35 lines
1.6 KiB
Text
35 lines
1.6 KiB
Text
import ..homotopy.spectrum ..homotopy.EM ..algebra.arrow_group ..algebra.direct_sum ..homotopy.fwedge ..choice ..homotopy.pushout ..move_to_lib
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open eq spectrum int pointed group algebra sphere nat equiv susp is_trunc
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function fwedge cofiber lift is_equiv choice algebra pi smash
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namespace homology
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/- homology theory -/
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structure homology_theory.{u} : Type.{u+1} :=
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(HH : ℤ → pType.{u} → AbGroup.{u})
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(Hh : Π(n : ℤ) {X Y : Type*} (f : X →* Y), HH n X →g HH n Y)
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(Hid : Π(n : ℤ) {X : Type*} (x : HH n X), Hh n (pid X) x = x)
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(Hcompose : Π(n : ℤ) {X Y Z : Type*} (g : Y →* Z) (f : X →* Y) (x : HH n X),
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Hh n (g ∘* f) x = Hh n g (Hh n f x))
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(Hsusp : Π(n : ℤ) (X : Type*), HH (succ n) (psusp X) ≃g HH n X)
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(Hsusp_natural : Π(n : ℤ) {X Y : Type*} (f : X →* Y),
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Hsusp n Y ∘ Hh (succ n) (psusp_functor f) ~ Hh n f ∘ Hsusp n X)
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(Hexact : Π(n : ℤ) {X Y : Type*} (f : X →* Y), is_exact_g (Hh n f) (Hh n (pcod f)))
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(Hadditive : Π(n : ℤ) {I : Set.{u}} (X : I → Type*), is_equiv (
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dirsum_elim (λi, Hh n (pinl i)) : dirsum (λi, HH n (X i)) → HH n (⋁ X))
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)
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section
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parameter (theory : homology_theory)
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open homology_theory
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definition HH_base_indep (n : ℤ) {A : Type} (a b : A)
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: HH theory n (pType.mk A a) ≃g HH theory n (pType.mk A b) :=
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calc HH theory n (pType.mk A a) ≃g HH theory (int.succ n) (psusp A) : by exact (Hsusp theory n (pType.mk A a)) ⁻¹ᵍ
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... ≃g HH theory n (pType.mk A b) : by exact Hsusp theory n (pType.mk A b)
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end
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end homology
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