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This means that we have a bijection ? Hom(X E; Y ) ' Hom X; Y E Moreover, this bijection is natural in the sense that it respects preperturbations X 0 ! X and postpertubations Y ! Y 0 . 5. Suppose C is a category in which products always exist. Y Y such that given any arrow X E ! Y , we have a unique arrow X ! Y E (called the transpose of ) such that the diagram "Y Y Y E E ????! (14) 1E x? X E ????! Y commutes. Then Y E is called an exponential object. Exponentials may not exist for a given pair of objects Y; E, but if an exponential Y E does exist, it is unique up to isomorphism.

This suggests that if C is itself a subcategory of D, then GrpC is a subcategory of GrpD as well as of C. Is it? Exercise: Fix a group G, a ring R, and a eld K, and let C be as above. Construct the categories Rng C , GGetC , RModC, and KLinC following the model of GrpC . Since toposes always have a nal object and products, these categories \know" the structure of the elementary algebraic categories, athough the theory of GrpT may very well be quite di erent from the theory of GrpSet. The most interesting toposes from the point of view of logic can model many more mathematical structures; in particular, they contain a notion of number.

14) 1E x? X E ????! Y commutes. Then Y E is called an exponential object. Exponentials may not exist for a given pair of objects Y; E, but if an exponential Y E does exist, it is unique up to isomorphism. Exercise: suppose C is a category with pullbacks, exponentials, and a nal object 1. X Z Y ' Exercise: suppose that exponentials always exist in C. Given an arrow X ! Y, E show that there is a unique arrow ' making the diagram "Y Y Y E E ????! x 'E 1E ? XE x? ' X E ????! "X commute. Verify that this de nes a functor, called the exponential functor, from C to itself, together with a natural transformation from the exponential functor onto the identity functor.

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