By Chris Hillman

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X to F ! X This de nes a category called the slice category over X , denoted C=X . If X is a topological space, the category Top=X is just Bn X. Exercise: show that there is a nal object in C=X. ) Observe how your argument blurs the distinction between objects and arrows of C=X. 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.

Y be two arrows in a category C. Then ' = i for all Z in C and all x 2Z X , ' x] = x]. Exercise: let C be a category. 1. Show that F is nal in C i for all objects Z of C, there exists a unique generalized element z 2Z F. 2. Show that A ! X is monic i for all Z and all z; z 0 2Z A, z] = z 0] implies z = z0. 3. Show that initial objects and epic arrows have no such characterizations. x X of X is in fact an element of the subobject ], a X such that x = exactly when there is an arrow 1 ! a. Examining the UMP for the classi er, we conclude this happens i >= A x This gives an internal criterion in T for when an element x of X is in fact an element of A.

X to ' : E ! Y . 40 CHRIS HILLMAN Exercise: suppose Y; E are objects of GGet with actions : G Y ! Y and : G E ! E. De ne Y E to be the set of mappings (not neccessarily G-homs) : E ! Y with the action de ned by letting g send to the map e 7! 1 e), which is easier to understand! Note that the G-homs in Y E are precisely the xed points or one-point subobjects under this action; that is, they are the (categorical) \elements" of Y E . Verify that the ev evaluation map Y E E ! Y is a G-hom. Conclude that exponentials always exist in GGet.

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A Categorical Primer by Chris Hillman

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