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Proof. 21 we can construct a commutative square of the form I 1 , such that the upper horizontal arrow is in G n WA and the right vertical one is an A-fibration. 29 we conclude immediately that the canonical morphism s: ~W --* ~W' x,~(,~ ),_r is in CNWA. Since both objects are fibrant over ,~2 we conclude that there is a m o r p h i s m f : ~ ' x ~ , ( . ~ ) ~ ~ ~" over (o~ such t h a t f o s = I d . 15). 7 we have to show that for any cartesian square ~~1 ) + "~2 1 ~"g"3 ' '~4 such that the right vertical arrow is in FA and the lower horizontal one is in WA, the upper horizontal one is also in WA.

The axioms MC1-MC3 are obvious from the definitions. The (trivial cofibration)/(fibration) part of MC4 is the definition of FA. The (cofibration)/(trivial fibration) part of MC5 follows immediately from the corresponding fact in the simplicial case since an trivial fibration is a trivial A-fibration. 5]. 10 by Joyal trick (see [18, p. 64]). 5. 22. 15(1). 7. Again, let A be a set of representatives for morphisms in A which satisfies the conditions of this theorem. 7. 23. - - Let p : ~ ~ ,)~ be a fibration such that , ~ that for any commutative diagram of the form gf Id is fibrant and suppose , in ,~efs(T ) such that i is in WA, there exists a morphism ~f" ~ ~ two triangles commutative.

5. More generally, any ringed site (T, ~') defines a site with interval. In particular we may consider the homotopy category associated with any subcategory in the category of schemes (over a base) which contains affine line. The functor Sing. 7 hold for the morphism z~ : pt ---+ I in any site with interval (T, I). In order to do it we construct an endofunctor Sing~. on the category of simplicial sheaves on a site with interval together with a natural transformation s : Id---+ Sin~. such that one has 1.

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A1-homotopy theory of schemes by Morel F.


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