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238 U. Bunke, T. Schick, M. Spitzweck, and A. Thom2.1.4 We can now define the notion of a strict Picard category. Instead of F will usethe symbol “C”.Definition 2.3. A Picard category is a groupoid P together with a bi-functor CW P P ! P and natural isomorphisms and as above such that .C;;/is a commutativeand associative functor, and such that for all X 2 P the functor Y 7! X C Y is anequivalence of categories.2.2 Examples of Picard categories2.2.1 Let G be an abelian group. We consider G as a category with only identitymorphisms. The addition CW G G ! G is a bi-functor. For and we choose theidentity transformations. Then .G; C;;/ is a strict Picard category which we willdenote again by G.2.2.2 Let us now consider the abelian group G as a category BG with one object and Mor BG .; / WD G. We let CW BG BG ! BG be the bi-functor which acts asaddition Mor BG ./Mor BG ./ ! Mor BG ./. For and we again choose the identitytransformations. We will denote this strict Picard category .BG; C;;/shortly by BG.2.2.3 Let G; H be abelian groups. We define the category EXT.G; H / as the categoryof short exact sequencesE W 0 ! H ! E ! G ! 0:Morphisms in EXT.G; H / are isomorphisms of complexes which reduce to the identityon G and H . We define the bi-functor CW EXT.G; H / EXT.G; H / ! EXT.G; H /as the Baer additionE 1 C E 2 W 0 ! H ! F ! G ! 0;where F WD zF=H, zF E 1 E 2 is the pre-image of the diagonal in G G, andthe action of H on zF is induced by the anti-diagonal action on E 1 E 2 . The associativitymorphism W .E 1 C E 2 / C E 3 ! E 1 C .E 2 C E 3 / is induced by the canonicalisomorphism .E 1 E 2 / E 3 ! E 1 .E 2 E 3 /. Finally, the transformation W E 1 CE 2 ! E 2 CE 1 is induced by the flip E 1 E 2 ! E 2 E 1 . The triple .C;;/defines on EXT.G; H / the structure of a Picard category.2.2.4 Let G be a topological abelian group, e.g. G WD T. On a space B we considerthe category of G-principal bundles BG.B/. Given two G-principal bundles Q ! B,P ! B we can define a new G-principal bundle Q ˝G P WD Q B P=G. Thequotient is taken by the anti-diagonal action, i.e. we identify .q; p/ .qg; pg 1 /. TheG-principal structure on Q ˝G P is induced by the action Œq; pg WD Œq; pg.We define the associativity morphism W .Q ˝G P/˝G R ! Q ˝G .P ˝G R/