814 lines
34 KiB
Text
814 lines
34 KiB
Text
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import ..algebra.exactness homotopy.cofiber homotopy.wedge
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open eq function is_trunc sigma prod lift is_equiv equiv pointed sum unit bool cofiber
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namespace pushout
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section
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variables {TL BL TR : Type*} {f : TL →* BL} {g : TL →* TR}
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{TL' BL' TR' : Type*} {f' : TL' →* BL'} {g' : TL' →* TR'}
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(tl : TL ≃ TL') (bl : BL ≃* BL') (tr : TR ≃ TR')
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(fh : bl ∘ f ~ f' ∘ tl) (gh : tr ∘ g ~ g' ∘ tl)
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definition ppushout_functor [constructor] (tl : TL → TL') (bl : BL →* BL') (tr : TR → TR')
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(fh : bl ∘ f ~ f' ∘ tl) (gh : tr ∘ g ~ g' ∘ tl) : ppushout f g →* ppushout f' g' :=
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begin
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fconstructor,
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{ exact pushout.functor tl bl tr fh gh },
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{ exact ap inl (respect_pt bl) },
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end
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definition ppushout_pequiv (tl : TL ≃ TL') (bl : BL ≃* BL') (tr : TR ≃ TR')
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(fh : bl ∘ f ~ f' ∘ tl) (gh : tr ∘ g ~ g' ∘ tl) : ppushout f g ≃* ppushout f' g' :=
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pequiv_of_equiv (pushout.equiv _ _ _ _ tl bl tr fh gh) (ap inl (respect_pt bl))
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end
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/-
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WIP: proving that satisfying the universal property of the pushout is equivalent to
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being equivalent to the pushout
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-/
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universe variables u₁ u₂ u₃ u₄
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variables {A : Type.{u₁}} {B : Type.{u₂}} {C : Type.{u₃}} {D D' : Type.{u₄}}
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{f : A → B} {g : A → C} {h : B → D} {k : C → D} (p : h ∘ f ~ k ∘ g)
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{h' : B → D'} {k' : C → D'} (p' : h' ∘ f ~ k' ∘ g)
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-- (f : A → B) (g : A → C) (h : B → D) (k : C → D)
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include p
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definition is_pushout : Type :=
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Π⦃X : Type.{max u₁ u₂ u₃ u₄}⦄ (h' : B → X) (k' : C → X) (p' : h' ∘ f ~ k' ∘ g),
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is_contr (Σ(l : D → X) (v : l ∘ h ~ h' × l ∘ k ~ k'),
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Πa, square (prod.pr1 v (f a)) (prod.pr2 v (g a)) (ap l (p a)) (p' a))
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definition cocone [reducible] (X : Type) : Type :=
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Σ(v : (B → X) × (C → X)), prod.pr1 v ∘ f ~ prod.pr2 v ∘ g
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definition cocone_of_map [constructor] (X : Type) (l : D → X) : cocone p X :=
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⟨(l ∘ h, l ∘ k), λa, ap l (p a)⟩
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-- definition cocone_of_map (X : Type) (l : D → X) : Σ(h' : B → X) (k' : C → X),
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-- h' ∘ f ~ k' ∘ g :=
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-- ⟨l ∘ h, l ∘ k, λa, ap l (p a)⟩
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omit p
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definition is_pushout2 [reducible] : Type :=
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Π(X : Type.{max u₁ u₂ u₃ u₄}), is_equiv (cocone_of_map p X)
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section
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open sigma.ops
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protected definition inv_left (H : is_pushout2 p) {X : Type} (v : cocone p X) :
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(cocone_of_map p X)⁻¹ᶠ v ∘ h ~ prod.pr1 v.1 :=
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ap10 (ap prod.pr1 (right_inv (cocone_of_map p X) v)..1)
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protected definition inv_right (H : is_pushout2 p) {X : Type} (v : cocone p X) :
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(cocone_of_map p X)⁻¹ᶠ v ∘ k ~ prod.pr2 v.1 :=
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ap10 (ap prod.pr2 (right_inv (cocone_of_map p X) v)..1)
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end
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section
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local attribute is_pushout [reducible]
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definition is_prop_is_pushout : is_prop (is_pushout p) :=
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_
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local attribute is_pushout2 [reducible]
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definition is_prop_is_pushout2 : is_prop (is_pushout2 p) :=
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_
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end
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definition ap_eq_apd10_ap {A B : Type} {C : B → Type} (f : A → Πb, C b) {a a' : A} (p : a = a') (b : B)
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: ap (λa, f a b) p = apd10 (ap f p) b :=
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by induction p; reflexivity
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variables (f g)
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definition is_pushout2_pushout : @is_pushout2 _ _ _ _ f g inl inr glue :=
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λX, to_is_equiv (pushout_arrow_equiv f g X ⬝e assoc_equiv_prod _)
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definition is_equiv_of_is_pushout2_simple [constructor] {A B C D : Type.{u₁}}
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{f : A → B} {g : A → C} {h : B → D} {k : C → D} (p : h ∘ f ~ k ∘ g)
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{h' : B → D'} {k' : C → D'} (p' : h' ∘ f ~ k' ∘ g)
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(H : is_pushout2 p) : D ≃ pushout f g :=
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begin
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fapply equiv.MK,
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{ exact (cocone_of_map p _)⁻¹ᶠ ⟨(inl, inr), glue⟩ },
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{ exact pushout.elim h k p },
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{ intro x, exact sorry
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},
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{ apply ap10,
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apply eq_of_fn_eq_fn (equiv.mk _ (H D)),
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fapply sigma_eq,
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{ esimp, fapply prod_eq,
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apply eq_of_homotopy, intro b,
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exact ap (pushout.elim h k p) (pushout.inv_left p H ⟨(inl, inr), glue⟩ b),
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apply eq_of_homotopy, intro c,
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exact ap (pushout.elim h k p) (pushout.inv_right p H ⟨(inl, inr), glue⟩ c) },
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{ apply pi.pi_pathover_constant, intro a,
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apply eq_pathover,
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refine !ap_eq_apd10_ap ⬝ph _ ⬝hp !ap_eq_apd10_ap⁻¹,
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refine ap (λx, apd10 x _) (ap_compose (λx, x ∘ f) pr1 _ ⬝ ap02 _ !prod_eq_pr1) ⬝ph _
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⬝hp ap (λx, apd10 x _) (ap_compose (λx, x ∘ g) pr2 _ ⬝ ap02 _ !prod_eq_pr2)⁻¹,
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refine apd10 !apd10_ap_precompose_dependent a ⬝ph _ ⬝hp apd10 !apd10_ap_precompose_dependent⁻¹ a,
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refine apd10 !apd10_eq_of_homotopy (f a) ⬝ph _ ⬝hp apd10 !apd10_eq_of_homotopy⁻¹ (g a),
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refine ap_compose (pushout.elim h k p) _ _ ⬝pv _,
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refine aps (pushout.elim h k p) _ ⬝vp (!elim_glue ⬝ !ap_id⁻¹),
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esimp, exact sorry
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},
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}
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end
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-- definition is_equiv_of_is_pushout2 [constructor] (H : is_pushout2 p) : D ≃ pushout f g :=
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-- begin
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-- fapply equiv.MK,
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-- { exact down.{_ u₄} ∘ (cocone_of_map p _)⁻¹ᶠ ⟨(up ∘ inl, up ∘ inr), λa, ap up (glue a)⟩ },
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-- { exact pushout.elim h k p },
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-- { intro x, exact sorry
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-- },
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-- { intro d, apply eq_of_fn_eq_fn (equiv_lift D), esimp, revert d,
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-- apply ap10,
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-- apply eq_of_fn_eq_fn (equiv.mk _ (H (lift.{_ (max u₁ u₂ u₃)} D))),
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-- fapply sigma_eq,
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-- { esimp, fapply prod_eq,
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-- apply eq_of_homotopy, intro b, apply ap up, esimp,
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-- exact ap (pushout.elim h k p ∘ down.{_ u₄})
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-- (pushout.inv_left p H ⟨(up ∘ inl, up ∘ inr), λa, ap up (glue a)⟩ b),
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-- exact sorry },
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-- { exact sorry },
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-- -- note q := @eq_of_is_contr _ H''
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-- -- ⟨up ∘ pushout.elim h k p ∘ down ∘ (center' H').1,
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-- -- (λb, ap (up ∘ pushout.elim h k p ∘ down) (prod.pr1 (center' H').2 b),
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-- -- λc, ap (up ∘ pushout.elim h k p ∘ down) (prod.pr2 (center' H').2 c))⟩
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-- -- ⟨up, (λx, idp, λx, idp)⟩,
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-- -- exact ap down (ap10 q..1 d)
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-- }
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-- end
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/- composing pushouts -/
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definition pushout_vcompose_to [unfold 8] {A B C D : Type} {f : A → B} {g : A → C} {h : B → D}
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(x : pushout h (@inl _ _ _ f g)) : pushout (h ∘ f) g :=
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begin
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induction x with d y b,
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{ exact inl d },
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{ induction y with b c a,
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{ exact inl (h b) },
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{ exact inr c },
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{ exact glue a }},
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{ reflexivity }
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end
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definition pushout_vcompose_from [unfold 8] {A B C D : Type} {f : A → B} {g : A → C} {h : B → D}
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(x : pushout (h ∘ f) g) : pushout h (@inl _ _ _ f g) :=
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begin
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induction x with d c a,
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{ exact inl d },
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{ exact inr (inr c) },
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{ exact glue (f a) ⬝ ap inr (glue a) }
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end
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definition pushout_vcompose [constructor] {A B C D : Type} (f : A → B) (g : A → C) (h : B → D) :
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pushout h (@inl _ _ _ f g) ≃ pushout (h ∘ f) g :=
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begin
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fapply equiv.MK,
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{ exact pushout_vcompose_to },
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{ exact pushout_vcompose_from },
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{ intro x, induction x with d c a,
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{ reflexivity },
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{ reflexivity },
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{ apply eq_pathover_id_right, apply hdeg_square,
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refine ap_compose pushout_vcompose_to _ _ ⬝ ap02 _ !elim_glue ⬝ _,
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refine !ap_con ⬝ !elim_glue ◾ !ap_compose'⁻¹ ⬝ !idp_con ⬝ _, esimp, apply elim_glue }},
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{ intro x, induction x with d y b,
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{ reflexivity },
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{ induction y with b c a,
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{ exact glue b },
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{ reflexivity },
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{ apply eq_pathover, refine ap_compose pushout_vcompose_from _ _ ⬝ph _,
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esimp, refine ap02 _ !elim_glue ⬝ !elim_glue ⬝ph _, apply square_of_eq, reflexivity }},
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{ apply eq_pathover_id_right, esimp,
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refine ap_compose pushout_vcompose_from _ _ ⬝ ap02 _ !elim_glue ⬝ph _, apply square_of_eq,
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reflexivity }}
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end
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definition pushout_hcompose {A B C D : Type} (f : A → B) (g : A → C) (h : C → D) :
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pushout (@inr _ _ _ f g) h ≃ pushout f (h ∘ g) :=
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calc
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pushout (@inr _ _ _ f g) h ≃ pushout h (@inr _ _ _ f g) : pushout.symm
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... ≃ pushout h (@inl _ _ _ g f) :
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pushout.equiv _ _ _ _ erfl erfl (pushout.symm f g) (λa, idp) (λa, idp)
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... ≃ pushout (h ∘ g) f : pushout_vcompose
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... ≃ pushout f (h ∘ g) : pushout.symm
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definition pushout_vcompose_equiv {A B C D E : Type} (f : A → B) {g : A → C} {h : B → D}
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{hf : A → D} {k : B → E} (e : E ≃ pushout f g) (p : k ~ e⁻¹ᵉ ∘ inl) (q : h ∘ f ~ hf) :
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pushout h k ≃ pushout hf g :=
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begin
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refine _ ⬝e pushout_vcompose f g h ⬝e _,
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{ fapply pushout.equiv,
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reflexivity,
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reflexivity,
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exact e,
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reflexivity,
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exact homotopy_of_homotopy_inv_post e _ _ p },
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{ fapply pushout.equiv,
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reflexivity,
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reflexivity,
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reflexivity,
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exact q,
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reflexivity },
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end
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definition pushout_hcompose_equiv {A B C D E : Type} {f : A → B} (g : A → C) {h : C → E}
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{hg : A → E} {k : C → D} (e : D ≃ pushout f g) (p : k ~ e⁻¹ᵉ ∘ inr) (q : h ∘ g ~ hg) :
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pushout k h ≃ pushout f hg :=
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calc
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pushout k h ≃ pushout h k : pushout.symm
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... ≃ pushout hg f : by exact pushout_vcompose_equiv _ (e ⬝e pushout.symm f g) p q
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... ≃ pushout f hg : pushout.symm
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definition pushout_of_equiv_left_to [unfold 6] {A B C : Type} {f : A ≃ B} {g : A → C}
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(x : pushout f g) : C :=
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begin
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induction x with b c a,
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{ exact g (f⁻¹ b) },
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{ exact c },
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{ exact ap g (left_inv f a) }
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end
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definition pushout_of_equiv_left [constructor] {A B C : Type} (f : A ≃ B) (g : A → C) :
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pushout f g ≃ C :=
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begin
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fapply equiv.MK,
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{ exact pushout_of_equiv_left_to },
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{ exact inr },
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{ intro c, reflexivity },
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{ intro x, induction x with b c a,
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{ exact (glue (f⁻¹ b))⁻¹ ⬝ ap inl (right_inv f b) },
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{ reflexivity },
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{ apply eq_pathover_id_right, refine ap_compose inr _ _ ⬝ ap02 _ !elim_glue ⬝ph _,
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apply move_top_of_left, apply move_left_of_bot,
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refine ap02 _ (adj f _) ⬝ !ap_compose⁻¹ ⬝pv _ ⬝vp !ap_compose,
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apply natural_square_tr }}
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end
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definition pushout_of_equiv_right [constructor] {A B C : Type} (f : A → B) (g : A ≃ C) :
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pushout f g ≃ B :=
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calc
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pushout f g ≃ pushout g f : pushout.symm f g
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... ≃ B : pushout_of_equiv_left g f
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-- todo: define pushout.equiv (renamed to pushout_equiv_pushout) using this
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variables {A₁ B₁ C₁ A₂ B₂ C₂ A₃ B₃ C₃ : Type} {f₁ : A₁ → B₁} {g₁ : A₁ → C₁}
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{f₂ : A₂ → B₂} {g₂ : A₂ → C₂} {f₃ : A₃ → B₃} {g₃ : A₃ → C₃}
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{h₂ : A₂ → A₃} {h₁ : A₁ → A₂}
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{i₂ : B₂ → B₃} {i₁ : B₁ → B₂}
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{j₂ : C₂ → C₃} {j₁ : C₁ → C₂}
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(p₂ : i₂ ∘ f₂ ~ f₃ ∘ h₂) (q₂ : j₂ ∘ g₂ ~ g₃ ∘ h₂)
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(p₁ : i₁ ∘ f₁ ~ f₂ ∘ h₁) (q₁ : j₁ ∘ g₁ ~ g₂ ∘ h₁)
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definition pushout_functor_compose :
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pushout.functor (h₂ ∘ h₁) (i₂ ∘ i₁) (j₂ ∘ j₁) (p₁ ⬝htyv p₂) (q₁ ⬝htyv q₂) ~
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pushout.functor h₂ i₂ j₂ p₂ q₂ ∘ pushout.functor h₁ i₁ j₁ p₁ q₁ :=
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begin
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intro x, induction x with b c a,
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{ reflexivity },
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{ reflexivity },
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{ apply eq_pathover, apply hdeg_square, esimp,
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refine !elim_glue ⬝ whisker_right _ (!ap_con ⬝ !ap_compose'⁻¹ ◾ idp) ◾
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(ap02 _ !con_inv ⬝ !ap_con ⬝ whisker_left _ (ap02 _ !ap_inv⁻¹ ⬝ !ap_compose'⁻¹)) ⬝ _ ⬝
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(ap_compose (pushout.functor h₂ i₂ j₂ p₂ q₂) _ _ ⬝ ap02 _ !elim_glue)⁻¹,
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refine _ ⬝ (!ap_con ⬝ (!ap_con ⬝ !ap_compose'⁻¹ ◾ !elim_glue) ◾ !ap_compose'⁻¹)⁻¹ᵖ,
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refine !con.assoc⁻¹ ⬝ whisker_right _ _,
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exact whisker_right _ !con.assoc ⬝ !con.assoc }
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end
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variables {p₁ q₁}
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definition pushout_functor_homotopy_constant {p₁' : i₁ ∘ f₁ ~ f₂ ∘ h₁} {q₁' : j₁ ∘ g₁ ~ g₂ ∘ h₁}
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(p : p₁ ~ p₁') (q : q₁ ~ q₁') :
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pushout.functor h₁ i₁ j₁ p₁ q₁ ~ pushout.functor h₁ i₁ j₁ p₁' q₁' :=
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begin
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induction p, induction q, reflexivity
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end
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definition pushout_functor_homotopy {h₁ h₂ : A₁ → A₂} {i₁ i₂ : B₁ → B₂} {j₁ j₂ : C₁ → C₂}
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{p₁ : i₁ ∘ f₁ ~ f₂ ∘ h₁} {q₁ : j₁ ∘ g₁ ~ g₂ ∘ h₁}
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{p₂ : i₂ ∘ f₁ ~ f₂ ∘ h₂} {q₂ : j₂ ∘ g₁ ~ g₂ ∘ h₂}
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(r : h₁ ~ h₂) (s : i₁ ~ i₂) (t : j₁ ~ j₂)
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(u : r ⬝htyh p₁ ~ p₂ ⬝htyh s) (v : r ⬝htyh q₁ ~ q₂ ⬝htyh t) :
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pushout.functor h₁ i₁ j₁ p₁ q₁ ~ pushout.functor h₂ i₂ j₂ p₂ q₂ :=
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begin
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induction r, induction s, induction t, apply pushout_functor_homotopy_constant,
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{ exact (rfl_hhconcat p₁)⁻¹ʰᵗʸ ⬝hty u ⬝hty hhconcat_rfl p₂ },
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exact (rfl_hhconcat q₁)⁻¹ʰᵗʸ ⬝hty v ⬝hty hhconcat_rfl q₂
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end
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/- pushout where one map is constant is a cofiber -/
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definition pushout_const_equiv_to [unfold 6] {A B C : Type} {f : A → B} {c₀ : C}
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(x : pushout f (const A c₀)) : cofiber (sum_functor f (const unit c₀)) :=
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begin
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induction x with b c a,
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{ exact !cod (sum.inl b) },
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{ exact !cod (sum.inr c) },
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{ exact glue (sum.inl a) ⬝ (glue (sum.inr ⋆))⁻¹ }
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end
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definition pushout_const_equiv_from [unfold 6] {A B C : Type} {f : A → B} {c₀ : C}
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(x : cofiber (sum_functor f (const unit c₀))) : pushout f (const A c₀) :=
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begin
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induction x with v v,
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{ induction v with b c, exact inl b, exact inr c },
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{ exact inr c₀ },
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{ induction v with a u, exact glue a, reflexivity }
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end
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definition pushout_const_equiv [constructor] {A B C : Type} (f : A → B) (c₀ : C) :
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pushout f (const A c₀) ≃ cofiber (sum_functor f (const unit c₀)) :=
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begin
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fapply equiv.MK,
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{ exact pushout_const_equiv_to },
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{ exact pushout_const_equiv_from },
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{ intro x, induction x with v v,
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{ induction v with b c, reflexivity, reflexivity },
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{ exact glue (sum.inr ⋆) },
|
||
{ apply eq_pathover_id_right,
|
||
refine ap_compose pushout_const_equiv_to _ _ ⬝ ap02 _ !elim_glue ⬝ph _,
|
||
induction v with a u,
|
||
{ refine !elim_glue ⬝ph _, apply whisker_bl, exact hrfl },
|
||
{ induction u, exact square_of_eq idp }}},
|
||
{ intro x, induction x with c b a,
|
||
{ reflexivity },
|
||
{ reflexivity },
|
||
{ apply eq_pathover_id_right, apply hdeg_square,
|
||
refine ap_compose pushout_const_equiv_from _ _ ⬝ ap02 _ !elim_glue ⬝ _,
|
||
refine !ap_con ⬝ !elim_glue ◾ (!ap_inv ⬝ !elim_glue⁻²) }}
|
||
end
|
||
|
||
/- wedge is the cofiber of the map 2 -> A + B -/
|
||
|
||
-- move to sum
|
||
definition sum_of_bool [unfold 3] (A B : Type*) (b : bool) : A + B :=
|
||
by induction b; exact sum.inl pt; exact sum.inr pt
|
||
|
||
definition psum_of_pbool [constructor] (A B : Type*) : pbool →* (A +* B) :=
|
||
pmap.mk (sum_of_bool A B) idp
|
||
|
||
-- move to wedge
|
||
definition wedge_equiv_pushout_sum [constructor] (A B : Type*) :
|
||
wedge A B ≃ cofiber (sum_of_bool A B) :=
|
||
begin
|
||
refine pushout_const_equiv _ _ ⬝e _,
|
||
fapply pushout.equiv,
|
||
exact bool_equiv_unit_sum_unit⁻¹ᵉ,
|
||
reflexivity,
|
||
reflexivity,
|
||
intro x, induction x: reflexivity,
|
||
intro x, induction x with u u: induction u; reflexivity
|
||
end
|
||
|
||
section
|
||
open prod.ops
|
||
/- products preserve pushouts -/
|
||
|
||
definition pushout_prod_equiv_to [unfold 7] {A B C D : Type} {f : A → B} {g : A → C}
|
||
(xd : pushout f g × D) : pushout (prod_functor f (@id D)) (prod_functor g id) :=
|
||
begin
|
||
induction xd with x d, induction x with b c a,
|
||
{ exact inl (b, d) },
|
||
{ exact inr (c, d) },
|
||
{ exact glue (a, d) }
|
||
end
|
||
|
||
definition pushout_prod_equiv_from [unfold 7] {A B C D : Type} {f : A → B} {g : A → C}
|
||
(x : pushout (prod_functor f (@id D)) (prod_functor g id)) : pushout f g × D :=
|
||
begin
|
||
induction x with bd cd ad,
|
||
{ exact (inl bd.1, bd.2) },
|
||
{ exact (inr cd.1, cd.2) },
|
||
{ exact prod_eq (glue ad.1) idp }
|
||
end
|
||
|
||
definition pushout_prod_equiv {A B C D : Type} (f : A → B) (g : A → C) :
|
||
pushout f g × D ≃ pushout (prod_functor f (@id D)) (prod_functor g id) :=
|
||
begin
|
||
fapply equiv.MK,
|
||
{ exact pushout_prod_equiv_to },
|
||
{ exact pushout_prod_equiv_from },
|
||
{ intro x, induction x with bd cd ad,
|
||
{ induction bd, reflexivity },
|
||
{ induction cd, reflexivity },
|
||
{ induction ad with a d, apply eq_pathover_id_right, apply hdeg_square,
|
||
refine ap_compose pushout_prod_equiv_to _ _ ⬝ ap02 _ !elim_glue ⬝ _, esimp,
|
||
exact !ap_prod_elim ⬝ !idp_con ⬝ !elim_glue }},
|
||
{ intro xd, induction xd with x d, induction x with b c a,
|
||
{ reflexivity },
|
||
{ reflexivity },
|
||
{ apply eq_pathover, apply hdeg_square,
|
||
refine ap_compose (pushout_prod_equiv_from ∘ pushout_prod_equiv_to) _ _ ⬝ _,
|
||
refine ap02 _ !ap_prod_mk_left ⬝ !ap_compose ⬝ _,
|
||
refine ap02 _ (!ap_prod_elim ⬝ !idp_con ⬝ !elim_glue) ⬝ _,
|
||
refine !elim_glue ⬝ !ap_prod_mk_left⁻¹ }}
|
||
end
|
||
end
|
||
|
||
/- interaction of pushout and sums -/
|
||
|
||
definition pushout_to_sum [unfold 8] {A B C : Type} {f : A → B} {g : A → C} (D : Type) (c₀ : C)
|
||
(x : pushout f g) : pushout (sum_functor f (@id D)) (sum.rec g (λd, c₀)) :=
|
||
begin
|
||
induction x with b c a,
|
||
{ exact inl (sum.inl b) },
|
||
{ exact inr c },
|
||
{ exact glue (sum.inl a) }
|
||
end
|
||
|
||
definition pushout_from_sum [unfold 8] {A B C : Type} {f : A → B} {g : A → C} (D : Type) (c₀ : C)
|
||
(x : pushout (sum_functor f (@id D)) (sum.rec g (λd, c₀))) : pushout f g :=
|
||
begin
|
||
induction x with x c x,
|
||
{ induction x with b d, exact inl b, exact inr c₀ },
|
||
{ exact inr c },
|
||
{ induction x with a d, exact glue a, reflexivity }
|
||
end
|
||
|
||
/- The pushout of B <-- A --> C is the same as the pushout of B + D <-- A + D --> C -/
|
||
definition pushout_sum_cancel_equiv [constructor] {A B C : Type} (f : A → B) (g : A → C)
|
||
(D : Type) (c₀ : C) : pushout f g ≃ pushout (sum_functor f (@id D)) (sum.rec g (λd, c₀)) :=
|
||
begin
|
||
fapply equiv.MK,
|
||
{ exact pushout_to_sum D c₀ },
|
||
{ exact pushout_from_sum D c₀ },
|
||
{ intro x, induction x with x c x,
|
||
{ induction x with b d, reflexivity, esimp, exact (glue (sum.inr d))⁻¹ },
|
||
{ reflexivity },
|
||
{ apply eq_pathover_id_right,
|
||
refine ap_compose (pushout_to_sum D c₀) _ _ ⬝ ap02 _ !elim_glue ⬝ph _,
|
||
induction x with a d: esimp,
|
||
{ exact hdeg_square !elim_glue },
|
||
{ exact square_of_eq !con.left_inv }}},
|
||
{ intro x, induction x with b c a,
|
||
{ reflexivity },
|
||
{ reflexivity },
|
||
{ apply eq_pathover_id_right, apply hdeg_square,
|
||
refine ap_compose (pushout_from_sum D c₀) _ _ ⬝ ap02 _ !elim_glue ⬝ !elim_glue }}
|
||
end
|
||
|
||
end pushout
|
||
|
||
namespace pushout
|
||
|
||
variables {A A' B B' C C' : Type} {f : A → B} {g : A → C} {f' : A' → B'} {g' : A' → C'}
|
||
definition sum_pushout_of_pushout_sum [unfold 11]
|
||
(x : pushout (sum_functor f f') (sum_functor g g')) : pushout f g ⊎ pushout f' g' :=
|
||
begin
|
||
induction x with b c a,
|
||
{ exact sum_functor inl inl b },
|
||
{ exact sum_functor inr inr c },
|
||
{ induction a with a a', exact ap sum.inl (glue a), exact ap sum.inr (glue a') }
|
||
end
|
||
|
||
definition pushout_sum_of_sum_pushout [unfold 11]
|
||
(x : pushout f g ⊎ pushout f' g') : pushout (sum_functor f f') (sum_functor g g') :=
|
||
begin
|
||
induction x with x x,
|
||
{ exact pushout.functor sum.inl sum.inl sum.inl homotopy.rfl homotopy.rfl x },
|
||
{ exact pushout.functor sum.inr sum.inr sum.inr homotopy.rfl homotopy.rfl x }
|
||
end
|
||
|
||
variables (f g f' g')
|
||
/-
|
||
do we want to define this in terms of sigma_pushout? One possible disadvantage is that the
|
||
computation on glue is less convenient
|
||
-/
|
||
definition pushout_sum_equiv_sum_pushout [constructor] :
|
||
pushout (sum_functor f f') (sum_functor g g') ≃ pushout f g ⊎ pushout f' g' :=
|
||
equiv.MK sum_pushout_of_pushout_sum pushout_sum_of_sum_pushout
|
||
abstract begin
|
||
intro x, induction x with x x,
|
||
{ induction x,
|
||
{ reflexivity },
|
||
{ reflexivity },
|
||
apply eq_pathover, apply hdeg_square, esimp,
|
||
exact ap_compose sum_pushout_of_pushout_sum _ _ ⬝
|
||
ap02 _ (!elim_glue ⬝ !con_idp ⬝ !idp_con) ⬝ !elim_glue },
|
||
{ induction x,
|
||
{ reflexivity },
|
||
{ reflexivity },
|
||
apply eq_pathover, apply hdeg_square, esimp,
|
||
exact ap_compose sum_pushout_of_pushout_sum _ _ ⬝
|
||
ap02 _ (!elim_glue ⬝ !con_idp ⬝ !idp_con) ⬝ !elim_glue },
|
||
end end
|
||
abstract begin
|
||
intro x, induction x with b c a,
|
||
{ induction b: reflexivity },
|
||
{ induction c: reflexivity },
|
||
{ apply eq_pathover_id_right,
|
||
refine ap_compose pushout_sum_of_sum_pushout _ _ ⬝ ap02 _ !elim_glue ⬝ph _,
|
||
induction a with a a':
|
||
(apply hdeg_square; refine !ap_compose'⁻¹ ⬝ !elim_glue ⬝ !con_idp ⬝ !idp_con) }
|
||
end end
|
||
|
||
variables {f g f' g'}
|
||
variables {D E F D' E' F' : Type} {h : D → E} {i : D → F} {h' : D' → E'} {i' : D' → F'}
|
||
{j : A → D} {k : B → E} {l : C → F} {j' : A' → D'} {k' : B' → E'} {l' : C' → F'}
|
||
{j₂ : A' → D} {k₂ : B' → E} {l₂ : C' → F}
|
||
(s : hsquare f h j k) (t : hsquare g i j l)
|
||
(s' : hsquare f' h' j' k') (t' : hsquare g' i' j' l')
|
||
(s₂ : hsquare f' h j₂ k₂) (t₂ : hsquare g' i j₂ l₂)
|
||
|
||
definition sum_rec_pushout_sum_equiv_sum_pushout :
|
||
sum.rec (pushout.functor j k l s t) (pushout.functor j₂ k₂ l₂ s₂ t₂) ∘
|
||
pushout_sum_equiv_sum_pushout f g f' g' ~
|
||
pushout.functor (sum.rec j j₂) (sum.rec k k₂) (sum.rec l l₂)
|
||
(sum_rec_hsquare s s₂) (sum_rec_hsquare t t₂) :=
|
||
begin
|
||
intro x, induction x with b c a,
|
||
{ induction b with b b': reflexivity },
|
||
{ induction c with c c': reflexivity },
|
||
{ exact abstract begin apply eq_pathover,
|
||
refine !ap_compose ⬝ ap02 _ !elim_glue ⬝ph _,
|
||
induction a with a a': exact hdeg_square (!ap_compose'⁻¹ ⬝ !elim_glue ⬝ !elim_glue⁻¹)
|
||
end end }
|
||
end
|
||
|
||
definition pushout_sum_equiv_sum_pushout_natural :
|
||
hsquare
|
||
(pushout.functor (j +→ j') (k +→ k') (l +→ l')
|
||
(sum_functor_hsquare s s') (sum_functor_hsquare t t'))
|
||
(pushout.functor j k l s t +→ pushout.functor j' k' l' s' t')
|
||
(pushout_sum_equiv_sum_pushout f g f' g')
|
||
(pushout_sum_equiv_sum_pushout h i h' i') :=
|
||
begin
|
||
intro x, induction x with b c a,
|
||
{ induction b with b b': reflexivity },
|
||
{ induction c with c c': reflexivity },
|
||
{ exact abstract begin apply eq_pathover,
|
||
refine !ap_compose ⬝ ap02 _ !elim_glue ⬝ph _ ⬝hp (!ap_compose ⬝ ap02 _ !elim_glue)⁻¹,
|
||
refine !ap_con ⬝ (!ap_con ⬝ !ap_compose'⁻¹ ◾ !elim_glue) ◾ (!ap_compose'⁻¹ ⬝ !ap_inv) ⬝ph _,
|
||
induction a with a a',
|
||
{ apply hdeg_square, refine !ap_compose'⁻¹ ◾ idp ◾ !ap_compose'⁻¹⁻² ⬝ _ ⬝ !ap_compose',
|
||
refine _ ⬝ (ap_compose sum.inl _ _ ⬝ ap02 _ !elim_glue)⁻¹,
|
||
exact (ap_compose sum.inl _ _ ◾ idp ⬝ !ap_con⁻¹) ◾ (!ap_inv⁻¹ ⬝ ap_compose sum.inl _ _) ⬝
|
||
!ap_con⁻¹ },
|
||
{ apply hdeg_square, refine !ap_compose'⁻¹ ◾ idp ◾ !ap_compose'⁻¹⁻² ⬝ _ ⬝ !ap_compose',
|
||
refine _ ⬝ (ap_compose sum.inr _ _ ⬝ ap02 _ !elim_glue)⁻¹,
|
||
exact (ap_compose sum.inr _ _ ◾ idp ⬝ !ap_con⁻¹) ◾ (!ap_inv⁻¹ ⬝ ap_compose sum.inr _ _) ⬝
|
||
!ap_con⁻¹ } end end }
|
||
end
|
||
|
||
end pushout
|
||
|
||
namespace pushout
|
||
open sigma sigma.ops
|
||
variables {X : Type} {A B C : X → Type} {f : Πx, A x → B x} {g : Πx, A x → C x}
|
||
definition sigma_pushout_of_pushout_sigma [unfold 7]
|
||
(x : pushout (total f) (total g)) : Σx, pushout (f x) (g x) :=
|
||
begin
|
||
induction x with b c a,
|
||
{ exact total (λx, inl) b },
|
||
{ exact total (λx, inr) c },
|
||
{ exact sigma_eq_right (glue a.2) }
|
||
end
|
||
|
||
definition pushout_sigma_of_sigma_pushout [unfold 7]
|
||
(x : Σx, pushout (f x) (g x)) : pushout (total f) (total g) :=
|
||
pushout.functor (dpair x.1) (dpair x.1) (dpair x.1) homotopy.rfl homotopy.rfl x.2
|
||
|
||
variables (f g)
|
||
definition pushout_sigma_equiv_sigma_pushout [constructor] :
|
||
pushout (total f) (total g) ≃ Σx, pushout (f x) (g x) :=
|
||
equiv.MK sigma_pushout_of_pushout_sigma pushout_sigma_of_sigma_pushout
|
||
abstract begin
|
||
intro x, induction x with x y, induction y with b c a,
|
||
{ reflexivity },
|
||
{ reflexivity },
|
||
{ apply eq_pathover, apply hdeg_square, esimp,
|
||
exact ap_compose sigma_pushout_of_pushout_sigma _ _ ⬝
|
||
ap02 _ (!elim_glue ⬝ !con_idp ⬝ !idp_con) ⬝ !elim_glue }
|
||
end end
|
||
abstract begin
|
||
intro x, induction x with b c a,
|
||
{ induction b, reflexivity },
|
||
{ induction c, reflexivity },
|
||
{ apply eq_pathover_id_right,
|
||
refine ap_compose pushout_sigma_of_sigma_pushout _ _ ⬝ ap02 _ !elim_glue ⬝ph _,
|
||
induction a with a a',
|
||
apply hdeg_square, refine !ap_compose'⁻¹ ⬝ !elim_glue ⬝ !con_idp ⬝ !idp_con }
|
||
end end
|
||
|
||
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.
|
||
The Pgluer is modified to get the right coherence
|
||
See https://github.com/HoTT/HoTT-Agda/blob/master/theorems/homotopy/elims/CofPushoutSection.agda
|
||
-/
|
||
|
||
open sigma.ops
|
||
definition cofiber_pushout_helper' {A : Type} {B : A → Type} {a₀₀ a₀₂ a₂₀ a₂₂ : A} {p₀₁ : a₀₀ = a₀₂}
|
||
{p₁₀ : a₀₀ = a₂₀} {p₂₁ : a₂₀ = a₂₂} {p₁₂ : a₀₂ = a₂₂} {s : square p₀₁ p₂₁ p₁₀ p₁₂}
|
||
{b₀₀ : B a₀₀} {b₂₀ : B a₂₀} {b₀₂ : B a₀₂} {b₂₂ b₂₂' : B a₂₂} {q₁₀ : b₀₀ =[p₁₀] b₂₀}
|
||
{q₀₁ : b₀₀ =[p₀₁] b₀₂} {q₂₁ : b₂₀ =[p₂₁] b₂₂'} {q₁₂ : b₀₂ =[p₁₂] b₂₂} :
|
||
Σ(r : b₂₂' = b₂₂), squareover B s q₀₁ (r ▸ q₂₁) q₁₀ q₁₂ :=
|
||
begin
|
||
induction s,
|
||
induction q₀₁ using idp_rec_on,
|
||
induction q₂₁ using idp_rec_on,
|
||
induction q₁₀ using idp_rec_on,
|
||
induction q₁₂ using idp_rec_on,
|
||
exact ⟨idp, idso⟩
|
||
end
|
||
|
||
definition cofiber_pushout_helper {A B C D : Type} {f : A → B} {g : A → C} {h : pushout f g → D}
|
||
{P : cofiber h → Type} {Pcod : Πd, P (cofiber.cod h d)} {Pbase : P (cofiber.base h)}
|
||
(Pgluel : Π(b : B), Pcod (h (inl b)) =[cofiber.glue (inl b)] Pbase)
|
||
(Pgluer : Π(c : C), Pcod (h (inr c)) =[cofiber.glue (inr c)] Pbase)
|
||
(a : A) : Σ(p : Pbase = Pbase), squareover P (natural_square cofiber.glue (glue a))
|
||
(Pgluel (f a)) (p ▸ Pgluer (g a))
|
||
(pathover_ap P (λa, cofiber.cod h (h a)) (apd (λa, Pcod (h a)) (glue a)))
|
||
(pathover_ap P (λa, cofiber.base h) (apd (λa, Pbase) (glue a))) :=
|
||
!cofiber_pushout_helper'
|
||
|
||
definition cofiber_pushout_rec {A B C D : Type} {f : A → B} {g : A → C} {h : pushout f g → D}
|
||
{P : cofiber h → Type} (Pcod : Πd, P (cofiber.cod h d)) (Pbase : P (cofiber.base h))
|
||
(Pgluel : Π(b : B), Pcod (h (inl b)) =[cofiber.glue (inl b)] Pbase)
|
||
(Pgluer : Π(c : C), Pcod (h (inr c)) =[cofiber.glue (inr c)] Pbase)
|
||
(r : C → A) (p : Πa, r (g a) = a)
|
||
(x : cofiber h) : P x :=
|
||
begin
|
||
induction x with d x,
|
||
{ exact Pcod d },
|
||
{ exact Pbase },
|
||
{ induction x with b c a,
|
||
{ exact Pgluel b },
|
||
{ exact (cofiber_pushout_helper Pgluel Pgluer (r c)).1 ▸ Pgluer c },
|
||
{ apply pathover_pathover, rewrite [p a], exact (cofiber_pushout_helper Pgluel Pgluer a).2 }}
|
||
end
|
||
|
||
/- universal property of cofiber -/
|
||
|
||
definition cofiber_exact_1 {X Y Z : Type*} (f : X →* Y) (g : pcofiber f →* Z) :
|
||
(g ∘* pcod f) ∘* f ~* pconst X Z :=
|
||
!passoc ⬝* pwhisker_left _ !pcod_pcompose ⬝* !pcompose_pconst
|
||
|
||
protected definition pcofiber.elim [constructor] {X Y Z : Type*} {f : X →* Y} (g : Y →* Z)
|
||
(p : g ∘* f ~* pconst X Z) : pcofiber f →* Z :=
|
||
begin
|
||
fapply pmap.mk,
|
||
{ intro w, induction w with y x, exact g y, exact pt, exact p x },
|
||
{ reflexivity }
|
||
end
|
||
|
||
protected definition pcofiber.elim_pcod {X Y Z : Type*} {f : X →* Y} {g : Y →* Z}
|
||
(p : g ∘* f ~* pconst X Z) : pcofiber.elim g p ∘* pcod f ~* g :=
|
||
begin
|
||
fapply phomotopy.mk,
|
||
{ intro y, reflexivity },
|
||
{ esimp, refine !idp_con ⬝ _,
|
||
refine _ ⬝ (!ap_con ⬝ (!ap_compose'⁻¹ ⬝ !ap_inv) ◾ !elim_glue)⁻¹,
|
||
apply eq_inv_con_of_con_eq, exact (to_homotopy_pt p)⁻¹ }
|
||
end
|
||
|
||
/-
|
||
The maps Z^{C_f} --> Z^Y --> Z^X are exact at Z^Y.
|
||
Here Y^X means pointed maps from X to Y and C_f is the cofiber of f.
|
||
The maps are given by precomposing with (pcod f) and f.
|
||
-/
|
||
definition cofiber_exact {X Y Z : Type*} (f : X →* Y) :
|
||
is_exact_t (@ppcompose_right _ _ Z (pcod f)) (ppcompose_right f) :=
|
||
begin
|
||
constructor,
|
||
{ intro g, apply eq_of_phomotopy, apply cofiber_exact_1 },
|
||
{ intro g p, note q := phomotopy_of_eq p,
|
||
exact fiber.mk (pcofiber.elim g q) (eq_of_phomotopy (pcofiber.elim_pcod q)) }
|
||
end
|
||
|
||
/- cofiber of pcod is suspension -/
|
||
|
||
definition pcofiber_pcod {A B : Type*} (f : A →* B) : pcofiber (pcod f) ≃* susp A :=
|
||
begin
|
||
fapply pequiv_of_equiv,
|
||
{ refine !pushout.symm ⬝e _,
|
||
exact pushout_vcompose_equiv f equiv.rfl homotopy.rfl homotopy.rfl },
|
||
reflexivity
|
||
end
|
||
|
||
-- definition pushout_vcompose [constructor] {A B C D : Type} (f : A → B) (g : A → C) (h : B → D) :
|
||
-- pushout h (@inl _ _ _ f g) ≃ pushout (h ∘ f) g :=
|
||
-- definition pushout_hcompose {A B C D : Type} (f : A → B) (g : A → C) (h : C → D) :
|
||
-- pushout (@inr _ _ _ f g) h ≃ pushout f (h ∘ g) :=
|
||
|
||
-- definition pushout_vcompose_equiv {A B C D E : Type} (f : A → B) {g : A → C} {h : B → D}
|
||
-- {hf : A → D} {k : B → E} (e : E ≃ pushout f g) (p : k ~ e⁻¹ᵉ ∘ inl) (q : h ∘ f ~ hf) :
|
||
-- pushout h k ≃ pushout hf g :=
|
||
|
||
end pushout
|
||
|
||
namespace pushout
|
||
/- define the quotient using pushout -/
|
||
section
|
||
open quotient sigma.ops
|
||
variables {A B : Type} (R : A → A → Type) {Q : B → B → Type}
|
||
(f : A → B) (k : Πa a' : A, R a a' → Q (f a) (f a'))
|
||
|
||
definition pushout_quotient {A : Type} (R : A → A → Type) : Type :=
|
||
@pushout ((Σa a', R a a') ⊎ (Σa a', R a a')) A (Σa a', R a a')
|
||
(sum.rec pr1 (λx, x.2.1)) (sum.rec id id)
|
||
|
||
variable {R}
|
||
definition pushout_quotient_of_quotient [unfold 3] (x : quotient R) : pushout_quotient R :=
|
||
begin
|
||
induction x with a a a' r,
|
||
{ exact inl a },
|
||
{ exact glue (sum.inl ⟨a, a', r⟩) ⬝ (glue (sum.inr ⟨a, a', r⟩))⁻¹ }
|
||
end
|
||
|
||
definition quotient_of_pushout_quotient [unfold 3] (x : pushout_quotient R) : quotient R :=
|
||
begin
|
||
induction x with a x x,
|
||
{ exact class_of R a },
|
||
{ exact class_of R x.2.1 },
|
||
{ induction x with x x, exact eq_of_rel R x.2.2, reflexivity }
|
||
end
|
||
|
||
variable (R)
|
||
definition quotient_equiv_pushout [constructor] : quotient R ≃ pushout_quotient R :=
|
||
equiv.MK pushout_quotient_of_quotient quotient_of_pushout_quotient
|
||
abstract begin
|
||
intro x, induction x with a x x,
|
||
{ reflexivity },
|
||
{ exact glue (sum.inr x) },
|
||
{ apply eq_pathover_id_right,
|
||
refine ap_compose pushout_quotient_of_quotient _ _ ⬝ ap02 _ !elim_glue ⬝ph _,
|
||
induction x with x x,
|
||
{ refine !elim_eq_of_rel ⬝ph _, induction x with a x, induction x with a' r,
|
||
exact whisker_bl _ hrfl },
|
||
{ exact square_of_eq idp }}
|
||
end end
|
||
abstract begin
|
||
intro x, induction x,
|
||
{ reflexivity },
|
||
{ apply eq_pathover_id_right, apply hdeg_square,
|
||
refine ap_compose quotient_of_pushout_quotient _ _ ⬝ ap02 _ !elim_eq_of_rel ⬝ _,
|
||
exact !ap_con ⬝ !elim_glue ◾ (!ap_inv ⬝ !elim_glue⁻²) }
|
||
end end
|
||
|
||
variable {R}
|
||
definition sigma_functor2 [unfold 7] : (Σ a a', R a a') → (Σ b b', Q b b') :=
|
||
sigma_functor f (λa, sigma_functor f (k a))
|
||
|
||
definition pushout_quotient_functor [unfold 7] : pushout_quotient R → pushout_quotient Q :=
|
||
let tf := sigma_functor2 f k in
|
||
pushout.functor (sum_functor tf tf) f tf
|
||
begin intro x, induction x: reflexivity end begin intro x, induction x: reflexivity end
|
||
|
||
definition quotient_equiv_pushout_natural :
|
||
hsquare (quotient.functor _ _ f k) (pushout_quotient_functor f k)
|
||
(quotient_equiv_pushout R) (quotient_equiv_pushout Q) :=
|
||
begin
|
||
intro x, induction x with a a a' r,
|
||
{ reflexivity },
|
||
{ apply eq_pathover, apply hdeg_square,
|
||
refine ap_compose pushout_quotient_of_quotient _ _ ⬝ _ ⬝
|
||
(ap_compose (pushout.functor _ _ _ _ _) _ _)⁻¹,
|
||
refine ap02 _ !elim_eq_of_rel ⬝ _ ⬝ (ap02 _ !elim_eq_of_rel)⁻¹,
|
||
refine !elim_eq_of_rel ⬝ _,
|
||
exact (!ap_con ⬝ (!pushout.elim_glue ⬝ !con_idp ⬝ !idp_con) ◾
|
||
(!ap_inv ⬝ (!pushout.elim_glue ⬝ !con_idp ⬝ !idp_con)⁻²))⁻¹ }
|
||
end
|
||
|
||
|
||
end
|
||
|
||
end pushout
|