Unless I've slipped up, every Grothendieck topos has a site with no idempotents (except identities). A proof is below. The trick I use for getting rid of idempotents also makes it impossible for the site to have pullbacks. So there is another question: Does every Grothendieck topos have some site with pullbacks but no idempotents? On the other hand it is clear that for some Grothendieck toposes, every subcanonical site has non-identity idempotents. For example, let E be a topos such that: every non-empty object of E has a global element ("nullstellensatz"). Then each non-empty object A of the topos has some constant x:1-->A and thus an idempotent A-->1-->A. This includes all non-empty representables A, and if the site is subcanonical then these idempotents exist in the site. This condition does not apply to the gros etale topos over a scheme, unless the scheme is the spectrum of an algebraically closed field. But it can probably be adapted using the fact that every scheme has points defined over algebraically closed fields. Just now, though, I want to post this: Theorem: Every Grothendieck topos has a site with no idempotents (except identities). Here "site" is understood as in Johnstone ELEPHANT A2.1 (and C2.1), not requiring pullbacks. Each object B in a site is equipped with a "coverage", a set of families of arrows to B such that each family is seen as a cover for B. These covering families are not assumed to be saturated. Proof: Let a topos have a site C,j. We get a site, for the same topos, where all non-identity idempotents are broken. Indeed, all non-identity endomorphisms are broken. Intuitively, we stack infinitely many copies of C above each other, but we insist that all non-identity arrows rise at least one level, and then we make each object cover its copies in every higher level so that from the viewpoint of the sheaves all those copies are iso. For simplicity, assume all the covering families in j are finite. At the end I'll describe the technicalities for avoiding that assumption. Define a category C@Z where an object is a pair (A,i) with A an object of C and i is any integer. Call i the "level" of the object (A,i). The arrows are triples (f,i,k):(A,i)-->(B,k) where either i=k and f is an identity arrow in C (so that also A=B), or i<k as integers and f:A-->B is any arrow in C. Composition is the obvious law (f,k,n)(g,i,k)=(fg,i,n). Call an arrow (f,i,k) of C@Z "vertical" iff f is an identity arrow. So every vertical arrow is monic and epic. There is a projection functor p:C@Z-->C. Define a coverage j@Z by saying a family of arrows (all with the same codomain) is a covering if and only if its image is a covering in C. Notice that since each covering family is finite, there is some lower bound to the levels i of objects (A,i) in the family. In particular every vertical arrow is a j@Z cover (and yet, if it is not an identity arrow in C@Z then it is not an effective epimorphism). Composing with the projection p gives a functor p* from presheaves on C to presheaves on C@Z. This functor is faithful since p is surjective. Direct verification shows that if F is a sheaf for j then p*F is a sheaf for j@Z. Each vertical arrow (f,i,k) in C@Z is a monic cover, so for any sheaf F on C@Z, the value F(f,i,k) is an isomorphism. In other words, all the vertical arrows become isomorphisms in any sheaf. Straightforward calculation shows that p* is also full as a functor from sheaves on C to sheaves on C@Z, basically because all the vertical arrows in p*F act as identities (and not just isos). It is also straightforward to check that each sheaf on C@Z is iso to p*F for some sheaf F on C. So p* is an equivalence functor from the topos of sheaves on C,j to the topos of sheaves on C@Z,j@Z. This completes the proof when j has only finite covering families. Now, to get rid of the restriction to finite covering families. The proof above used two properties of Z. First, Z has no greatest member (this is needed so that non-identity arrows always have somewhere to go). Second, and more complicated, no finite covering family can map co-initially into Z. We stated this above as saying the levels of any covering family in C@Z are bounded below in Z. This was used to verify that the procedure gives a coverage on C@Z and of course it would fail if j had infinite covering families. To extend the proof to work for all sites we must replace Z. Exactly how we do this depends on exactly what kind of sites we allow. It seems to me the most natural formulation for the theorem is to use all small sites. (I.e. we pick a single universe U, let C,j be any U-small sites, and it turns out that we get a U-small site with no idempotents but the same topos as C,j). Given any small site C,j let K be the next cardinal above the cardinality of the set of arrows of C. Then no function from the arrows of C into K will have image cofinal in K. Let K* be the opposite (or dual) order to K, and let K*+N be K* with a copy of the natural numbers N put 'above' it. Then K*+N has no maximal element, and no covering family in j can map co-initially into it. So it will serve in place of Z above. I hope the details are clear to anyone who wants to see them. best, Colin 15-Feb-2005 16:23:56 -0400,1679;000000000001-00000000