shorten proof of spi_compose_left
This commit is contained in:
Floris van Doorn
parent
df54ac858e
commit
a4c4da36df
3 changed files with 90 additions and 97 deletions
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@ -657,52 +657,13 @@ set_option pp.coercions true
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spectrum.MK (λn, Π*a, E a n)
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(λn, !loop_pppi_pequiv⁻¹ᵉ* ∘*ᵉ ppi_pequiv_right (λa, equiv_glue (E a) n))
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definition spi_compose_left_topsq
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{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
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: psquare
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(ppi_compose_left (λ a, f a n))
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(ppi_compose_left (λ a, Ω→ (f a (S n))))
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(ppi_pequiv_right (λ a, equiv_glue (E a) n))
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(ppi_pequiv_right (λ a, equiv_glue (F a) n))
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:=
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begin
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fapply psquare_of_ppi_compose_left,
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intro a, exact glue_square (f a) n,
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end
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definition spi_compose_left_botsq
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{N : succ_str} {A : Type*} {E F : A → gen_spectrum N} (f : Π a, (E a) →ₛ (F a)) (n : N)
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: psquare
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(ppi_compose_left (λ a, Ω→ (to_fun (f a) (S n))))
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(Ω→ (ppi_compose_left (λ a, to_fun (f a) (S n))))
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(!loop_pppi_pequiv⁻¹ᵉ*)
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(!loop_pppi_pequiv⁻¹ᵉ*)
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:=
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begin
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refine (_)⁻¹ᵛ*,
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fapply psquare_loop_ppi_compose_left,
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end
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definition spi_compose_left [constructor] {N : succ_str} {A : Type*} {E F : A -> gen_spectrum N}
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(f : Πa, E a →ₛ F a) : spi A E →ₛ spi A F :=
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smap.mk (λn, ppi_compose_left (λa, f a n))
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begin
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intro n,
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fapply psquare_of_phomotopy,
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refine
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(passoc _ _ (ppi_compose_left (λ a, to_fun (f a) n)))
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⬝* _ ⬝*
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(passoc (!loop_pppi_pequiv⁻¹ᵉ*)
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(ppi_compose_left (λ a, Ω→ (f a (S n))))
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(ppi_pequiv_right (λa, equiv_glue (E a) n)))⁻¹*
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⬝* _ ⬝*
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(passoc (Ω→ (ppi_compose_left (λ a, to_fun (f a) (S n)))) _ _),
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{ refine (pwhisker_left (!loop_pppi_pequiv⁻¹ᵉ*) _),
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fapply spi_compose_left_topsq},
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{ refine (pwhisker_right (ppi_pequiv_right (λ a, equiv_glue (E a) n)) _),
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fapply spi_compose_left_botsq},
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end
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begin
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intro n,
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exact psquare_ppi_compose_left (λa, (glue_square (f a) n)) ⬝v* !loop_pppi_pequiv_natural⁻¹ᵛ*
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end
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-- unpointed spi
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definition supi [constructor] {N : succ_str} (A : Type) (E : A → gen_spectrum N) :
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@ -251,4 +251,23 @@ namespace pointed
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A →* pointed.MK B (f pt) :=
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pmap.mk f idp
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/- TODO: computation rule -/
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open pi
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definition fiberwise_pointed_map_rec {A : Type} {B : A → Type*}
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(P : Π(C : A → Type*) (g : Πa, B a →* C a), Type)
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(H : Π(C : A → Type) (g : Πa, B a → C a), P _ (λa, pmap_of_map_pt (g a))) :
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Π⦃C : A → Type*⦄ (g : Πa, B a →* C a), P C g :=
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begin
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refine equiv_rect (!sigma_pi_equiv_pi_sigma ⬝e
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arrow_equiv_arrow_right A !pType.sigma_char⁻¹ᵉ) _ _,
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intro R, cases R with R r₀,
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refine equiv_rect (!sigma_pi_equiv_pi_sigma ⬝e
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pi_equiv_pi_right (λa, !pmap.sigma_char⁻¹ᵉ)) _ _,
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intro g, cases g with g g₀, esimp at (g, g₀),
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revert g₀, change (Π(g : (λa, g a (Point (B a))) ~ r₀), _),
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refine homotopy.rec_idp _ _, esimp,
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apply H
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end
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end pointed
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121
pointed_pi.hlean
121
pointed_pi.hlean
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@ -39,9 +39,6 @@ namespace pointed
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((Σa, B a) →ᵘ* C) ≃* Πᵘ*a, B a →ᵘ* C :=
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pequiv_of_equiv (equiv_sigma_rec _)⁻¹ᵉ idp
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definition loop_pupi [constructor] {A : Type} (B : A → Type*) : Ω (Πᵘ*a, B a) ≃* Πᵘ*a, Ω (B a) :=
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pequiv_of_equiv eq_equiv_homotopy idp
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definition phomotopy_mk_pupi [constructor] {A : Type*} {B : Type} {C : B → Type*}
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{f g : A →* (Πᵘ*b, C b)} (p : f ~2 g)
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(q : p pt ⬝hty apd10 (respect_pt g) ~ apd10 (respect_pt f)) : f ~* g :=
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@ -115,8 +112,40 @@ namespace pointed
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(Πᵘ*a, B a) ≃* (Πᵘ*a, B' a) :=
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pupi_pequiv erfl f
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definition loop_pumap [constructor] (A : Type) (B : Type*) : Ω (A →ᵘ* B) ≃* A →ᵘ* Ω B :=
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loop_pupi (λa, B)
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definition loop_pupi [constructor] {A : Type} (B : A → Type*) : Ω (Πᵘ*a, B a) ≃* Πᵘ*a, Ω (B a) :=
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pequiv_of_equiv eq_equiv_homotopy idp
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-- definition loop_pupi_natural [constructor] {A : Type} {B B' : A → Type*} (f : Πa, B a →* B' a) :
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-- psquare (Ω→ (pupi_functor_right f)) (pupi_functor_right (λa, Ω→ (f a)))
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-- (loop_pupi B) (loop_pupi B') :=
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definition ap1_gen_pi {A A' : Type} {B : A → Type} {B' : A' → Type}
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{h₀ h₁ : Πa, B a} {h₀' h₁' : Πa', B' a'} (f : A' → A) (g : Πa, B (f a) → B' a)
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(p₀ : pi_functor f g h₀ ~ h₀') (p₁ : pi_functor f g h₁ ~ h₁') (r : h₀ = h₁) (a' : A') :
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apd10 (ap1_gen (pi_functor f g) (eq_of_homotopy p₀) (eq_of_homotopy p₁) r) a' =
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ap1_gen (g a') (p₀ a') (p₁ a') (apd10 r (f a')) :=
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begin
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induction r, induction p₀ using homotopy.rec_idp, induction p₁ using homotopy.rec_idp, esimp,
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exact ap (λx, apd10 (ap1_gen _ x x _) _) !eq_of_homotopy_idp
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end
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-- definition ap1_gen_pi_idp {A A' : Type} {B : A → Type} {B' : A' → Type}
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-- {h₀ : Πa, B a} (f : A' → A) (g : Πa, B (f a) → B' a) (a' : A') :
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-- ap1_gen_pi f g (homotopy.refl (pi_functor f g h₀)) (homotopy.refl (pi_functor f g h₀)) idp a' =
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-- begin esimp [ap1_gen] end :=
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-- begin
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-- end
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definition loop_pupi_natural [constructor] {A : Type} {B B' : A → Type*} (f : Πa, B a →* B' a) :
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psquare (Ω→ (pupi_functor_right f)) (pupi_functor_right (λa, Ω→ (f a)))
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(loop_pupi B) (loop_pupi B') :=
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begin
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fapply phomotopy_mk_pupi,
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{ intro p a, exact ap1_gen_pi id f (λa, respect_pt (f a)) (λa, respect_pt (f a)) p a },
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{ intro a, revert B' f, refine fiberwise_pointed_map_rec _ _, intro B' f,
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exact sorry }
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end
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definition phomotopy_mk_pumap [constructor] {A C : Type*} {B : Type} {f g : A →* (B →ᵘ* C)}
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(p : f ~2 g) (q : p pt ⬝hty apd10 (respect_pt g) ~ apd10 (respect_pt f))
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@ -152,6 +181,9 @@ namespace pointed
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(A →ᵘ* B) ≃* (A' →ᵘ* B) :=
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pumap_pequiv f pequiv.rfl
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definition loop_pumap [constructor] (A : Type) (B : Type*) : Ω (A →ᵘ* B) ≃* A →ᵘ* Ω B :=
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loop_pupi (λa, B)
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/- the pointed sigma type -/
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definition psigma_gen [constructor] {A : Type*} (P : A → Type) (x : P pt) : Type* :=
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pointed.MK (Σa, P a) ⟨pt, x⟩
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@ -652,23 +684,6 @@ namespace pointed
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exact !pmap_compose_ppi2_refl⁻¹
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end
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/- TODO: computation rule -/
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definition fiberwise_pointed_map_rec {A : Type} {B : A → Type*}
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(P : Π(C : A → Type*) (g : Πa, B a →* C a), Type)
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(H : Π(C : A → Type) (g : Πa, B a → C a), P _ (λa, pmap_of_map_pt (g a))) :
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Π⦃C : A → Type*⦄ (g : Πa, B a →* C a), P C g :=
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begin
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refine equiv_rect (!sigma_pi_equiv_pi_sigma ⬝e
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arrow_equiv_arrow_right A !pType.sigma_char⁻¹ᵉ) _ _,
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intro R, cases R with R r₀,
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refine equiv_rect (!sigma_pi_equiv_pi_sigma ⬝e
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pi_equiv_pi_right (λa, !pmap.sigma_char⁻¹ᵉ)) _ _,
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intro g, cases g with g g₀, esimp at (g, g₀),
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revert g₀, change (Π(g : (λa, g a (Point (B a))) ~ r₀), _),
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refine homotopy.rec_idp _ _, esimp,
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apply H
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end
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definition ppi_assoc_ppi_const_right (g : Πa, Q a →* R a) (f : Πa, P a →* Q a) :
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ppi_assoc g f (ppi_const P) ⬝*'
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(pmap_compose_ppi_phomotopy_right _ (pmap_compose_ppi_ppi_const f) ⬝*'
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@ -700,7 +715,7 @@ namespace pointed
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apply eq_of_phomotopy, exact p a
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end
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definition psquare_of_ppi_compose_left {A : Type*} {B C D E : A → Type*}
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definition psquare_ppi_compose_left {A : Type*} {B C D E : A → Type*}
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{ftop : Π (a : A), B a →* C a} {fbot : Π (a : A), D a →* E a}
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{fleft : Π (a : A), B a →* D a} {fright : Π (a : A), C a →* E a}
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(psq : Π (a :A), psquare (ftop a) (fbot a) (fleft a) (fright a))
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@ -714,7 +729,6 @@ namespace pointed
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refine (ppi_compose_left_pcompose fright ftop)⁻¹* ⬝* _ ⬝*
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(ppi_compose_left_pcompose fbot fleft),
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exact ppi_compose_left_phomotopy psq
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-- the last step is probably an unnecessary application of function extensionality.
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end
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definition ppi_pequiv_right [constructor] (g : Π(a : A), P a ≃* Q a) :
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@ -886,33 +900,6 @@ namespace pointed
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sorry
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-- psigma_gen_pequiv_psigma_gen (pppi_pequiv_ppmap A B) (λf, begin esimp, exact ppi_gen_equiv_ppi_gen_right (λa, _) _ end) _
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definition pfiber_ppcompose_left (f : B →* C) :
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pfiber (@ppcompose_left A B C f) ≃* ppmap A (pfiber f) :=
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calc
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pfiber (@ppcompose_left A B C f) ≃*
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psigma_gen (λ(g : ppmap A B), f ∘* g = pconst A C)
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proof (eq_of_phomotopy (pcompose_pconst f)) qed :
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by exact !pfiber.sigma_char'
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... ≃* psigma_gen (λ(g : ppmap A B), f ∘* g ~* pconst A C) proof (pcompose_pconst f) qed :
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by exact psigma_gen_pequiv_psigma_gen_right (λa, !pmap_eq_equiv)
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!phomotopy_of_eq_of_phomotopy
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... ≃* psigma_gen (λ(g : ppmap A B), ppi_gen (λa, f (g a) = pt)
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(transport (λb, f b = pt) (respect_pt g)⁻¹ (respect_pt f)))
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(ppi_const _) :
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begin
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refine psigma_gen_pequiv_psigma_gen_right
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(λg, ppi_gen_equiv_ppi_gen_basepoint (_ ⬝ !eq_transport_Fl⁻¹)) _,
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intro g, refine !con_idp ⬝ _, apply whisker_right,
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exact ap02 f !inv_inv⁻¹ ⬝ !ap_inv,
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apply ppi_eq, fapply ppi_homotopy.mk,
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intro x, reflexivity,
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refine !idp_con ⬝ _, symmetry, refine !ap_id ◾ !idp_con ⬝ _, apply con.right_inv
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end
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... ≃* ppmap A (psigma_gen (λb, f b = pt) (respect_pt f)) :
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by exact (ppmap_psigma _ _)⁻¹ᵉ*
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... ≃* ppmap A (pfiber f) : by exact pequiv_ppcompose_left !pfiber.sigma_char'⁻¹ᵉ*
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definition pfiber_ppi_compose_left {B C : A → Type*} (f : Πa, B a →* C a) :
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pfiber (ppi_compose_left f) ≃* Π*(a : A), pfiber (f a) :=
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calc
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@ -939,6 +926,33 @@ namespace pointed
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by exact (ppi_psigma _ _)⁻¹ᵉ*
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... ≃* Π*(a : A), pfiber (f a) : by exact ppi_pequiv_right (λa, !pfiber.sigma_char'⁻¹ᵉ*)
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/- TODO: proof the following as a special case of pfiber_ppi_compose_left -/
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definition pfiber_ppcompose_left (f : B →* C) :
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pfiber (@ppcompose_left A B C f) ≃* ppmap A (pfiber f) :=
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calc
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pfiber (@ppcompose_left A B C f) ≃*
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psigma_gen (λ(g : ppmap A B), f ∘* g = pconst A C)
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proof (eq_of_phomotopy (pcompose_pconst f)) qed :
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by exact !pfiber.sigma_char'
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... ≃* psigma_gen (λ(g : ppmap A B), f ∘* g ~* pconst A C) proof (pcompose_pconst f) qed :
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by exact psigma_gen_pequiv_psigma_gen_right (λa, !pmap_eq_equiv)
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!phomotopy_of_eq_of_phomotopy
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... ≃* psigma_gen (λ(g : ppmap A B), ppi_gen (λa, f (g a) = pt)
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(transport (λb, f b = pt) (respect_pt g)⁻¹ (respect_pt f)))
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(ppi_const _) :
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begin
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refine psigma_gen_pequiv_psigma_gen_right
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(λg, ppi_gen_equiv_ppi_gen_basepoint (_ ⬝ !eq_transport_Fl⁻¹)) _,
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intro g, refine !con_idp ⬝ _, apply whisker_right,
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exact ap02 f !inv_inv⁻¹ ⬝ !ap_inv,
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apply ppi_eq, fapply ppi_homotopy.mk,
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intro x, reflexivity,
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refine !idp_con ⬝ _, symmetry, refine !ap_id ◾ !idp_con ⬝ _, apply con.right_inv
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end
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... ≃* ppmap A (psigma_gen (λb, f b = pt) (respect_pt f)) :
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by exact (ppmap_psigma _ _)⁻¹ᵉ*
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... ≃* ppmap A (pfiber f) : by exact pequiv_ppcompose_left !pfiber.sigma_char'⁻¹ᵉ*
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-- definition pppi_ppmap {A C : Type*} {B : A → Type*} :
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-- ppmap (/- dependent smash of B -/) C ≃* Π*(a : A), ppmap (B a) C :=
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@ -978,13 +992,12 @@ namespace pointed
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exact (pppi.sigma_char (Ω ∘ B))⁻¹ᵉ*
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end
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definition psquare_loop_ppi_compose_left {A : Type*} {X Y : A → Type*} (f : Π (a : A), X a →* Y a) :
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definition loop_pppi_pequiv_natural {A : Type*} {X Y : A → Type*} (f : Π (a : A), X a →* Y a) :
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psquare
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(Ω→ (ppi_compose_left f))
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(ppi_compose_left (λ a, Ω→ (f a)))
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(loop_pppi_pequiv X)
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(loop_pppi_pequiv Y)
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:=
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(loop_pppi_pequiv Y) :=
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begin
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refine ap1_psquare (pppi_sigma_char_natural f) ⬝v* _,
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refine !loop_psigma_gen_natural ⬝v* _,
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