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248 U. Bunke, T. Schick, M. Spitzweck, and A. Thom3 Sheaf theory on big sites of topological spaces3.1 Topological spaces and sites3.1.1 In this paper a topological space will always be compactly generated and Hausdorff.We will define categorical limits and colimits in the category of compactlygenerated Hausdorff spaces. Furthermore, we will equip mapping spaces Map.X; Y /with the compactly generated topology obtained from the compact-open topology. Inthis category we have the exponential lawMap.X Y; Z/ Š Map.X; Map.Y; Z//:By Map.X; Y / ı we will denote the underlying set.category of topological spaces we refer to [Ste67].For details on this convenient3.1.2 The sheaf theory of the present paper refers to the Grothendieck site S. Theunderlying category of S is the category of compactly generated topological Hausdorffspaces. The covering families of a space X 2 S are coverings by families of opensubsets.We will also need the sites S lc and S lc-acyc given by the full subcategories of locallycompact and locally compact locally acyclic spaces (see 3.4.6).3.1.3 We let Pr S and ShS denote the category of presheaves and sheaves of sets on S.Then we have an adjoint pair of functorsi ] W Pr S , ShS W iwhere i is the inclusion of sheaves into presheaves, and i ] is the sheafification functor.If F 2 Pr S, then sometimes we will write F ] WD i ] F .As before, by Pr Ab S and Sh Ab S we denote the categories of presheaves and sheavesof abelian groups.3.2 Sheaves of topological groups3.2.1 In this subsection we collect some general facts about sheaves generated byspaces and topological groups. We formulate the results for the site S. But they remaintrue if one replaces S by S lc or S lc-acyc .3.2.2 Every object X 2 S represents a presheaf X 2 Pr S of sets defined byon objects, and byon morphisms.S 3 U 7! X.U / WD Hom S .U; X/ 2 SetsHom S .U; V / 3 f 7! f W X.V / ! X.U /