Let me just say I now have a cleaner, more explicit, LaTeXed proof that every Grothendieck topos has a site with no non-identity idempotents. I can send it to anyone who wants it. It is one and a half pages in pdf. Proper use of Peter Johnstone's idea of a "coverage" makes the general case nearly as simple as the case where all covering families are finite. The stuff I posted about co-finality in transfinite ordinals is not needed. The awkward and nameless notation C@Z has been replaced by [CxZ] and named the "slanted product" because it is like the cartesian product CxZ except that all non-identity arrows are required to "slant upwards". I describe this as "blowing up" all non-identity endomorphisms since it turns each one into an infinite spiral. best, colin 16-Feb-2005 10:53:49 -0400,1331;000000000000-00000000