RE: stacks (was: size_question_encore)

[Eduardo Dubuc <edubuc@dm.uba.ar>]

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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