149 lines
7 KiB
Text
149 lines
7 KiB
Text
/-
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Copyright (c) 2017 Yuri Sulyma, Favonia
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Yuri Sulyma, Favonia
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Reduced homology theories
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-/
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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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(Hpid : Π(n : ℤ) {X : Type*} (x : HH n X), Hh n (pid X) x = x)
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(Hpcompose : Π(n : ℤ) {X Y Z : Type*} (f : Y →* Z) (g : X →* Y),
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Hh n (f ∘* g) ~ Hh n f ∘ Hh n g)
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(Hpsusp : Π(n : ℤ) (X : Type*), HH (succ n) (psusp X) ≃g HH n X)
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(Hpsusp_natural : Π(n : ℤ) {X Y : Type*} (f : X →* Y),
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Hpsusp n Y ∘ Hh (succ n) (psusp_functor f) ~ Hh n f ∘ Hpsusp 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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structure ordinary_homology_theory.{u} extends homology_theory.{u} : Type.{u+1} :=
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(Hdimension : Π(n : ℤ), n ≠ 0 → is_contr (HH n (plift (psphere 0))))
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section
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parameter (theory : homology_theory)
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open homology_theory
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theorem 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 (Hpsusp theory n (pType.mk A a)) ⁻¹ᵍ
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... ≃g HH theory n (pType.mk A b) : by exact Hpsusp theory n (pType.mk A b)
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theorem Hh_homotopy' (n : ℤ) {A B : Type*} (f : A → B) (p q : f pt = pt)
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: Hh theory n (pmap.mk f p) ~ Hh theory n (pmap.mk f q) := λ x,
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calc Hh theory n (pmap.mk f p) x
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= Hh theory n (pmap.mk f p) (Hpsusp theory n A ((Hpsusp theory n A)⁻¹ᵍ x))
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: by exact ap (Hh theory n (pmap.mk f p)) (equiv.to_right_inv (equiv_of_isomorphism (Hpsusp theory n A)) x)⁻¹
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... = Hpsusp theory n B (Hh theory (succ n) (pmap.mk (susp.functor f) !refl) ((Hpsusp theory n A)⁻¹ x))
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: by exact (Hpsusp_natural theory n (pmap.mk f p) ((Hpsusp theory n A)⁻¹ᵍ x))⁻¹
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... = Hh theory n (pmap.mk f q) (Hpsusp theory n A ((Hpsusp theory n A)⁻¹ x))
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: by exact Hpsusp_natural theory n (pmap.mk f q) ((Hpsusp theory n A)⁻¹ x)
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... = Hh theory n (pmap.mk f q) x
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: by exact ap (Hh theory n (pmap.mk f q)) (equiv.to_right_inv (equiv_of_isomorphism (Hpsusp theory n A)) x)
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theorem Hh_homotopy (n : ℤ) {A B : Type*} (f g : A →* B) (h : f ~ g) : Hh theory n f ~ Hh theory n g := λ x,
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calc Hh theory n f x
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= Hh theory n (pmap.mk f (respect_pt f)) x : by exact ap (λ f, Hh theory n f x) (pmap.eta f)⁻¹
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... = Hh theory n (pmap.mk f (h pt ⬝ respect_pt g)) x : by exact Hh_homotopy' n f (respect_pt f) (h pt ⬝ respect_pt g) x
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... = Hh theory n g x : by exact ap (λ f, Hh theory n f x) (@pmap_eq _ _ (pmap.mk f (h pt ⬝ respect_pt g)) _ h (refl (h pt ⬝ respect_pt g)))
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definition HH_isomorphism (n : ℤ) {A B : Type*} (e : A ≃* B)
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: HH theory n A ≃g HH theory n B :=
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begin
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fapply isomorphism.mk,
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{ exact Hh theory n e },
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fapply adjointify,
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{ exact Hh theory n e⁻¹ᵉ* },
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{ intro x, exact calc
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Hh theory n e (Hh theory n e⁻¹ᵉ* x)
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= Hh theory n (e ∘* e⁻¹ᵉ*) x : by exact (Hpcompose theory n e e⁻¹ᵉ* x)⁻¹
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... = Hh theory n !pid x : by exact Hh_homotopy n (e ∘* e⁻¹ᵉ*) !pid (to_right_inv e) x
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... = x : by exact Hpid theory n x
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},
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{ intro x, exact calc
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Hh theory n e⁻¹ᵉ* (Hh theory n e x)
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= Hh theory n (e⁻¹ᵉ* ∘* e) x : by exact (Hpcompose theory n e⁻¹ᵉ* e x)⁻¹
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... = Hh theory n !pid x : by exact Hh_homotopy n (e⁻¹ᵉ* ∘* e) !pid (to_left_inv e) x
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... = x : by exact Hpid theory n x
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}
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end
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definition Hfwehce (n : ℤ) {I : Set} (X : I → Type*): HH theory n (⋁ X) ≃g dirsum (λi, HH theory n (X i)) :=
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(isomorphism.mk _ (Hadditive theory n X))⁻¹ᵍ
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definition Hpwedge (n : ℤ) (A B : Type*) : HH theory n (pwedge A B) ≃g HH theory n A ×g HH theory n B :=
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calc HH theory n (pwedge A B) ≃g HH theory n (fwedge (bool.rec A B)) : by exact sorry
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... ≃g dirsum (λb, HH theory n (bool.rec A B b)) : by exact sorry
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... ≃g HH theory n A ×g HH theory n B : by exact sorry
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end
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/- homology theory associated to a prespectrum -/
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definition homology (X : Type*) (E : prespectrum) (n : ℤ) : AbGroup :=
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pshomotopy_group n (smash_prespectrum X E)
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/- an alternative definition, which might be a bit harder to work with -/
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definition homology_spectrum (X : Type*) (E : spectrum) (n : ℤ) : AbGroup :=
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shomotopy_group n (smash_spectrum X E)
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definition parametrized_homology {X : Type*} (E : X → spectrum) (n : ℤ) : AbGroup :=
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sorry
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definition ordinary_homology [reducible] (X : Type*) (G : AbGroup) (n : ℤ) : AbGroup :=
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homology X (EM_spectrum G) n
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definition ordinary_homology_Z [reducible] (X : Type*) (n : ℤ) : AbGroup :=
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ordinary_homology X agℤ n
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notation `H_` n `[`:0 X:0 `, ` E:0 `]`:0 := homology X E n
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notation `H_` n `[`:0 X:0 `]`:0 := ordinary_homology_Z X n
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notation `pH_` n `[`:0 binders `, ` r:(scoped E, parametrized_homology E n) `]`:0 := r
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definition unpointed_homology (X : Type) (E : spectrum) (n : ℤ) : AbGroup :=
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H_ n[X₊, E]
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definition homology_functor [constructor] {X Y : Type*} {E F : spectrum} (f : X →* Y) (g : E →ₛ F) (n : ℤ)
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: homology X E n →g homology Y F n :=
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pshomotopy_group_fun n (smash_prespectrum_fun f g)
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definition homology_theory_spectrum.{u} [constructor] (E : spectrum.{u}) : homology_theory.{u} :=
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begin
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fapply homology_theory.mk,
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exact (λn X, H_ n[X, E]),
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exact (λn X Y f, homology_functor f (sid E) n),
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exact sorry, -- Hid is uninteresting but potentially very hard to prove
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exact sorry,
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exact sorry,
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exact sorry,
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exact sorry,
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exact sorry
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-- sorry
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-- sorry
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-- sorry
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-- sorry
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-- sorry
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-- sorry
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-- (λn A, H^n[A, Y])
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-- (λn A B f, cohomology_isomorphism f Y n)
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-- (λn A, cohomology_isomorphism_refl A Y n)
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-- (λn A B f, cohomology_functor f Y n)
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-- (λn A B f g p, cohomology_functor_phomotopy p Y n)
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-- (λn A B f x, cohomology_functor_phomotopy_refl f Y n x)
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-- (λn A x, cohomology_functor_pid A Y n x)
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-- (λn A B C g f x, cohomology_functor_pcompose g f Y n x)
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-- (λn A, cohomology_psusp A Y n)
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-- (λn A B f, cohomology_psusp_natural f Y n)
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-- (λn A B f, cohomology_exact f Y n)
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-- (λn I A H, spectrum_additive H A Y n)
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end
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end homology
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