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X , there is a unique such that the diagram 1 ????! N ????! N ?? y ?? y 1 ????! N ????! N As usual, a natural object may not exist, but if it does it is unique up to isomorphism. The idea is that a natural number object contains the \recursive structure" needed to carry out nite induction using the logic of T. We can then proceed to de ne addition on the natural numbers (the elements of N), and so forth; see 17]. Exercise: nd a natural number object for a topos other than Set. Exercise: suppose C is a category with a nal object, products, sums, coequalizers, and a natural number object.

Y Verify that here ; are the components of a natural transformation. (The rst functor takes the nonidentity arrows of the category Pair to the arrows X ';! Y of; C, while the second takes the nonidentity arrows of Pair to the ' arrows A ! ) Verify that the map taking an arrow X ! 1???! ?? y' '? 1???! Y Y ;1Y de nes a functor, called the diagonal functor, from C to C## . 2. De ne a category with three objects U; V; W and ve arrows 1U ; 1V ; 1W , and U ! W V . Verify that this is in fact a category.

However, note that for many topoi, the classi er has elements (truth values) other than >; ?. Now for the suprise: the law of the excluded middle :: = fails for general topoi T. Thus, cannot, in general, be a Boolean algebra. In fact, it is in general a Heyting algebra; these are generalizations of Boolean algebras which rst arose in the formal study of intuitionistic logic (where the law of the excluded middle is denied). A topos T whose classi er is a Boolean algebra is called a Boolean topos; otherwise T will have a weaker logic than the classical rst order logic: in particular, the law of the excluded middle fails, although the De Morgan laws may still hold.

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A Categorical Primer by Chris Hillman

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