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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 . 3. Show that + is isomorphic to + in C=X. 4. Show that Hom(S + T; X) is in bijection with Hom(S; X) ] Hom(T; X). 5. Now suppose that there is an initial object I in C. Show that S must be isomorphic to S + I. (Hint: show that 1S + , where I ! X S + I. ) 6. Conclude that the objects + 1X ; are isomorphic in C=X.

Suppose that E is an object in C. Show that we can de ne a functor E from C=E to C as follows. Take the object : X ! E of C=E to dom = X, and take the arrow ' : X ! Y of C=E to itself (considered as an arrow of C). 44 CHRIS HILLMAN Recall that if products exist in C, we can de ne the slice functor E from C to C=E. Show that E is the left adjoint of E . 2. Suppose that E ! F is an arrow in C. Show that we can de ne a functor from C=E to C=F as follows. Take the object of C=E to the object of C=B, and take the arrow ' : X !

Y ?? A / . ; / to denote the mutually inverse pair of natural bijections associated with an adjunction generalizes their use in 12] to denote the maps de ning' a galois connection. Exercise: suppose X ! Y is a map. We can consider the powerset P X (ordered by inclusion) as a preorder category, denoted here by P. Likewise, we can consider P Y as a preorder category, denoted here by Q. 1 (B) of P de nes a functor, the preimage functor; similarly taking an object A of P to an object '(A) of Q de nes a functor, the image functor.

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A Categorical Primer by Chris Hillman

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