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Duality for topological abelian group stacks and T -duality 2573.3.12 Note that a site S which has finite products satisfies the assumption of Proposition3.12. In fact, for A 2 S we have by Lemma 3.6 thatA jY Š A Y ! Y :Therefore the restriction functor .:::/ jY preserves representables.Corollary 3.14. Let S be as in Proposition 3.12. For sheaves F;G 2 Sh Ab S and Y 2 Swe haveExt i Sh Ab S .F; G/ jY Š Ext i ShAb S=Y .F jY ;G jY /:Proof. Since .:::/ jY is exact (Lemma 3.8) we haveExt i Sh Ab S .F; G/ jY Š .H i RHom ShAb S .F; G// jYŠ H i .RHom ShAb S .F; G/ jY /Š H i .RHom ShAb S=Y .F jY ;G jY // (by Proposition 3.12)Š Ext i ShAbS=Y .F jY ;G jY /:3.3.13 Let us assume that S has finite fibre products. Fix U 2 S. Then we can definea morphism of sites U W S ! S=U which maps A 2 S to .U A ! U/2 S=U. Oneeasily checks the conditions given in [Tam94, 1.2.2]. This morphism of sites inducesan adjoint pair of functorsU W ShS , ShS=U W U :Let f W S=U ! S be the restriction defined in 3.3.2 (with Y replaced by U ) and letf W ShS=U , ShS W f be the corresponding pair of adjoint functors.Lemma 3.15. We assume that S has finite products. Then we have a canonical isomorphismf Š U .Proof. We first define this isomorphism on representable sheaves. Since every sheafcan be written as a colimit of representable sheaves, U commutes with colimits asa left-adjoint, and f commutes with colimits by Lemma 3.8, the isomorphism thenextends to all sheaves. Let W 2 S and F 2 ShS=U/. Then we haveHom Sh S=U . U W ;F/Š Hom Sh S .W ; U .F //Š U .F /.W /Š F.W U ! U/Š Hom Sh S=U ..W U ! U/;F/ (by Lemma 3.6)Š Hom Sh S=U .f W ;F/This isomorphism is represented by a canonical isomorphismU .W / Š f .W /in ShS=U.