On Tue, 12 Jul 2011, Eduardo Dubuc wrote:
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.
The answer is no. Let M be the free monoid on a proper class of generators (indexed by the ordinals, say), and consider the category of M-sets (that is, sets equipped with a proper class of independent endomorphisms e_\alpha, \alpha \in Ord). This satisfies all the Giraud axioms except for the generating set, but doesn't have a subobject classifier. To see this, let \kappa be a cardinal, and consider the M-set A_\kappa whose underlying set is the set of \kappa-sequences of natural numbers with all but finitely many terms zero, with e_\alpha acting as the successor map on the \alpha-th term for \alpha < \kappa, and as the identity otherwise. Clearly, A_\kappa has \kappa distinct sub-M-sets, so it has \kappa distinct morphisms to \Omega if the latter exists. But it's generated as an M-set by the single element (0,0,0,...), so this must map to \kappa distinct elements of \Omega, and in particular \Omega has cardinality at least \kappa. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]