process

Duality for topological abelian group stacks and T -duality 237objects in two-categories. We will not be interested in the most general case. For ourpurpose it suffices to consider a notion which includes sheaves of abelian groups andgerbes with band given by a sheaf of abelian groups. We choose to work with sheavesof strictly commutative Picard categories.2.1.2 Let C be a category,F W C C ! Cbe a bi-functor, andbe a natural isomorphism of tri-functors. W F.F.X;Y/;Z/ ! F.X;F.Y;Z//Definition 2.1. The pair .F; / is called an associative functor if the following holds.For every family .X i / i2I of objects of C let e W I ! M.I/ denote the canonicalmap into the free monoid (without identity) on I . We require the existence of amap F W M.I/ ! Ob.C / and isomorphisms a i W F .e.i// ! X i , and isomorphismsa g;h W F .gh/ ! F.F.g/; F .h// such that the following diagram commutes:F .f .gh// a f;ghF.F.f /; F .gh//a g;hF.F.f /; F .F .g/; F .h//F ..fg/h/ afg;hF.F.fg/; F .h// F.F.F.f /; F .g//; F .h//.a f;g(6)2.1.3 Let .F; / be as above. Let in addition be given a natural transformation ofbi-functors W F.X;Y/ ! F.Y;X/:Definition 2.2. .F;;/is called a commutative and associative functor if the followingholds. For every family .X i / i2I of objects in C let e W I ! N.I/ denote the canonicalmap into the free abelian monoid (without identity) on I . We require that there existF W N.I/ ! C , isomorphisms a i W F .e.i// ! X i and isomorphisms a g;h W F .gh/ !F.F.g/; F .h// such that (6) andF .gh/ a g;hF .hg/ ah;gF.F.g/; F .h//F.F.h/; F .g//(7)commute.