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By Chris Hillman

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A0 B 0 ????! B0 B 0 A0 ??? ) We have established a natural bijection Hom (DX; (A; B)) ' Hom (X; P (A; B)) C C C where D is the diagonal functor from C to C C (taking the object X of C to the object (X; X) of C C) and P is the product functor from C C back to C (taking the object (A; B) of C C to the object A B of C). Such natural bijections are quite important and they occur throughout mathematics. 1. Suppose F is a functor from A to B and G is a functor from B back to A, such that there is a natural bijection .

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 ! X of F=C. Similarly, given an arrow of E=C; that is, an arrow : X ! Y such that E E ?? y (10) ?? y X ????! Y commutes, we can pushout out (10) to obtain an arrow of F=C. 2. Suppose that pullbacks always exist in C. Show that we obtain a cofunctor from C=F to C=E, called the slice change cofunctor, as follows.

Exercise: suppose products always exist in C. Fix an object E of C. Show that we can de ne a functor E , the slice functor, from C to C=E as follows. Take X to the canonical arrow E X ! E and take ' : X ! Y to 1E '. Exercise: suppose C is a category in which sums always exist. 1. Show that we can \add" the objects of the slice category C=X. More precisely, + given S ! X and T ! X, de ne an arrow S + T ! X, where S + T is the sum of the objects S; T of C. ) 2. Prove the identity 1S +T = 1S + 1T .

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A Categorical Primer by Chris Hillman

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