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250 U. Bunke, T. Schick, M. Spitzweck, and A. ThomLet now A 2 S and f 2 colim i2I X.i/.A/DHom S .A; colim i2I X.i//. Ascolim i2I X.i/is discrete and f is continuous the family of subsets ¹f1 .x/ A j x 2 colim i2I X.i/ºis an open covering of A by disjoint open subsets. The familyYf jf 1 .x/ 2 Y p colim i2I X.i/.f1 .x//xxrepresents a section over A of the sheafification colim i2I X.i/ of p colim i2I X.i/ whichof course maps to f under the inclusion (13). Therefore (13) is also surjective.3.2.5 If G 2 S is a topological abelian group, then Hom S .U; G/ WD G.U / has thestructure of a group by point-wise multiplications. In this case G is a sheaf of abeliangroups G 2 Sh Ab S.3.2.6 We can pass back and forth between (pre)sheaves of abelian groups and(pre)sheaves of sets using the following adjoint pairs of functorsp Z.:::/W Pr S , Pr Ab S W F ; Z.:::/W ShS , Sh Ab S W F :The functor F forgets the abelian group structure. The functors p Z.:::/and Z.:::/are called the linearization functors. The presheaf linearization associates to a presheafof sets H 2 Pr S the presheaf p Z.H / 2 Pr Ab S which sends U 2 S to the free abeliangroup Z.H /.U / generated by the set H.U/. The linearization functor for sheaves is thecomposition Z.:::/WD i ] ı p Z.:::/of the presheaf linearization and the sheafification.Lemma 3.4. Consider an exact sequence of topological groups in S1 ! G ! H ! L ! 1:Then 1 ! G ! H ! L is an exact sequence of sheaves of groups.The map of topological spaces H ! L has local sections if and only ifis an exact sequence of sheaves of groups.1 ! G ! H ! L ! 1Proof. Exactness at G and H is clear.Assume the existence of local sections of H ! L. This implies exactness at L.On the other hand, evaluating the sequence at the object L 2 S we see that theexactness of the sequence of sheaves implies the existence of local sections to H ! L.3.2.7 For sheaves G; H 2 ShS we define the sheafHom Sh S .G; H / 2 ShS; Hom Sh S .G; H /.U / WD Hom Sh S .G jU ;H jU /:Again, this a priori defines a presheaf, but one checks the sheaf conditions in a straightforwardmanner.