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HTTのススメ
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rorijo
June 16, 2019
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HTTのススメ
Higher Topos Theoryの概略モドキ
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Transcript
HTT ͷεεϝ @R O R I J O June 14, 2019
ࣗݾհ 1. HTT ΛಡΉ. 2. HA ʹखΛग़࢝͠Ί͍ͯΔ. 3. εΩʔϜʹڵຯ͕͋Δ. 4. ҙͷू߹ʹ͍ͭͯͦΕΛؚΉ Grothendieck Ӊ͕ଘࡏ͢Δ ͜ͱΛԾఆ͍ͯ͠Δ.
࣍ simplicial set quasi category ∞-topos
simplicial set quasi category ∞-topos
simplicial set Definition. ۭͰͳ͍༗ݶશॱংू߹ͷݍͷࠎ֨Λ △ ͱ͓͘. n + 1 ݩ͔ΒͳΔ ༗ݶશॱংू߹ʹରԠ͢ΔରΛ [n] ͱॻ͘. ͜ͷͱ͖, Set△op Λ sSet ͱॻ͖ɺͦͷରΛ simplicial set ͱݺͿ. simplicial set X ʹ͍ͭͯ, X([n]) ͷ͜ͱΛ Xn ͱॻ͘. y([n]) Λ n-simplex ͱݺͼ, ∆n ͱॻ͘.
ಛผͳ simplicial set Definition. simplicial set Λn i Λ࣍ͷਤࣜͷ༨ۃݶͱͯ͠ఆٛ͢Δ. ⨿ 0≤k<k′≤n k,k′̸=i ∆n−2 ⇒ ⨿ 0≤k≤n k̸=i ∆n−1 ·ͨ, ∂∆n Λ࣍ͷਤࣜͷ༨ۃݶͱͯ͠ఆٛ͢Δ. ⨿ 0≤k<k′≤n ∆n−2 ⇒ ⨿ 0≤k≤n ∆n−1
simplicial set quasi category ∞-topos
quasi category Definition. simplicial set X ͕ quasi category Ͱ͋Δͱ, ҙͷ n, 0 < i < n ͱҙͷ࣍ͷܗͷਤࣜʹ͍ͭͯઢͷࣹ͕ଘࡏͯ͠ਤࣜΛՄʹ͢ Δͱ͖Λݴ͏. i = 0, n ͷͱ͖͜Ε͕Γཱͭ࣌, Kan complex ͱݺͿ. Λn i ∆n X Remark. Kan complex ʹ͓͍ͯϗϞτϐʔ͕ల։Ͱ͖Δ. ͜ͷϗϞτ ϐʔҐ૬ۭؒͷϗϞτϐʔͱϞσϧݍతʹಉͰ͋Δ͜ ͱ͕ΒΕ͍ͯΔ.
ྫ Example. 1. Kan complex શମͷͳ͢ quasi category S 2. quasi category શମͷͳ͢ quasi category 3. Kan complex 4. ถాຒΊࠐΈ △ → Set△op ͷ inclusion △ → Cat ʹΑΔ Kan ֦ுʹΑͬͯ, ݍ quasi category ͱࢥ͑Δ. 5. simplicial set K ͔Β quasi category C ͷࣹͷͳ͢ quasi category Hom(K, C)
slice Definition. simplicial set X, Y ʹ͍ͭͯ, ͦͷ join X ⋆ Y Λ (X ⋆ Y )n = Xn ∪ Yn ∪ ∪ 0≤k≤n−1 Xk × Yn−k−1 ͱͳΔΑ͏ʹఆٛ͢Δ. Definition. simplicial set K, X, ࣹ f : K → X ʹ͍ͭͯ, Xf / Λ, ࣍ͷಉܕ͕ࣗ વʹΓཱͭΑ͏ʹఆٛ͢Δ. Hom(S, Xf / ) ∼ = Hom(K ⋆ S, X) ×Hom(K,X) {f } X/f ରతʹఆٛ͢Δ.
limit Definition. quasi category X ʹ͍ͭͯ, ͦͷର x ͕ X ͷ initial object Ͱ͋Δ ͱ, ͲΜͳ࣍ͷܗͷਤࣜʹ͍ͭͯઢͷࣹ͕ଘࡏͯ͠ਤࣜΛՄ ʹ͢Δͱ͖Λݴ͏. Xx/ X ∂∆n ∆n Definition. simplicial set K, quasi category X, ࣹ f : K → X ʹ͍ͭͯ, ͋Δର x ͕ f ͷ colimit Ͱ͋Δͱ, x ͕ Xf / ͷ initial object Ͱ͋Δͱ ͖Λݴ͏. limit ରతʹఆٛ͞ΕΔ.
Hom Definition. quasi category C ͱͦͷର x, y ʹ͍ͭͯ, HomR(x, y) Λ C/y ×C {x} ͱఆٛ͢Δ. Proposition. HomR(x, y) Kan complex Ͱ͋Δ. Definition. quasi category C ͱͦͷର x ʹ͍ͭͯ, x ʹΑͬͯ corepresent ͞ Εͨؔख C → S Λ, C ͷର y ʹରͯ͠ HomR(x, y) ΛׂΓͯ ΔΑ͏ʹఆٛ͢Δ.
filtered Definition. ਖ਼ଇج κ ʹ͍ͭͯ, simplicial set K ͕ κ-small Ͱ͋Δͱ, શͯ ͷ n ʹ͍ͭͯ Kn ͷೱ͕ κ ະຬͰ͋Δͱ͖Λݴ͏. Definition. ਖ਼ଇج κ ʹ͍ͭͯ, quasi category C ͕ κ-filtered Ͱ͋Δͱ, ҙͷ κ-small ͳ K ͱ K → C ʹ͍ͭͯ, ͦΕ͕ K ⋆ ∆0 → C ʹ֦ு Ͱ͖Δͱ͖Λݴ͏. Definition. quasi category C ͷର x ͕ κ-compact ͱ, x ʹΑͬͯ corepresent ͞ΕΔؔख͕ κ-filtered colimit Λอͭͱ͖Λ͍͏.
presentable category Definition. quasi category C ͕ presentable category Ͱ͋Δͱ, ࣍ͷ݅Λຬ ͨ͢ͱ͖Λݴ͏. 1. ༨උͰ͋Δ. 2. Hom ͕খ͞ͳ Kan complex ͱ homotopy equivalent Ͱ͋Δ. 3. ਖ਼ଇج κ ͱ κ-compact ͳ C ͷର͔ΒͳΔখ͞ͳू߹ S ͕ ͋Γ, C ͷͯ͢ͷର S ͷݩͷ colimit ͱͯ͠ॻ͚Δ. Theorem. presentable category උͰ͋Δ.
simplicial set quasi category ∞-topos
groupoid Definition. quasi category C ʹର͠, ࣹ N(△op) → C ͷ͜ͱΛ C ͷ simplicial object ͱݺͿ. Definition. C Λ quasi category ͱ͢Δ. simplicial object U : N(△op) → C ͕ groupoid Ͱ͋Δͱ, ҙͷ S ∪ S′ = [n], S ∩ S′ = s ͳΔ S, S′ ʹ ͍ͭͯ, ࣍ͷਤ͕ࣜ pullback ʹͳ͍ͬͯΔͱ͖Λݴ͏. U([n]) U(S) U(S′) U({s})
effective groupoid Definition. ݍ △+ Λ, ༗ݶશॱংू߹ͷݍͷࠎ֨ͱ͢Δ. ۭͳશॱংू߹ʹର Ԡ͢ΔରΛ [−1] ͱॻ͘. groupoid U : N(△op) → C ͕ effective Ͱ͋Δͱ, ͦΕ͕ U+ : N((△+)op) → C Ͱ͋ͬͯ U ͷ colimit Ͱ͋Γ, ͔ͭ࣍ͷਤࣜ ͕ pullback Ͱ͋ΔΑ͏ͳͷʹ֦ுͰ͖Δͱ͖Λݴ͏. U+([1]) U+([0]) U+([0]) U+([−1])
∞-topos Definition. presentable category C ͕ ∞-topos Ͱ͋Δͱ, ࣍ͷ݅Λຬͨ͢ ͱ͖Λݴ͏. 1. X ×Z colim Yλ ∼ = colim(X ×Z Yλ) 2. X ×X ⨿ Y Y = 0 3. groupoid ͕શͯ effective Example. SCop ∞-topos Ͱ͋Δ. ಛʹ S ∞-topos Ͱ͋Δ.
presentable category abelian group ͷྨࣅ Proposition. presentable category ͷͳ͢ quasi category ʹ͓͍ͯ, ͱੵ Ұக͢Δ. Definition. presentable category C ͱͦͷର X, Kan complex K ʹ͍ͭͯ, K → C Λ X ͷఆؔखͱ͢Δͱ͖, ͦͷ colimit Λ X ⊗ K ͱॻ͘.
S ମͷྨࣅ Proposition. ඇࣗ໌ͳ ∞-topos C ͷ༨ۃݶͱ༗ݶۃݶΛอͭؔख F : S → C ʹ͍ͭͯ, F(x) ∼ = 0 ͱͳΔΑ͏ͳ x શମͷͳ͢ॆຬ෦ݍ I Λߟ͑ ͨ࣌, I = {0} Ͱ͋Δ. Definition. ∞-topos ͷΛ, S ͷ༨ۃݶͱ༗ݶۃݶΛอͭؔखͱͯ͠ఆٛ ͢Δ.
presentable category ϕΫτϧۭؒͷྨࣅ Definition. presentable category C, D ʹ͍ͭͯ, C ⊗ D Λ Cop ͔Β D ͷۃ ݶΛอͭؔखશମͷͳ͢ quasi category ͱ͢Δ. Proposition. C, D Λ presentable category ͱ͢Δͱ͖, C ⊗ D presentable Ͱ ͋Δ. ·ͨ, E presentable category Ͱ͋Δͱ͖, FunL(C, FunL(D, E)) ∼ = FunL(C ⊗ D, E) Ͱ͋Δ. ͨͩ͠, FunL(X, Y ) X ͔Β Y ͷ༨ۃݶΛอͭؔखશମͷͳ͢ quasi category ͷ͜ͱͱ͢Δ. Proposition. SCop ⊗ SDop ∼ = S(C×D)op
·ͱΊ 1. presentable category abelian group ϕΫτϧۭؒͷΑ͏ʹ ৼΔ͏ʂ 2. ∞-topos ͷΑ͏ʹৼΔ͏ʂ
ࢀߟจݙ Lurie, Jacob, Higher Topos Theory Lurie, Jacob, Derived Algebraic Geometry II: Noncommutative Algebra