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According to Tierney, the monic logical operator J : ! should be interpreted as \it is locally true that"; in other words, we have a notion of a \local neighborhood" of the value taken on by some variable x appearing in the proposition. Exercise: show that the following de ne Lawvere-Tierney operators: 1. j = :: (this gives the double negation topology), 2. j = ) ( ), where : 1 ! is any truth value (the open topology), A CATEGORICAL PRIMER 61 3. j = ^ ( ), where : 1 ! is any truth value (the closed toplogy).

6. Recall that G rm, the sheaf of germs cofunctor, takes Ps X to Et X. Since every object of Sh X is in Ps X and every object of Et X is in BnX, we can consider S ec and G rm to be functors between Bn X and Ps X. Verify that S ec is a left adjoint of G rm. 7. Verify that Et X and Sh X are isomorphic categories. 10. Topoi Any topos may be regarded as a mathematical domain of discourse or \world" in which mathematical concepts can be interpreted and mathematical constructions performed. J. L. Bell 2] One of the most remarkable properties of Set is that we can represent the subobjects (subsets) A of X not just as equivalence classes of monic arrows into X, but as characteristic functions A : X !

Now suppose there is a nal object F in C, and show that M F is always isomorphic to M. 6. Conclude that the objects 1M ; 1M 1F are isomorphic in X=C. 7. Show that the isomorphism classes of objects in X=C form a commutative monoid. Exercise: Fix an arrow E ! F of C. A CATEGORICAL PRIMER 31 1. Suppose that pushouts always exist in C. Show that we obtain a functor from E=C to F=C, called the coslice change functor, as follows. Given an object : E ! X of E=C, we have the situation E ????! F ?? y X so we can push out along to obtain the object : F !

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