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add naturality of sigma's commuting with pushouts

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Floris van Doorn 2017-08-22 22:34:13 +01:00
parent 6e2d8807f4
commit b9c2145fab
2 changed files with 38 additions and 5 deletions
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39
homotopy/pushout.hlean
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@ -277,7 +277,16 @@ namespace pushout
pushout.functor (h₂ ∘ h₁) (i₂ ∘ i₁) (j₂ ∘ j₁) (p₁ ⬝htyv p₂) (q₁ ⬝htyv q₂) ~
pushout.functor h₂ i₂ j₂ p₂ q₂ ∘ pushout.functor h₁ i₁ j₁ p₁ q₁ :=
begin
exact sorry
intro x, induction x with b c a,
{ reflexivity },
{ reflexivity },
{ apply eq_pathover, apply hdeg_square, esimp,
refine !elim_glue ⬝ whisker_right _ (!ap_con ⬝ !ap_compose'⁻¹ ◾ idp) ◾
(ap02 _ !con_inv ⬝ !ap_con ⬝ whisker_left _ (ap02 _ !ap_inv⁻¹ ⬝ !ap_compose'⁻¹)) ⬝ _ ⬝
(ap_compose (pushout.functor h₂ i₂ j₂ p₂ q₂) _ _ ⬝ ap02 _ !elim_glue)⁻¹,
refine _ ⬝ (!ap_con ⬝ (!ap_con ⬝ !ap_compose'⁻¹ ◾ !elim_glue) ◾ !ap_compose'⁻¹)⁻¹ᵖ,
refine !con.assoc⁻¹ ⬝ whisker_right _ _,
exact whisker_right _ !con.assoc ⬝ !con.assoc }
end
variables {p₁ q₁}
@ -593,8 +602,32 @@ equiv.MK sigma_pushout_of_pushout_sigma pushout_sigma_of_sigma_pushout
induction a with a a',
apply hdeg_square, refine !ap_compose'⁻¹ ⬝ !elim_glue ⬝ !con_idp ⬝ !idp_con }
end end
-- apply eq_pathover_id_right, refine ap_compose sigma_pushout_of_pushout_sigma _ _ ⬝ ap02 _ !elim_glue ⬝ph _
-- apply eq_pathover_id_right, refine ap_compose pushout_sigma_of_sigma_pushout _ _ ⬝ ap02 _ !elim_glue ⬝ph _
variables {f g}
variables {X' : Type} {A' B' C' : X' → Type} {f' : Πx, A' x → B' x} {g' : Πx, A' x → C' x}
{s : X → X'} {h₁ : Πx, A x → A' (s x)} {h₂ : Πx, B x → B' (s x)} {h₃ : Πx, C x → C' (s x)}
(p : Πx, h₂ x ∘ f x ~ f' (s x) ∘ h₁ x) (q : Πx, h₃ x ∘ g x ~ g' (s x) ∘ h₁ x)
definition pushout_sigma_equiv_sigma_pushout_natural :
hsquare
(pushout.functor (sigma_functor s h₁) (sigma_functor s h₂) (sigma_functor s h₃)
(λa, sigma_eq_right (p a.1 a.2)) (λa, sigma_eq_right (q a.1 a.2)))
(sigma_functor s (λx, pushout.functor (h₁ x) (h₂ x) (h₃ x) (p x) (q x)))
(pushout_sigma_equiv_sigma_pushout f g) (pushout_sigma_equiv_sigma_pushout f' g') :=
begin
intro x, induction x with b c a,
{ reflexivity },
{ reflexivity },
{ exact abstract begin apply eq_pathover, apply hdeg_square,
refine !ap_compose ⬝ ap02 _ !elim_glue ⬝ !ap_con ⬝
(!ap_con ⬝ (!ap_compose'⁻¹ ⬝ !ap_compose'⁻¹) ◾ !elim_glue) ◾
(!ap_compose'⁻¹ ⬝ ap02 _ !ap_inv⁻¹ ⬝ !ap_compose'⁻¹) ⬝ _,
exact
(ap_compose (sigma_functor s (λ x, pushout.functor (h₁ x) (h₂ x) (h₃ x) (p x) (q x))) _ _ ⬝
ap02 _ !elim_glue ⬝ !ap_compose'⁻¹ ⬝ ap_compose (dpair _) _ _ ⬝ ap02 _ !elim_glue ⬝
!ap_con ⬝ (!ap_con ⬝ !ap_compose'⁻¹ ◾ idp) ◾ !ap_compose'⁻¹)⁻¹ end end }
end
/- an induction principle for the cofiber of f : A → B if A is a pushout where the second map has a section.

4
move_to_lib.hlean
View file

@ -137,10 +137,10 @@ namespace eq
λx, !idp_con ⬝ ap_id (q x)
definition rfl_hvconcat (q : hsquare f₃₀ f₃₂ f₂₁ f₄₁) : homotopy.rfl ⬝htyv q ~ q :=
λx, inv_inv (q x)
λx, !idp_con
definition hvconcat_rfl (q : hsquare f₃₀ f₃₂ f₂₁ f₄₁) : q ⬝htyv homotopy.rfl ~ q :=
λx, (!idp_con ⬝ !ap_id)⁻² ⬝ inv_inv (q x)
λx, !ap_id
end hsquare
definition homotopy_group_succ_in_natural (n : ℕ) {A B : Type*} (f : A →* B) :