517 lines
21 KiB
Text
517 lines
21 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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/- 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 ⋆) },
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{ apply eq_pathover_id_right,
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refine ap_compose pushout_const_equiv_to _ _ ⬝ ap02 _ !elim_glue ⬝ph _,
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induction v with a u,
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{ refine !elim_glue ⬝ph _, apply whisker_bl, exact hrfl },
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{ induction u, exact square_of_eq idp }}},
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{ intro x, induction x with c b 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_const_equiv_from _ _ ⬝ ap02 _ !elim_glue ⬝ _,
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refine !ap_con ⬝ !elim_glue ◾ (!ap_inv ⬝ !elim_glue⁻²) }}
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end
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/- wedge is the cofiber of the map 2 -> A + B -/
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-- move to sum
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definition sum_of_bool [unfold 3] (A B : Type*) (b : bool) : A + B :=
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by induction b; exact sum.inl pt; exact sum.inr pt
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definition psum_of_pbool [constructor] (A B : Type*) : pbool →* (A +* B) :=
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pmap.mk (sum_of_bool A B) idp
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-- move to wedge
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definition wedge_equiv_pushout_sum [constructor] (A B : Type*) :
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wedge A B ≃ cofiber (sum_of_bool A B) :=
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begin
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refine pushout_const_equiv _ _ ⬝e _,
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fapply pushout.equiv,
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exact bool_equiv_unit_sum_unit⁻¹ᵉ,
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reflexivity,
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reflexivity,
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intro x, induction x: reflexivity,
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intro x, induction x with u u: induction u; reflexivity
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end
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section
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open prod.ops
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/- products preserve pushouts -/
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definition pushout_prod_equiv_to [unfold 7] {A B C D : Type} {f : A → B} {g : A → C}
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(xd : pushout f g × D) : pushout (prod_functor f (@id D)) (prod_functor g id) :=
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begin
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induction xd with x d, induction x with b c a,
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{ exact inl (b, d) },
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{ exact inr (c, d) },
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{ exact glue (a, d) }
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end
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definition pushout_prod_equiv_from [unfold 7] {A B C D : Type} {f : A → B} {g : A → C}
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(x : pushout (prod_functor f (@id D)) (prod_functor g id)) : pushout f g × D :=
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begin
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induction x with bd cd ad,
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{ exact (inl bd.1, bd.2) },
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{ exact (inr cd.1, cd.2) },
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{ exact prod_eq (glue ad.1) idp }
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end
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definition pushout_prod_equiv {A B C D : Type} (f : A → B) (g : A → C) :
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pushout f g × D ≃ pushout (prod_functor f (@id D)) (prod_functor g id) :=
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begin
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fapply equiv.MK,
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{ exact pushout_prod_equiv_to },
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{ exact pushout_prod_equiv_from },
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{ intro x, induction x with bd cd ad,
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{ induction bd, reflexivity },
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{ induction cd, reflexivity },
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{ induction ad with a d, apply eq_pathover_id_right, apply hdeg_square,
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refine ap_compose pushout_prod_equiv_to _ _ ⬝ ap02 _ !elim_glue ⬝ _, esimp,
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exact !ap_prod_elim ⬝ !idp_con ⬝ !elim_glue }},
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{ intro xd, induction xd with x d, 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,
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refine ap_compose (pushout_prod_equiv_from ∘ pushout_prod_equiv_to) _ _ ⬝ _,
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refine ap02 _ !ap_prod_mk_left ⬝ !ap_compose ⬝ _,
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refine ap02 _ (!ap_prod_elim ⬝ !idp_con ⬝ !elim_glue) ⬝ _,
|
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refine !elim_glue ⬝ !ap_prod_mk_left⁻¹ }}
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||
end
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||
end
|
||
|
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/- interaction of pushout and sums -/
|
||
|
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definition pushout_to_sum [unfold 8] {A B C : Type} {f : A → B} {g : A → C} (D : Type) (c₀ : C)
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(x : pushout f g) : pushout (sum_functor f (@id D)) (sum.rec g (λd, c₀)) :=
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begin
|
||
induction x with b c a,
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||
{ exact inl (sum.inl b) },
|
||
{ exact inr c },
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||
{ 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)
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||
(x : pushout (sum_functor f (@id D)) (sum.rec g (λd, c₀))) : pushout f g :=
|
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begin
|
||
induction x with x c x,
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||
{ induction x with b d, exact inl b, exact inr c₀ },
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{ exact inr c },
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{ induction x with a d, exact glue a, reflexivity }
|
||
|
||
end
|
||
|
||
definition pushout_sum_equiv [constructor] {A B C : Type} (f : A → B) (g : A → C) (D : Type)
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||
(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
|
||
|
||
/- 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
|
||
|
||
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) ≃* psusp 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
|