[FOM] Fraenkel-Mostowski-Specker method and category theory

Andreas Blass ablass at umich.edu
Tue Apr 4 09:48:20 EDT 2006

	In his message of April 3, Solovay listed two situations in which  
changing a group-and-filter pair (G,Gamma) leaves the resulting  
Fraenkel-Mostowski-Specker topos C(G,Gamma) unchanged up to  
equivalence of categories, namely:

>       (a) If there is an isomorphism of G_1 with G_2 that carries
> Gamma_1 onto Gamma_2 then the two categories are equivalent.
>       (b) Let G be a group and Gamma a normal filter of subgroups of
> G. Let H be a subgroup of G lying in Gamma, and let Gamma' be the  
> collection
> of all subgroups of H lying in Gamma. Then C(G, Gamma) and C 
> (H,Gamma') are equivalent.

I believe there are two more situations of this sort.

(c) Let N be the intersection of all the groups in the filter Gamma.   
Then N is a normal subgroup of G, and the category C(G,Gamma) is  
equivalent to C(G/N,Gamma'), where Gamma' consists of the groups H/N  
for H in Gamma.

(d) Suppose G' is a subgroup of G with the property that, for each g  
in G and each H in Gamma, the coset gH meets G'.  (If we make G a  
topological group by taking Gamma as a base of neighborhoods of the  
identity, then this property is just that G' is dense in G.)  Then C 
(G,Gamma) is equivalent to C(G',Gamma'), where Gamma' consists of the  
groups (H intersect G') for H in Gamma.

Of course, these observations don't address the "real" issue behind  
Solovay's question, namely whether two categories of the form C 
(G,Gamma) can be equivalent for "interesting" reasons.

By the way, situations (c) and (d) could be eliminated by insisting  
that G be a complete, Hausdorff group with respect to the topology  
mentioned in (d).  According to (c) and (d), we can always replace G  
by the completion of its universal Hausdorff quotient, so restricting  
to complete, Hausdorff groups involves no loss of generality.

Andreas Blass

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