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Duality for topological abelian group stacks and T -duality 265Consider a point a 2 A and s 2 H i .AI R W .I //. We must show that there existsa neighbourhood a 2 U A of a such that s jU D 0. Since I is an exact sequenceof sheaves in degree 1 there exists an open covering ¹Y r º r2R of A W such thatQs jYr D 0. After refining this covering we can assume that Y r Š U r V r for suitable opensubsets U r A and V r W . Consider the set R a WD ¹r 2 R j a 2 U r º. Since W iscompact the covering ¹V r º r2Ra of W admits a finite subcovering indexed by Z R a .The set U WD T r2Z U r A is an open neighbourhood of a. By further restriction weget Qs jU Vr D 0 for all r 2 Z. By (18) this means that 0 D s jU jV r2 .W I H i R U .I //.Therefore s jU vanishes locally on W and therefore globally. This implies s jU D 0.3.4.4 Let f W S lc ! S be the inclusion of the full subcategory of locally compacttopological spaces. Since an open subset of a locally compact space is again locallycompact we can define the topology on S lc bycov Slc .A/ WD cov S .A/; A 2 S lc :The compactly generated topology and the product topology on products of locallycompact spaces coincides. The same applies to fibre products. Furthermore, a fibreproduct of locally compact spaces is locally compact. The functor f preserves fibreproducts. In view of the definition of the topology S lc the inclusion functor f W S lc ! Sis a morphism of sites.Lemma 3.31. Restriction to the site S lc commutes with sheafification, i.e.i ] ı p f Š f ı i ] :Proof. (Compare with the proof of Lemma 3.7.) This follows from cov Slc .A/ Šcov S .A/ for all A 2 S and the explicit construction of i ] in terms of the set cov Slc(see [Tam94, Section 3.1]).Lemma 3.32. The restriction f W ShS ! ShS lc is exact.Proof. (Compare with the proof of Lemma 3.8.) The functor f is a right-adjoint andthus commutes with limits. Colimits of presheaves are defined object-wise, i.e. for adiagramC ! Pr S; c 7! F cof presheaves we have. p colim c2C F c /.U / D colim c2C F c .U /:It follows from the explicit description of p f that this functor commutes with colimitsof presheaves. For a diagram of sheaves F W C ! ShS we havecolim c2C F c D i ]p colim c2C F c :By Lemma 3.31 we get f colim c2C F c Š f i ]p colim c2C F c Š i ]p f p colim c2C F c Ši ]p colim c2C p f F c Š colim c2C f F c :