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Duality for topological abelian group stacks and T -duality 2411.F.x C y/F./F.y C x/F.x/C F.y/ F.y/C F.x/;2.F ..x C y/ C z/F.x C y/ C F.z/ .F .x/ C F.y//C F.z/F./F.x C .y C z// F.x/C F.y C z// F.x/C .F .y/ C F.z//.Definition 2.6. An isomorphism between additive functors uW F ! G is an isomorphismof functors such thatF.x C y/u xCy G.x C y/F.x/C F.y/ u xCu y G.x/ C G.y/commutes.Definition 2.7. We let Hom.P 1 ;P 2 / denote the groupoid of additive functors from P 1to P 2 .ByPIC we denote the two-category of Picard categories.2.4.2 The groupoid Hom.P 1 ;P 2 / has a natural structure of a Picard category. We setand define the transformationsuch that.F 1 C F 2 /.x/ WD F 1 .x/ C F 2 .x/.F 1 C F 2 /.x C y/ ! .F 1 C F 2 /.x/ C .F 1 C F 2 /.y/.F 1 C F 2 /.x C y/ .F1 C F 2 /.x/ C .F 1 C F 2 /.y/F 1 .x C y/ C F 2 .x C y/ .F1 .x/ C F 1 .y// C .F 2 .x/ C F 2 .y// ˛ıı˛ F1 .x/ C F 2 .x/ C F 1 .y/ C F 2 .y/commutes. The associativity and commutativity constraints are induced by those of P 2 .
242 U. Bunke, T. Schick, M. Spitzweck, and A. Thom2.4.3 Let now P 1 ;P 2 be two Picard stacks on S.Definition 2.8. An additive functor F W P 1 ! P 2 is a morphism of stacks togetherwith a two-morphismP 1 P C 1 P 1FF F P 2 P C 2 P 2which induces for each U 2 S an additive functor F U W P 1 .U / ! P 2 .U / of Picard categories.An isomorphism between additive functors uW F 1 ! F 2 is a two-isomorphismof morphisms of stacks which induces for each U 2 S an isomorphism of additivefunctors u U W F 1;U ! F 2;U .As in the case of Picard categories, the additive functors between Picard stacks formagain a Picard category.Definition 2.9. We let Hom.P 1 ;P 2 / denote the groupoid of additive functors from P 1 ,P 2 .We get a two-category PIC.S/ of Picard stacks on S.Definition 2.10. We let HOM.P 1 ;P 2 / denote the Picard category of additive functorsbetween Picard stacks P 1 and P 2 .2.4.4 Let P;Q be Picard stacks on the site S. By localization we define a Picard stackHOM.P; Q/.Definition 2.11. The Picard stack HOM.P; Q/ is the sheaf of Picard categories givenbyHOM PIC.S/ .P; Q/.U / WD HOM.P jU ;Q jU /:A priori this describes a prestack, but one easily checks the stack conditions.2.5 Representation of Picard stacks by complexes of sheaves of groups2.5.1 There is an obvious notion of a Picard prestack. Furthermore, there is an associatedPicard stack construction a such that for a Picard prestack P and a Picard stackQ we have a natural equivalenceHom.aP; Q/ Š Hom.P; Q/ (8)2.5.2 Let S be a site. We follow Deligne, SGA 4.3, Expose XVIII, [Del]. Let C.S/ bethe two-category of complexes of sheaves of abelian groupsK W 0 ! K 1 d ! K 0 ! 0which live in degrees 1; 0. Morphisms in C.S/ are morphisms of complexes, andtwo-isomorphisms are homotopies between morphisms.To such a complex we associate a Picard stack ch.K/ on S as follows. We firstdefine a Picard prestack p ch.K/.
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EMS Series of Congress ReportsEMS S
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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42 A. Bartels, S. Echterhoff, and W
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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82 H. Emerson and R. MeyerAs a resu
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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On K 1 of a Waldhausen category 93t
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On K 1 of a Waldhausen category 95(
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On K 1 of a Waldhausen category 992
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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142 M. KaroubiWe would like to poin
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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148 M. Karoubi[6] M. F. Atiyah, R.
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Equivariant cyclic homology for qua
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174 C. Voigtand using ˇ.x; y/ D .S
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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A Schwartz type algebra for the tan
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292 U. Bunke, T. Schick, M. Spitzwe
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Deformations of gerbes on smooth ma
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Parshin’s conjecture revisitedTho
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Parshin’s conjecture revisited 41
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428 C. WeibelGiven (0.4) and axiom
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430 C. WeibelAs pointed out in the
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432 C. WeibelConsider the cohomolog
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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