb. Phil. Soc. 107 (1990) 59-79), where I prove, in Corollary 5.4 to
the main Theorem 5.1, the following, which is, oin the case of groupoids, what
you call the third way. It says that, if W is the class of isomorphisms classes
of weak equivalences of etale complete groupoids (ECG), then W admits a
calculus of right fractions, and the functor B from Gpd to Top induces an
equivalence ECG[W-1] \iso [Top], where [Top} denotes the category of
Grotehndieck toposes (over a base S not necessarily Sets) and isomorphism
classes of geometric morphisms. So, the purpose of the third way, in my  view,
is to prove classification theorems. However, I am not au courant of more
recent developments.

 

 

 

As for the other two approaches I
mentioned in my correspondence with Andre Joyal, their equivalence is not  that
obvious. In the 1-dimensional case, this is done in Bunge-Pare (1979)
Proposition 2.7, and in the 2-dimensional case it is done in Bunge-Hermida
(2010) Theorem 4-9.

 

 

 

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. 

 

 

 

Good luck with your projects.

Marta



> Date: Mon, 11 Jul 2011 15:06:38 +0930
> From: david.roberts@adelaide.edu.au
> To: joyal.andre@uqam.ca; martabunge@hotmail.com
> CC: categories@mta.ca
> Subject: stacks (was: size_question_encore)
> 
> Dear Marta, André, and others,
> 
> this is perhaps a bit cheeky, because I am writing this in reply to Marta's
> email to André, quoted below. To me it almost feels like reading anothers' mail;
> please forgive the stretch of etiquette.
> 
> ---
> 
> Marta raised an interesting point that stacks can be described in (at least) two
> ways: via a model structure and via descent. The former implicitly (in the case
> of topoi: take all epis) or explicitly needs a pretopology on the base category
> in question. This is to express the notion of essential surjectivity.
> 
> However, I would advertise a third way, and that is to localise the (or a!)
> 2-category of categories internal to the base directly, rather than using a
> model category, which is a tool (among other things) to localise the 1-category
> of internal categories. Dorette Pronk proved a few special cases of this in her
> 1996 paper discussing bicategorical localisations, namely algebraic,
> differentiable and topological stacks, all of a fixed sort.
> 
> By this I mean she took a static definition of said stacks, rather than  working
> with a variable notion of cover, as one finds, for example in algebraic
> geometry: Artin stacks, Deligne-Mumford stacks, orbifolds etc, or as in Behrang
> Noohi's 'Foundations of topological stacks', where one can have a variable class
> of 'local fibrations', which control the behaviour of the fibres of
> source/target maps of a presenting groupoid.
> 
> With enough structure on the base site (say, existence and stability under
> pullback of reflexive coequalisers), then one can define (in roughly historical
> order, as far as I know):
> 
> representable internal distributors/profunctors
> = meriedric morphisms (generalising Pradines)
> = Hilsum-Skandalis morphisms
> = (internal) saturated anafunctors
> = (incorrectly) Morita morphisms
> = right principal bibundles/bitorsors
> 
> and then (it is at morally true that) the 2-category of stacks of groupoids is
> equivalent to the bicategory with objects internal groupoids and 1-arrows  the
> above maps (which have gathered an interesting collection of names), both of
> which are a localisation of the same 2-category at the 'weak equivalences'.
> 
> Without existence of reflexive coequalisers (say for example when working  in
> type-theoretic foundations), then one can consider ordinary (as opposed  to
> saturated) anafunctors. Whether these also present the 2-category of stacks is a
> (currently stalled!) project of mine. The question is a vast generalisation of
> this: without the 'clutching' construction associating to a Cech cocyle a  actual
> principal bundle, is a stack really a stack of bundles, or a stack of
> cocycles/descent data.
> 
> The link to the other two approaches mentioned by Marta is not too obscure: the
> class of weak equivalences in the 2-categorical and 1-categorical approaches are
> the same, and if one has enough projectives (of the appropriate variety), then
> an internal groupoid A (say) with object of objects projective satisfies
> 
> Gpd(S)(A,B) ~~> Gpd_W(S)(A,B)
> 
> for all other objects B, and where Gpd_W(S) denotes the 2-categorical
> localisation of Gpd(S) at W.
> 
> One more point: Marta mentioned the need to have a generating family. While in
> the above approach one keeps the same objects (the internal
> categories/groupoids), there is a need to have a handle on the size of the
> hom-categories, to keep local smallness. One achieves this by demanding  that for
> every object of the base site there is a *set* of covers for that object cofinal
> in all covers for that object. Then the hom-categories for the localised
> 2-category are essentially small.
> 
> All the best,
> 
> David
> 

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stacks (was: size_question_encore)

[David Roberts]

Dear Marta, André, and others,

this is perhaps a bit cheeky, because I am writing this in reply to Marta's
email to André, quoted below. To me it almost feels like reading anothers' mail;
please forgive the stretch of etiquette.

---

Marta raised an interesting point that stacks can be described in (at least) two
ways: via a model structure and via descent. The former implicitly (in the case
of topoi: take all epis) or explicitly needs a pretopology on the base category
in question. This is to express the notion of essential surjectivity.

However, I would advertise a third way, and that is to localise the (or a!)
2-category of categories internal to the base directly, rather than using  a
model category, which is a tool (among other things) to localise the 1-category
of internal categories. Dorette Pronk proved a few special cases of this in her
1996 paper discussing bicategorical localisations, namely algebraic,
differentiable and topological stacks, all of a fixed sort.

By this I mean she took a static definition of said stacks, rather than working
with a variable notion of cover, as one finds, for example in algebraic
geometry: Artin stacks, Deligne-Mumford stacks, orbifolds etc, or as in Behrang
Noohi's 'Foundations of topological stacks', where one can have a variable class
of 'local fibrations', which control the behaviour of the fibres of
source/target maps of a presenting groupoid.

With enough structure on the base site (say, existence and stability under
pullback of reflexive coequalisers), then one can define (in roughly historical
order, as far as I know):

representable internal distributors/profunctors
= meriedric morphisms (generalising Pradines)
= Hilsum-Skandalis morphisms
= (internal) saturated anafunctors
= (incorrectly) Morita morphisms
= right principal bibundles/bitorsors

and then (it is at morally true that) the 2-category of stacks of groupoids is
equivalent to the bicategory with objects internal groupoids and 1-arrows  the
above maps (which have gathered an interesting collection of names), both  of
which are a localisation of the same 2-category at the 'weak equivalences'.

Without existence of reflexive coequalisers (say for example when working  in
type-theoretic foundations), then one can consider ordinary (as opposed to
saturated) anafunctors. Whether these also present the 2-category of stacks is a
(currently stalled!) project of mine. The question is a vast generalisation of
this: without the 'clutching' construction associating to a Cech cocyle a  actual
principal bundle, is a stack really a stack of bundles, or a stack of
cocycles/descent data.

The link to the other two approaches mentioned by Marta is not too obscure: the
class of weak equivalences in the 2-categorical and 1-categorical approaches are
the same, and if one has enough projectives (of the appropriate variety),  then
an internal groupoid A (say) with object of objects projective satisfies

Gpd(S)(A,B) ~~> Gpd_W(S)(A,B)

for all other objects B, and where Gpd_W(S) denotes the 2-categorical
localisation of Gpd(S) at W.

One more point: Marta mentioned the need to have a generating family. While in
the above approach one keeps the same objects (the internal
categories/groupoids), there is a need to have a handle on the size of the
hom-categories, to keep local smallness. One achieves this by demanding that for
every object of the base site there is a *set* of covers for that object cofinal
in all covers for that object. Then the hom-categories for the localised
2-category are essentially small.

All the best,

David

Quoting André Joyal <joyal.andre@uqam.ca>:

> dear Marta,
>
> I apologise, I had forgoten our conversation!
> My memory was never good, and it is getting worst.
>
> You wrote:
>
>  >No, I am not thinking of the analogue of Steve Lack's model
> structure since, strictly speaking,
>  >it has nothing to do with stacks. Comments to that effect (with
> which Steve agrees) are included
>  >in the Bunge-Hermida paper. It was actually a surprise to discover
> that after trying to do what
>  >you suggest and failing.
>
> I disagree with your conclusion. I looked at your paper with Hermida.
> We are not talking about the same model structure. The fibrations in
> 2Cat(S) defined
> by Steve Lack (your definition 7.1) are too weak when the topos S does
> not satify the axiom of choice.
> Equivalently, his generating set of trivial cofibrations is too small.
>
> Nobody has read my paper with Myles
> <A.Joyal, M.Tierney: Classifying spaces for sheaves of simplicial
> groupoids, JPAA, Vol 89, 1993>.
>
> Best,
> André
>

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