By H. Jacquet, R. P. Langlands
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Extra info for Automorphic Forms on GL(2) (part 2)
Xl ACxCDxDE o×I ADxDE 1×o lxo 1lxlx° 1XO ABxBCxCE ACxCE AE commutes. ) BC7 The (degenerate) cube 39 for c o h e r e n c e 1,3 1 X AA×AB×BB lx o ....................... rI AB×BB 3 ×I× II > AA×AB -"eA~k o/¢ r n IB .... o III ~ ~ l |~ IB . IxAB ABxl commutes. 2. between A pseudo-functor bicategories together (or m o r p h i s m ) is an o b j e c t function F F : ~ ~ ~' : Ob ~ ~ Ob with PFI functors PF2 natural FA, B : ~(A,B) ~ ~' (FA,FB) , transformations o fB(A,B) × ~ ( B , C ) c > I o ~' (FA,FB) ×~' (FB,FC) PF3 PF4 l-cells ~A ~(A,C) : I F, A two conditions: The cube ~ ~ F ( I A) 40 in ~' (FA,FC) ~' (FA,FA) , subject to ~' 1,3 4 I X° ABxBC×CD x l / ABxBD FABXFBcXFcD FAB×FBD AC×CD o > AD FAD FAc×FcD i×o t (A)F(B) ×~'(B) F(C) x~ (C) F (D) ----c | e - - .
B) A i-cell from b imodu I e RMs (A,S) to . A 2-cell modules. ~M®N Mo~ is a coequalizer 2-cells (B ,R) S in ~(A,C) . The composition is the induced morphism. is That of The identity for RRR . ~ • eR~R~M . RoM ..... / ~eR°M IAoR*M is a split coequalizer. ) M 7 / IA~,M Associativity follows since composition w i t h a fixed i-cell preserves coequalizers. , as a b i c a t e g o r y with one object, Bim(Ab) = gory V ~ X , then Spans %(V,V) whose objects look like are diagrams 48 then Bim(Spans %) is the m u l t i p l i c a t i v e E do dl morphisms ~ Bim 3) In this work we shall be concerned with If Ab, ~ V cate- ~ and whose 1,3 12 > E where both as above triangles commute.
Com- are t h o s e o f the form ~ is a (B,b) (2-Cat)-category~ closed category is n e e d e d 2-Cat is a n o t a t i o n T h e y are c o m b i n a t i o n s , where A°P(A,B) = ~(B,A) b) o p ~ , where °PA(A,B) = ~(A,B) Op op A 3-functor A , where is a transformations, modifications, t h e r e are v a r i o u s A(A,B) modifications There of such, of small and 3 - c e l l s corresponding to the a n a l o g o u s comma category. Fun(A,~) 31 closed 3-categories. 3). = °P[~(A,B) ] (2-Cat)-functor. , of ~op c) in (gf,F(f),~-~) : (A,F(f)b)~ A 3-cateqory In this w o r k all that a) (B,b) e : a ~ F(f)b morphisms (f,idF(f) b) possible as above, w h i l e is g i v e n by (g,~) (f,~) enriched (A,a) The objects are p a i r s (f,~) where category.