Re: RE: stacks (was: size_question_encore)

Re: RE: stacks (was: size_question_encore)

[André Joyal]

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?


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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)

[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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