V satisfies some coassociative conditions. If we want to categorify this notation, we need to replace functions by sheaves. There are two kinds of sheaf theories: quasi-coherent sheaves and D-modules. Given a finite type G, * -pullback along with the multiplication map provides QCoh(G) a comonoidal category structure, Definition A.4. For a DG category C with a weak G-action, we mean C is a comodule category of (QCoh(G), m * ). By duality, The coalgebra structure is given by the pullback along with the multiplication map m. A structure of G-representation on V is a coaction of the coalgebra ? ,
, But when we define actions of pro-finite group scheme, such as G(O), weak action definition is different
, The category of categories with weak G(O)-action is defined to be: G(O) ? mod weak := colim n QCoh(K n ) ? mod ?
, Given an ind-pro group scheme G. According to Definition A.3, we could define two DG-categories D * (G) and D ! (G). The following lemma in, Remark The functor G(O) m od weak ?? QCoh
, The * -direct image functor along with m : G × G ? G defines D * (G) a monoidal category structure m * : D * (G)
,
, Dually, m ! provides D ! a structure of comonoidal category
, For a DG category C with a strong G-action(or infinitesimal trivialized action), we mean C is a module category of (D * (G), m * ). Dually
,
1. Given a smooth scheme X and an effective Cardier s divisor Y inside. Then, the following functor is bijective. k/Z ?? tame gerbes on X dR with a trivialization on ,
, What's more
2. a).(descend along with finite surjective morphism) If X ?? Y is a finite, surjective map between smooth schemes, then, the pullback defines an equivalence, Grb reg (Y ) Grb reg (X) ,
, A 1 -homotopy) There is a canonical equivalence: Grb reg (X) Grb reg (X × A 1 )
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, Nous notons par KL?(?) la catégorie des modules Kac-Moody?(O) -intégrables. Ici,?? désigne l'extension centrale de?((t)) par?. Et selon Kazhdan-Lusztig