alternative lecture notes - Rational points and algebraic cycles

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Define the nerve N ∆ ((U α )) of a covering U := ∐ α∈A U α → X as the simplicial set with ⎛ ⎞ N ∆ ((U α )) n := π 0 ⎜ ⎝ U × · · · × U⎟ X X ⎠ . } {{ } n+1 Given f, g : X → Y , we say that f ∼ g if there exists H : I × X → Y such that H| 0 = f and H| 1 = g. Define the simplicial set ∆ n : ∆ op → Set by ∆ n ([m]) := Hom ∆ ([m], [n]). For example, ∆ 0 is a point. Definition 5.12. Let A and B be simplicial sets, and let f : A → B. We say that f ∼ st g if there exists H : A × ∆ 1 → B such that H| 0 = f and H| 0 = g. This is reflexive, but not symmetric or transitive. Define the relation ∼ as the equivalence relation generated by ∼. st Definition 5.13. Let A and B be simplicial sets. Define the mapping space Map(A, B) ∈ Set ∆op as the simplicial set given by Map(A, B) n := Hom Set ∆ op (∆ n × A, B). For X, Y, Z ∈ CGHS, we have the exponential law Map(X × Y, Z) ≈ Map(Y, Map(X, Z)). Let A, B be simplicial sets. Let f, g : A → B. The following are equivalent: (1) f ∼ g (2) f and g (or rather, the realization of the corresponding two points in Map(A, B) 0 ) lie in the same connected component of | Map(A, B)|. If A and B are simplicial sets, there is a map φ: | Map(A, B)| → Map Top (|A|, |B|). The map from a circle to a circle that wraps around twice does not come from a simplicial map (unless one subdivides). Dan Kan solved this problem as follows. Definition 5.14. Let Λ n k be ∆n minus the interior and k th face. A Kan simplicial set is a simplicial set A such that for every diagram of the form the diagonal map exists. Λ n k ∆ n A ∃ Theorem 5.15. If B is Kan, then φ above is a homotopy equivalence. 6. July 18 (Harpaz) If C and D are categories with finite limits, and F : C → D is a functor that preserves them, then Pro(F ): Pro(C) → Pro(D) preserves all limits. The inclusion functor FinGr → Groups preserves only finite limits, but we get an adjoint P L: Pro(Groups) → Pro(FinGr). 12
Let F : C → Set be a functor. Suppose that F preserves all limits. Then F has a left adjoint L: Set → C. Let ∗ be a singleton. What is L(∗)? We have This proves the nontrivial half of Hom C (L(∗), c) ≃ Hom Set (∗, F (c)) ≃ F (c). Theorem 6.1. Let C be a category with all limits. A functor F : C → Set is representable if and only F it preserves all limits. What if F preserves only finite limits? Then Pro(F ): Pro(C) → Pro(Set) preserves all limits, and we get P L: Pro(Set) → Pro(C). For c ∈ C ↩→ Pro(C), Hom Pro(C) (P L(∗), c) ≃ Hom Pro(Set) (∗, Pro(F )(c)) ≃ F (c). A functor F : C → Set is pro- Theorem 6.2. Let C be a category with finite limits. representable if and only if F preserves finite limits. Claim 1: The functor T : Pro(C) → (Set C ) op sending (C i ) to Hom Pro(C) (C i , −) is fully faithful. Claim 2: The essential image of T consists of all left exact functors. (Left exact means “preserves finite limits”.) Claim 3: For every C with finite limits, the category of left exact functors C → Set is complete and cocomplete. Proof. (a) Set C is complete and cocomplete. (b) If E is a complete cocomplete category and E ′ ⊆ E is a full subcategory closed under limits, then E ′ is cocomplete. □ Back to simplicial sets. Take a convenient category of nice spaces, such as Top. Let S be the category of simplicial sets. We have the realization functor S → Top. Definition 6.3. For X, Y ∈ S , let [X, Y ] S classes for ∼). be the set of homotopy classes (equivalence How does this compare with [|X|, |Y |] Top ? There is an obvious map [X, Y ] S → [|X|, |Y |] Top . Theorem 6.4. If B is Kan, this map is a bijection of sets. Theorem 6.5. There exists a functor Ex ∞ : S → S and a natural transformation id → Ex ∞ such that 1) Ex ∞ X is Kan 2) The map X → Ex ∞ (X) induces a homotopy equivalence on realizations. Example 6.6. If X is the circle having one nondegenerate 0-simplex and one nondegenerate 1-simplex, Ex ∞ (X) adds a 1-simplex that wraps around the circle n times, for each n, and adds appropriate 2-simplices, etc. 13
Page 1 and 2: ARITHMETIC APPLICATIONS OF ÉTALE H
Page 3 and 4: A functorial obstruction is given b
Page 5 and 6: We have 1 = π 2 (EG/G) → π 1 (X
Page 7 and 8: Proof. We have Conclude by Yoneda.
Page 9 and 10: (non-faithful) essentially surjecti
Page 11: Definition 5.5. Let X be a space an
Page 15 and 16: A relative functor F : (D, W 1 )
Page 17 and 18: Example 7.5. C = Top, W is the subc
Page 19 and 20: The diagonal map must be in W . (2)
Page 21 and 22: Definition 9.3. A model category st
Page 23 and 24: sending E = (· · · → E 1 → E
Page 25 and 26: Proof. It suffices to check that co
Page 27 and 28: Example 11.11. (Colimits do not pre
Page 29 and 30: If G is a group, then we have the c
Page 31 and 32: which we would like to be Quillen.
Page 33 and 34: Definition 13.9. Let U • , V •
Page 35 and 36: 15. August 20 (Schlank) Let k be a
Page 37 and 38: 16. August 22 (Harpaz) Let K be a f
Page 39 and 40: Also prolong the forgetful functor
Page 41: Our complex C U almost satisfies th

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