Re: RE: stacks (was: size_question_encore)

["André Joyal" <joyal.andre@uqam.ca>]

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/ ]