alternative lecture notes - Rational points and algebraic cycles

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(3) F ◦ W = C. Examples 12.4. (1) Any model category. (2) ΓSet ∆op where W consists of the morphisms inducing weak equivalences on underlying simplicial sets, and F consists of the morphisms inducing fibrations on underlying simplicial sets. (3) More generally: let C be a Grothendieck site with enough points. Then we give Sh(C) ∆op the structure of a weak fibration category as follows. Let W C be the morphisms that give weak equivalences on the stalks, and F C the morphisms that give Kan fibrations on the stalks. The next object is, given a weak fibration category (C, W, F ), to put a model structure on Pro(C). We require the morphisms in W and F to become weak equivalences and fibrations in Pro(C) via the full embedding C ↩→ Pro(C). The other weak equivalences and fibrations we want to add in a minimal way: Definition 12.5. Let (C, W, F ) be a weak fibration category. Pro(C) is said to have a (F, W )-generated model structure if there exists a model structure on Pro(C) such that COF = ⊥ (F ∩ W ). If Pro(C) has such a model structure, it follows that further triv FIB = ( ⊥ (F ∩ W )) ⊥ , triv COF = ⊥ F and FIB = ( ⊥ F ) ⊥ . Definition 12.6. Let (C, W ) be a relative category and let X, Y ∈ Pro(C). A morphism f : X → Y is called a levelwise weak equivalence if there exist isomorphisms X ∼= → X ′ and Y ∼= → Y ′ in Pro(C) and f ′ ∈ Hom(X ′ , Y ′ ) such that the diagram X ∼= X ′ f ′ Y f ∼= Y ′ commutes, X ′ , Y ′ have the same indexing set I, f ′ is defined levelwise for each object in I (i.e. is induced by a natural transformation), and the morphisms f ′ (i) : X ′ (i) → Y ′ (i) are in W for all i ∈ I. The set of all levelwise weak equivalences is denoted Lw ∼= (W ). Theorem 12.7 (Schlank, Barnea). Let (C, W, F ) be a weak fibration category. If the class Lw ∼= (W ) of levelwise weak equivalences is closed under composition and satisfies the 2-outof-3 property, then the (F, W )-generated model structure on Pro(C) exists. In this case, the class of weak equivalences coincides with Lw ∼= (W ). Example 12.8. The levelwise weak equivalences in Pro(ΓSet ∆op ) satisfy the hypotheses of this theorem, if W consists of the levelwise weak equivalences in ΓSet ∆op . Starting from the adjunction we get an adjunction between Pro-categories • Γ : ΓSet ∆op SSet : const Pro(• Γ ): Pro(ΓSet ∆op ) 30 Pro(SSet) : Pro(const)
which we would like to be Quillen. We have to check that Pro(const) preserves fibrations and trivial fibrations. This is taken care of by: Proposition 12.9 (Schlank, Barnea). Let F : C → D be a functor between weak fibration categories, and assume that Pro(C), Pro(D) have (F, W )-generated model structures. If F preserves fibrations, trivial fibrations and finite limits, then Pro(F ) : Pro(C) → Pro(D) preserves fibrations, trivial fibrations and all limits. Let C be a Grothendieck site with enough points that is locally connected, i.e., there exists a functor π 0 fitting into an adjunction π 0 : Sh(C) ∆op SSet : const (Example: if C = Spec(k) et , then π 0 is the lower-shriek functor f ! .) Here, the category on the left is considered with the weak fibration category structure mentioned in Example 12.4(3). Since const surely preserves fibrations and trivial fibrations, and also limits since it is a right adjoint, is a Quillen adjunction. Pro(π 0 ): Pro(Sh(C) ∆op ) Pro(SSet) : Pro(const) Remark 12.10. For the result to hold it is not necessary to require that C has enough points. However, we only had a definition of weak fibration category structure on Sh(C) ∆op in that case (Example 12.4(3)). Also, the local connectedness requirement is not essential. Definition 12.11. L Pro(π 0 )(∗) ∈ Pro(SSet) is called the shape or realization of C. 13. August 13 (Schlank) Let C be a Grothendieck site with enough points. Let M C = Pro(Sh(C) ∆op ). Example 13.1. If C is trivial (one object), then M C = Pro(Set ∆op ) (pro-spaces). Example 13.2. If C = (Spec k) et , then M C = Pro((Γ k -set) ∆op ) (pro-Γ k -spaces). Theorem 13.3 (Barnea, S.). There exists a model structure (M C , W, COF, FIB) on M C such that (1) W = Lw ≃ W C , where W C are the stalkwise weak equivalences (2) COF = ⊥ (F C ∩ W C ), where F C are the stalkwise fibrations. Theorem 13.4 (Barnea, Schlank). Let f : C → D be a map of sites (we use the convention that this means that f is a functor C → D). Then there is a Quillen adjunction L: M D M C : Pro(f ∗ ) Every site has a unique map Γ: ∗ → C so we get L: M C Pro(Set ∆op ) : Pro(Γ) Define the shape of C as |C| := LL(∗ MC ) ∈ Pro(Set ∆op ). Given a (pro) space X and an abelian group A, we have H n (X, A) ≃ [X, K(A, n)] Pro-Set ∆ op . 31
Page 1 and 2: ARITHMETIC APPLICATIONS OF ÉTALE H
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Page 9 and 10: (non-faithful) essentially surjecti
Page 11 and 12: Definition 5.5. Let X be a space an
Page 13 and 14: Let F : C → Set be a functor. Sup
Page 15 and 16: A relative functor F : (D, W 1 )
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Page 21 and 22: Definition 9.3. A model category st
Page 23 and 24: sending E = (· · · → E 1 → E
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Page 27 and 28: Example 11.11. (Colimits do not pre
Page 29: If G is a group, then we have the c
Page 33 and 34: Definition 13.9. Let U • , V •
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Page 37 and 38: 16. August 22 (Harpaz) Let K be a f
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