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Y Exercise: let T be a topos. Fix an object E of T. Show that the slice functor E from T to T=E has a right adjoint E , de ned as follows. Take : X ! E to the object E de ned by pulling back E : X E ! E E along d1E e : 1 ! E E , so that E ????! X E ?? y ?? y E d1E e 1 ????! EE is a pullback diagram. Take ' : X ! Y (considered as an arrow of T=E to the arrow E ' de ned by pulling back the diagram X ????! E ? '? y Y ????! ) Combining with a previous exercise gives the adjunctions E a E a E , as indicated in the following diagram: along d1E e : 1 !

Y in Set, for each x 2 X we can de ne (x) to be the map taking e 7! '(x; e). This gives an arrow : X ! Y E , where we de ne Y E = f : E ! Y g. Conversely, given we can recover ' by observing that '(x; e) = (x)(e) = ev( (x); e) where setting ev( ; e) = (e) for all : E ! Y in Y E de nes the evaluation map ev : Y E E ! Y . 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 !

G 1 ???? '? y commutes, "H ! H 1 ???? 2. e. G G G G ????! ' '? y commutes. '? y H H H H ????! Exercise: verify that Grp-arrows between Grp-objects form a subcategory of C, denoted GrpC . Show that this category has products and equalizers (and thus kernels in the usual sense of group theory). Verify that GrpTop is the category of A CATEGORICAL PRIMER 59 topological groups, while GrpMan is the category of Lie groups, GrpSh X is the category of sheaves of groups over X, and AbgSh X is the category of sheaves of abelian groups over X.

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A Categorical Primer by Chris Hillman

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