process

Duality for topological abelian group stacks and T -duality 2513.2.8 If X; H 2 S and H is in addition a topological abelian group, then Map.X; H /is again a topological abelian group. For a topological abelian group G 2 S we letHom top-Ab .G; H / Map.G; H / be the closed subgroup of homomorphisms. Recallthe construction of the internal Hom ShAb S .:::;:::/, compare 3.2.7.Lemma 3.5. Assume that G; H 2 S are topological abelian groups. Then we have acanonical isomorphism of sheaves of abelian groupse W Hom top-Ab .G; H / ! Hom ShAb S .G;H/:Proof. We first describe the morphism e. Let 2 Hom.G; H /.U /. We define theelement e./ 2 Hom ShAb S .G;H/.U /, i.e. a morphism of sheaves e./W G jU ! H jU ,such that it sends f 2 G. W V ! U/to ¹V 3 v 7! ..v//f .v/º2 H. W V ! U/.Let us now describe the inverse e 1 . Let 2 Hom ShAb S .G;H/.U / be given. Since.pr U W U G ! U/2 S=U it gives rise to a mapG W G.pr U W U G ! U/! H .pr U W U G ! U/:We now consider .pr G W U G ! G/ 2 G.pr U W U G ! U/ and set G WDG.pr G / 2 H .pr U W U G ! U/ D Hom S .U G; H/. We now invoke the exponentiallaw isomorphism expW Hom S .U G; H/ ! Hom S .U; Map.G; H //. Sincewas a homomorphism and using the sheaf property, we actually get an elemente 1 . / WD exp. G / 2 Hom S .U; Hom top-Ab .G; H // D Hom top-Ab .G; H /.U /. (Detailsof the argument for this fact are left to the reader.)If S is replaced by one of the sub-sites S lc or S lc-acyc , then it may happen thatHom top-Ab .G; H / does not belong to the sub-site. In this case Lemma 3.5 remains trueif one interprets Hom top-Ab .G; H / as the restriction of the sheaf on S represented byHom top-Ab .G; H / to the corresponding sub-site.3.3 Restriction3.3.1 In this subsection we prove a general result in sheaf theory (Proposition 3.12)which is probably well-known, but which we could not locate in the literature. Let S bea site. For Y 2 S we can consider the relative site S=Y . Its objects are maps .U ! Y/in S, and its morphisms are diagramsU V Y .The covering families for S=Y are induced by the covering families of S, i.e. for.U ! Y/ 2 S=Y a covering family WD .U i ! U/ i2I induces in S=Y a covering