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7. If T is a topos, then for any category C, the category TC is a topos. See 17] 20] for a proof. In particular, if P is a poset, let P be the corresponding preorder category. Then SetP is a topos. It is hardly ever a Boolean topos; indeed, the classifying object usually has many more than two elements. Cohen's idea for forcing a certain property to be true in a \nonstandard set theory" now comes down to this. We interpret the elements of P as stages of \knowlege", where p q means that q is a later (and more extensive) stage of knowledge than p.

Fourth edition. New York: Macmillan, 1977. 4.

It is called the sheaf of germs cofunctor from Ps X to Et X. Exercise: ; 1. De ne a category with two objects U; V and four arrows, 1U ; 1V and U ?! V. Verify that this is a preorder category. It is denoted Pair or ##. Verify that the objects of the functor category C## , the category of pairs, are diagrams '; '; in C with the \shape" X ?! Y . Verify that an arrow in C## from X ?! Y ; to A ?! B is a pair of commuting diagrams ' X ????! Y X ????! Y A ????! B A ????! B ?? y ?? y ?? y ?? y Verify that here ; are the components of a natural transformation.

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