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Extra resources for A1-homotopy theory of schemes
Proof. 4 it is sufficient to show that the compostion pr ~ Id_~io,~;' x I is elementary I-homotopic to the identity. This homotopy ,(,6~' X A t is given by the morphism Id x g : ,~7" x I x I ~ ,~i;" x I. 6. - - Any strict I-homotopy equivalence is an I-weak equivalence. Proof. - - Let f : t~---+~" be a strict I-homotopy equivalence and g be a I-homotopy inverse to f We have to show that the compositions f o g and g o f are equal to the corresponding identity morphisms in the I-homotopy category.
Since A~ is a proper model category the big square of this diagram is isomorphic to the original one in ,~'s(T) and in particular the left vertical arrow is in Wa. Decomposing it into a cofibration and a trivial fibration and using the fact that p is an A-fibration we get a morphism ~ " ~ ~" in ~,fs(T) with the required property. 26 we get the following corollary. 27. - - Let p" ,~ ~ ~ be a fibration such that B is fibrant and suppose that p is isomorphic in ~oZ, fs(T ) to an Afibration. Then p is an Afibration.
For any pointed morphism f and any pointed connected simplicial sheaf 1 ,Sg', the simplicial sheafLx,;f/(,,,~,~a;)/s connected. 2~ ) is thusf-local. Then the canonical induced morphism: I~j(R~l(,Sg'~)) ---+Rg~ Lz~(a'/(--~"). is a weak equivalence. In the case T = Sets this theorem was proven by Bousfield and independently by Dror [10, 3. 1]. Proof. - - One may assume ,~g' 0-reduced and set G ' = G(Sg)'). 32. ~;. 35. - - Let f : MI ---+ M,~ be a homomorphism of simplicial monoids which is a f-weak equivalence as a morphism of simplicial sheaves of sets.