On Mon, 8 Jun 2009, Eduardo J. Dubuc wrote:
Ross Street wrote:
Problem. Suppose A is a locally small site whose category E of Set- valued sheaves is also locally small. Is E a topos? (see (*) below)
This is one of (probably) many problems of Girau topoi [satisfy all conditions in Girau's Theorem exept (may be) the set of generators] which are not known to be a topos.
(*) It seems Not: Take a Girau (really faux but locally small) topos E, with the canonical topology. Then the topos of sheaves should be E again, which is not a topos (am I missing something ?).
I presume that Ross was using the word "topos" to mean "elementary topos". But in any case, Eduardo was missing something: the proof that, if E is an \infty-pretopos (my preferred name for what he calls a "Girau(d) topos"), then every canonical sheaf on E is representable, requires the existence of a generating set (see C2.2.7 in the Elephant). For a counterexample in the absence of generators, let G be the "large" group of all functions from the ordinals to {0,1} having finite support, the group operation being pointwise addition mod 2 (or, if you prefer, the group of finite subsets of the ordinals under symmetric difference), and let E be the (elementary) topos of G-sets. For each ordinal \alpha, let A_\alpha be the set {0,1} with G acting via its \alpha-th factor; then any G-set admits morphisms into only a set of the A_\alpha, from which it follows that the coproduct of all the A_\alpha exists as a (set-valued) canonical sheaf on E, though it clearly isn't a set. Moreover, this coproduct admits a proper class of maps to itself, so the category of sheaves on E isn't locally small; hence it doesn't violate Ross's conjecture. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]