By Chris Hillman

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7. A proposition is said to be valid if the corresponding truth set is the entire domain. For example, the statement that \ implies " is valid; on the other hand, the statement \ implies " is not, because the contradiction of this statement, ^ : , has the nonempty truth set just computed. A CATEGORICAL PRIMER 53 8. It is convenient to introduce a third binary logical operator, called material implication and written ): ! , de ned such that ) is the proposition which is false only for those values x such that (x) = 1 but (x) = 0.

9 '? 8 ? = 9 ? = 8 ? 9' = 9 ? 8' = 8 ? The relation between the six cofunctors a a and 9 a ? a 8 , and the four functors LE a IE and LF a IF is given in the following Lemma. 2. In the following diagram (called the doctrinal diagram), Sub(E ) x ?? yIE T=E 9 ??????! ?????? 8 ??????! ?????? Sub(F ) x??? yIF T=F we have the following four natural isomorphisms: 9 LF IF 8 IF ? LF ' LE ' IE ' IE ? ' LE Exercise: verify the claims made in Figure 4. 12. Models in a Topos A startling aspect of topos theory is that it uni es two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory.

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. Note that each element of SetP is a sort of \net" of sets indexed by P. There is a natural notion of asymptotic agreement between two such elements of SetP ; moding out by this equivalence relation we obtain Set, the Cohen extension of Set. This will be a Boolean topos. Another way of describing this construction is to note that SetP is essentially the presheaf category over P , and Set is P:: .