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Duality for topological abelian group stacks and T -duality 2613.3.18 To abbreviate, let us introduce the following notation.Definition 3.23. If S has finite products, then for W 2 S we introduce the functorR W WD W ı W W ShS ! ShS:We denote its restriction to the category of sheaves of abelian groups by the samesymbol. Since W is exact and W is left-exact, it admits a right-derived functorRR W W D C .Sh Ab S/ ! D C .Sh Ab S/.Lemma 3.24. Let F 2 Sh Ab S and W 2 S. Then we have a canonical isomorphismRHom ShAb S .Z.W /; F / Š RR W .F /.Proof. This follows from the isomorphism of functors Hom ShAb S .Z.W /;:::/ ŠR W .:::/from Sh Ab S to Sh Ab S. In fact, we have for F 2 Sh Ab S thatHom ShAb S .Z.W /; F / Š Hom Sh S .W ; F .F //Š W . W .F // (Equation 16) (17)Š R W .F / (Definition 3.23):3.4 Application to sites of topological spaces3.4.1 In this subsection we consider the site S of compactly generated topologicalspaces as in 3.1.2 and some of its sub-sites. We are interested in proving that restrictionto sub-sites preserve Ext i -sheaves.We will further study properties of the functor R W . In particular, we are interestedin results asserting that the higher derived functors R i R W .F /, i 1 vanish undercertain conditions on F and W .Lemma 3.25. If C 2 S is compact and H is a discrete space, then Map.C; H / isdiscrete, andR C .H / D Map.C; H /:Proof. We first show that Map.C; H / is a discrete space in the compact-open topology.Let f 2 Map.C; H /. Since C is compact, the image f.C/ is compact, hence finite.We must show that ¹f º Map.C; H / is open. Let h 1 ;:::;h r be the finite set of valuesof f . The sets f 1 .h i / C are closed and therefore compact and their union is C .The sets ¹h i º H are open. Therefore U i WD ¹g 2 Map.C; H / j g.f 1 .h i // ¹h i ººare open subsets of Map.C; H /. We now see that ¹f ºD T riD1 U i is open.We have by the exponential lawR C .H /.A/ Š H .A C/ Š Hom S .A C;H/Š Hom S .A; Map.C; H // Š Map.C; H /.A/: