By Morel F.

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81 is A-fibrant; is A-locaL Proof. 25. 8 to verify that if we have a morphism i: ,r ~ i~" in t3 N WA then the morphism g o m ( ~ , N') ~ H o m ( , Z ' , ~') is a trivial fibration. 15, by adjointness. 7. 29. Then the projection . r ~ is in W A n C. be a morphism in FA. Proof. - - Consider the class G of morphisms f ~ ~ " in WA n C such that for any A-fibration ~" --+ ~ the projection , ~ ' x ~ kf ---* ~ is in W A n Cl. This class has the following properties: 1. if two out of three morphisms f , g , f o g E CI n WA are in G then so is the third; 2.

Recall that the left adjoint to the forgeffull functor A~ ~ A~ is the functor ,r H ,N'+ where ,Sg'+ is the simplicial sheaf ,~" Hpt pointed by the canonical embedding pt--~ ,'2~2"IIpt. Both functors preserve weak equivalences and thus induce a pair of adjoint functors between 3r and '~fs((Srn/S)x~). 5~)', x), (~/" ,y) define their wedge (d~;', x)V (5~r ,y) and their smash product (~,q~, x) A (~" ,y) in the usual way (, : ~ , x) V (~" ,y) = (,r x) A Ilpt ~ d , x =y) x x) V ,y), x xy). At-HOMOTOPY THEORY OF SCHEMES 83 Note that (,317, x)V ( ~ / , y ) is the sheaf associated to the presheaf which takes an object U of T to the wedge of pointed simplicial sets (,~Y(U), xu) and (~" (U),yu) and (5ig', x)A (~" ,y) is the sheaf associated to the presheaf which takes an object U of T to the smash product of pointed simplicial sets (,~g'(U), xu) and (~r The functor A~ ~ Aa~176 ( ~ ' , x) ~-+ (2g', x)A ( ~ ,y) has as fight adjoint the functor (o~; , z) H H0m ((~" ,y), (g, z)) whose value is the fiber over the base point of ~ of the evaluation morphism y* : Hom(~J , ~ ) --+ Hom(pt, o~; ) ~- ~ .

We have to show that the upper horizontal arrow in the cartesian square x. g" x I) 1 , ld• 0 5g" , 1 ,Z'xI is an I-weak equivalence. Applying the functor Sing). to this diagram we get a cartesian square (by (1)) which is I-weak equivalent to the original one (by (4)). (Id x/0) is a simplicial weak equivalence. (~f) is a simplicial weak equivalence since the simplicial model structure is proper. D e f n e a cosimplicial object A~ 9 A ~ ShvO" ) as follows. O n objects we set A~ = I". , m) be a morphism in the standard simplicial category A.

### A1-homotopy theory of schemes by Morel F.

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