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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 .

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. Given an object : X ! E of E=C, we have the situation X ?? y (11) E ????! F so we can pull back along to obtain the object : F^ ! E of E=C. Similarly, given an arrow of F=C; that is, an arrow : X ! Y such that X ????! Y ?? y ?? y F F commutes, we can pull back (11) along to obtain an arrow of C=S.

Show that the isomorphisms of DC are precisely the natural isomorphisms between functors from C to D. Exercise: let C be a small category and x an object E of C. De ne a map taking a cofunctor F from C to Set to the set F X and taking a natural transformation ! X : F X ! G X. Verify that this de nes a cofunctor from SetC to Set. 32 CHRIS HILLMAN Exercise: observe that for the trival category Z, TopZ is the category of pointed topological spaces (X; x), where an arrow from (X; x) to (Y; y) is a continous map taking x to y.

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A Categorical Primer by Chris Hillman

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