process

328 U. Bunke, T. Schick, M. Spitzweck, and A. ThomBy Prin B .G/ we denote the category of G-principal bundles over B. The setH 0 .Prin B .G// of isomorphism classes in Prin B .G/ is in one-to-one correspondencewith homotopy classes ŒB; BG of maps from B to the classifying space BG of G. Ifwe fix a universal bundle EG ! BG, then the bijectionŒB; BG ! H 0 .Prin B .G//is given byŒf W B ! BG 7! ŒB f;BG EG ! B:6.2.3 We now specialize to G WD T n . The classifying space BT n of T n has thehomotopy type of the Eilenberg–MacLane space K.Z n ;2/. We thus have a naturalisomorphismH 2 .BI Z n / Def.Š ŒB; K.Z n ; 2/ Š ŒB; BT n Š H 0 .Prin B .T n //:So, T n -principal bundles are classified by the characteristic class c.E/ 2 H 2 .BI Z n /.Using the decompositionH 2 .BI Z n / Š H 2 .BI Z/ ˚ :::H 2 .BI Z/„ ƒ‚ …n summandswe can writec.E/ D .c 1 .E/;:::;c n .E//:Definition 6.9. The class c.E/ is called the Chern class of E. The c i .E/ are called thecomponents of c.E/.In fact, if n D 1, then c.E/ is the classical first Chern class of the T-principalbundle E.6.2.4 Let S be a site and F 2 Sh Ab S be a sheaf of abelian groups.Definition 6.10. An F -torsor T is a sheaf of sets T 2 ShS together with an actionT F ! T such that the natural map T F ! T T is an isomorphism of sheaves.An isomorphism of F -torsors T ! T 0 is an isomorphism of sheaves which commuteswith the action of F .By Tors.F / we denote the category of F -torsors.6.2.5 In 2.3.4 we have introduced the category EXT.Z;F/whose objects are extensionsof sheaves of groupsW W 0 ! F ! W ! w Z ! 0; (51)and whose morphisms are isomorphisms of extensions. We have furthermore definedthe equivalence of categoriesgiven by U.W/ WD w 1 .1/.U W EXT.Z;F/ ! Tors.F /