 By Andreas Blass

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The proof of the lemma uses some machinery that it did not seem worthwhile to develop for just this one application. The proof will not be referred to later, so readers unfamiliar with the machinery can safely skip it. Proof. Diaconescu's description of pullbacks of bounded geometric morphisms of topoi [9,29] shows that the pullback of ~>op p op S —* % along sh-i-i (S) <—> % is, on the one hand fsh- i - » (8)] , •* v and on the other hand a sheaf subtopos in S . The lemma asserts that eop sh. (S ] )] for some is j 38 A.

For the purposes of the present paper, it serves as a tool for formulating, in a suitably strong way, the connection between topoi (like Freyd's examples) and familiar models. 42 A. BLASS, A. SCEDROV 3C4. If a Grothendieck topos M' represents a model M of ZFA, then the pure part M n of M is represented by the subtopos M' of M' constructed as follows. First do the Fourman construction, within M', of the cumulative hierachy starting with the empty object in place of A, then take all subobjects of the objects obtained in this way, and finally let M' be the full subcategory of M' determined by these.

For any b e 3, it is clear that no element of P can be below b - \/ (p e p|p £ b} , so this difference 49 FREYD'S MODELS must be 0; thus each b e ft is covered by a subset of the lemme de comparaison , the topos of sheaves on P. e. an arbitrary downward-closed set below the following are equivalent: covers p with respect to The downward-closure ft of J. ft in SB covers p with SB is respect to the canonical topology. The supremum of ft , or equivalently of ft, in p. p - V ft = 0. No element of P is £ p - \/ ft.

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