process

Duality for topological abelian group stacks and T -duality 249Lemma 3.1. X is a sheaf.Proof. Straightforward.Note that a topology is called sub-canonical if all representable presheaves aresheaves. Hence S carries a sub-canonical topology.3.2.3 We will also need the relative version. For Y 2 S we consider the relative siteS=Y of spaces over Y . Its objects are morphisms A ! Y , and its morphisms arecommutative diagramsA B Y .The covering families of A ! Y are induced from the coverings of A by open subsets.An object X ! Y 2 S=Y represents the presheaf X ! Y 2 Pr S=Y (by a similardefinition as in the absolute case 3.2.2). The induced topology on S=Y is again subcanonical.In fact, we have the following lemma.Lemma 3.2. For all .X ! Y/2 S the presheaf X ! Y is a sheaf.Proof. Straightforward.3.2.4 Let I be a small category and X 2 S I . Then we havelim i2I X.i/ Š lim i2I X.i/:One can not expect a similar property for arbitrary colimits. But we have the followingresult.Lemma 3.3. Let I be a directed partially ordered set and X 2 S I be a direct systemof discrete sets such that X.i/ ! X.j / is injective for all i j . Then the canonicalmapcolim i2I X.i/ ! colim i2I X.i/is an isomorphism.Proof. Let p colim i2I X.i/ denote the colimit in the sense of presheaves. Then we havean inclusionp colim i2I X.i/ ,! colim i2I X.i/(this uses injectivity of the structure maps of the system X). Since the target is a sheafand sheafification preserves injections it induces an inclusioncolim i2I X.i/ ,! colim i2I X.i/: (13)