feat(category.pushout): finish universal property of pushout
In the previous commit there was still one step missing: that the natural isomorphisms are also unique.
This commit is contained in:
Floris van Doorn
parent
fe1fbae540
commit
fd5adb831b
16 changed files with 300 additions and 24 deletions
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@ -82,6 +82,12 @@ namespace functor
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(θ : Πc, G c ⟶ F c) (r : Πc, θ c ∘ η c = id) (q : Πc, η c ∘ θ c = id) : F ≅ G :=
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iso.mk (nat_trans.mk η p) (@(is_natural_iso _) (λc, is_iso.mk (θ c) (r c) (q c)))
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definition natural_iso.mk' [constructor]
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(η : Πc, F c ≅ G c) (p : Π(c c' : C) (f : c ⟶ c'), G f ∘ to_hom (η c) = to_hom (η c') ∘ F f) :
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F ≅ G :=
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natural_iso.MK (λc, to_hom (η c)) p (λc, to_inv (η c))
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(λc, to_left_inverse (η c)) (λc, to_right_inverse (η c))
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end
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section
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@ -29,4 +29,8 @@ namespace category
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fundamental_groupoid A ⇒ fundamental_groupoid B :=
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functor.mk f (λa a', tap f) (λa, tap_tidp f) (λa₁ a₂ a₃ q p, tap_tcon f p q)
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notation `Π₁` := fundamental_groupoid
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notation `Π₁⇒` := fundamental_groupoid_functor
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end category
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@ -196,7 +196,7 @@ namespace category
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end
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include η
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parameters {F G}
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parameters {F G H K}
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definition Cpushout_functor_reduction_rule [unfold 12] (i : R x x') :
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Cpushout_functor_ob x ⟶ Cpushout_functor_ob x' :=
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begin
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@ -361,7 +361,34 @@ namespace category
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{ exact ap010 natural_map (to_right_inverse η₂) e}},
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end
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end
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open bpushout_prehom_index prod prod.ops is_equiv equiv
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definition Cpushout_universal {C D E : Precategory} {X : Category} (F : C ⇒ D) (G : C ⇒ E)
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(H : D ⇒ X) (K : E ⇒ X) (η : H ∘f F ≅ K ∘f G) :
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is_contr (Σ(L : Cpushout F G ⇒ X) (θ : L ∘f Cpushout_inl F G ≅ H × L ∘f Cpushout_inr F G ≅ K),
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Πs, natural_map (to_hom θ.2) (to_fun_ob G s) ∘ to_fun_hom L (class_of [DE (λ c, c) F G s]) ∘
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natural_map (to_inv θ.1) (to_fun_ob F s) = natural_map (to_hom η) s) :=
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begin
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fapply is_contr.mk,
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{ exact ⟨Cpushout_functor η, (Cpushout_functor_inl η, Cpushout_functor_inr η),
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Cpushout_functor_coh η⟩},
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intro v₁, induction v₁ with L v₂, induction v₂ with θ p, induction θ with θ₁ θ₂,
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fapply sigma_eq,
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{ esimp, apply eq_of_iso, symmetry, exact Cpushout_functor_unique η L θ₁ θ₂ p},
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fapply sigma_pathover,
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{ apply prod_pathover: esimp,
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{ apply iso_pathover,
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apply hom_pathover_functor_left_constant_right (precomposition_functor _ _),
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apply nat_trans_eq, intro d,
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xrewrite [↑[hom_of_eq], to_right_inv !eq_equiv_iso, ▸*],
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exact (ap010 natural_map (to_right_inverse θ₁) d)⁻¹},
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{ apply iso_pathover,
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apply hom_pathover_functor_left_constant_right (precomposition_functor _ _),
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apply nat_trans_eq, intro e,
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xrewrite [↑[hom_of_eq], to_right_inv !eq_equiv_iso, ▸*],
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exact (ap010 natural_map (to_right_inverse θ₂) e)⁻¹}},
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apply is_prop.elimo
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end
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/- Pushout of groupoids with a type of basepoints -/
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@ -416,7 +443,7 @@ namespace category
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Groupoid_paths (bpushout_hom_rel_index k F G) (bpushout_index_inv k F G)
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(bpushout_index_reverse k F G) (bpushout_index_li k F G) (bpushout_index_ri k F G)
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definition Groupoid_pushout [constructor] : Groupoid :=
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definition Gpushout [constructor] : Groupoid :=
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Groupoid_bpushout (λc, c) F G
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end
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@ -10,7 +10,7 @@ Adjoint functors, isomorphisms and equivalences have their own file
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import .basic function arity
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open eq functor trunc prod is_equiv iso equiv function is_trunc
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open eq functor trunc prod is_equiv iso equiv function is_trunc sigma
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namespace category
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variables {C D E : Precategory} {F : C ⇒ D} {G : D ⇒ C}
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@ -20,8 +20,6 @@ namespace category
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definition fully_faithful [class] (F : C ⇒ D) := Π(c c' : C), is_equiv (@(to_fun_hom F) c c')
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definition split_essentially_surjective [class] (F : C ⇒ D) := Π(d : D), Σ(c : C), F c ≅ d
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definition essentially_surjective [class] (F : C ⇒ D) := Π(d : D), ∃(c : C), F c ≅ d
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definition is_weak_equivalence [class] (F : C ⇒ D) :=
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fully_faithful F × essentially_surjective F
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definition is_equiv_of_fully_faithful [instance] (F : C ⇒ D)
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[H : fully_faithful F] (c c' : C) : is_equiv (@(to_fun_hom F) c c') :=
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@ -136,8 +134,9 @@ namespace category
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: is_prop (essentially_surjective F) :=
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by unfold essentially_surjective; exact _
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theorem is_prop_is_weak_equivalence [instance] (F : C ⇒ D) : is_prop (is_weak_equivalence F) :=
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by unfold is_weak_equivalence; exact _
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definition essentially_surjective_of_split_essentially_surjective [instance] (F : C ⇒ D)
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[H : split_essentially_surjective F] : essentially_surjective F :=
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λd, tr (H d)
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definition fully_faithful_equiv (F : C ⇒ D) : fully_faithful F ≃ (faithful F × full F) :=
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equiv_of_is_prop (λH, (faithful_of_fully_faithful F, full_of_fully_faithful F))
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@ -161,4 +160,36 @@ namespace category
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(λc', !is_embedding_equiv_is_injective)))
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-/
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definition fully_faithful_compose (G : D ⇒ E) (F : C ⇒ D) [fully_faithful G] [fully_faithful F] :
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fully_faithful (G ∘f F) :=
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λc c', is_equiv_compose (to_fun_hom G) (to_fun_hom F)
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definition full_compose (G : D ⇒ E) (F : C ⇒ D) [full G] [full F] : full (G ∘f F) :=
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λc c', is_surjective_compose (to_fun_hom G) (to_fun_hom F) _ _
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definition faithful_compose (G : D ⇒ E) (F : C ⇒ D) [H₁ : faithful G] [H₂ : faithful F] :
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faithful (G ∘f F) :=
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λc c' f f' p, H₂ (H₁ p)
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definition essentially_surjective_compose (G : D ⇒ E) (F : C ⇒ D) [H₁ : essentially_surjective G]
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[H₂ : essentially_surjective F] : essentially_surjective (G ∘f F) :=
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begin
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intro e,
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induction H₁ e with v, induction v with d p,
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induction H₂ d with w, induction w with c q,
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exact exists.intro c (to_fun_iso G q ⬝i p)
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end
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definition split_essentially_surjective_compose (G : D ⇒ E) (F : C ⇒ D)
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[H₁ : split_essentially_surjective G] [H₂ : split_essentially_surjective F]
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: split_essentially_surjective (G ∘f F) :=
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begin
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intro e, induction H₁ e with d p, induction H₂ d with c q,
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exact ⟨c, to_fun_iso G q ⬝i p⟩
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end
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/- we get the fact that the identity functor satisfies all these properties via the fact that it
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is an isomorphism -/
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end category
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@ -273,4 +273,29 @@ namespace functor
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ap010_assoc K (H ∘f G) F a],
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end
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definition hom_pathover_functor_left_functor_right {c₁ c₂ : C} {p : c₁ = c₂} (F G : C ⇒ D)
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{f₁ : F c₁ ⟶ G c₁} {f₂ : F c₂ ⟶ G c₂}
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(q : to_fun_hom G (hom_of_eq p) ∘ f₁ = f₂ ∘ to_fun_hom F (hom_of_eq p)) : f₁ =[p] f₂ :=
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hom_pathover (hom_whisker_right _ (respect_hom_of_eq G _)⁻¹ ⬝ q ⬝
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hom_whisker_left _ (respect_hom_of_eq F _))
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definition hom_pathover_constant_left_functor_right {c₁ c₂ : C} {p : c₁ = c₂} {d : D} (F : C ⇒ D)
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{f₁ : d ⟶ F c₁} {f₂ : d ⟶ F c₂} (q : to_fun_hom F (hom_of_eq p) ∘ f₁ = f₂) : f₁ =[p] f₂ :=
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hom_pathover_constant_left (hom_whisker_right _ (respect_hom_of_eq F _)⁻¹ ⬝ q)
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definition hom_pathover_functor_left_constant_right {c₁ c₂ : C} {p : c₁ = c₂} {d : D} (F : C ⇒ D)
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{f₁ : F c₁ ⟶ d} {f₂ : F c₂ ⟶ d} (q : f₁ = f₂ ∘ to_fun_hom F (hom_of_eq p)) : f₁ =[p] f₂ :=
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hom_pathover_constant_right (q ⬝ hom_whisker_left _ (respect_hom_of_eq F _))
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definition hom_pathover_id_left_functor_right {c₁ c₂ : C} {p : c₁ = c₂} (F : C ⇒ C)
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{f₁ : c₁ ⟶ F c₁} {f₂ : c₂ ⟶ F c₂} (q : to_fun_hom F (hom_of_eq p) ∘ f₁ = f₂ ∘ hom_of_eq p) :
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f₁ =[p] f₂ :=
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hom_pathover_id_left (hom_whisker_right _ (respect_hom_of_eq F _)⁻¹ ⬝ q)
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definition hom_pathover_functor_left_id_right {c₁ c₂ : C} {p : c₁ = c₂} (F : C ⇒ C)
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{f₁ : F c₁ ⟶ c₁} {f₂ : F c₂ ⟶ c₂} (q : hom_of_eq p ∘ f₁ = f₂ ∘ to_fun_hom F (hom_of_eq p)) :
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f₁ =[p] f₂ :=
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hom_pathover_id_right (q ⬝ hom_whisker_left _ (respect_hom_of_eq F _))
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end functor
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@ -8,7 +8,7 @@ Functors which are equivalences or isomorphisms
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import .adjoint
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open eq functor iso prod nat_trans is_equiv equiv is_trunc
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open eq functor iso prod nat_trans is_equiv equiv is_trunc sigma.ops
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namespace category
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variables {C D : Precategory} {F : C ⇒ D} {G : D ⇒ C}
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@ -18,6 +18,9 @@ namespace category
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(is_iso_unit : is_iso η)
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(is_iso_counit : is_iso ε)
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definition is_weak_equivalence [class] (F : C ⇒ D) :=
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fully_faithful F × essentially_surjective F
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abbreviation inverse := @is_equivalence.G
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postfix ⁻¹ := inverse
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--a second notation for the inverse, which is not overloaded (there is no unicode superscript F)
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@ -33,8 +36,16 @@ namespace category
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(to_functor : C ⇒ D)
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(struct : is_isomorphism to_functor)
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structure weak_equivalence (C D : Precategory) :=
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mk' :: (intermediate : Precategory)
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(left_functor : intermediate ⇒ C)
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(right_functor : intermediate ⇒ D)
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[structl : is_weak_equivalence left_functor]
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[structr : is_weak_equivalence right_functor]
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infix ` ≃c `:25 := equivalence
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infix ` ≅c `:25 := isomorphism
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infix ` ≃w `:25 := weak_equivalence
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attribute equivalence.struct isomorphism.struct [instance] [priority 1500]
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attribute equivalence.to_functor isomorphism.to_functor [coercion]
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@ -51,8 +62,8 @@ namespace category
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definition iso_counit (F : C ⇒ D) [H : is_equivalence F] : F ∘f F⁻¹ᴱ ≅ 1 :=
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@(iso.mk _) !is_iso_counit
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definition split_essentially_surjective_of_is_equivalence (F : C ⇒ D)
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[H : is_equivalence F] : split_essentially_surjective F :=
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definition split_essentially_surjective_of_is_equivalence [instance] (F : C ⇒ D)
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[is_equivalence F] : split_essentially_surjective F :=
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begin
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intro d, fconstructor,
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{ exact F⁻¹ d},
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@ -131,6 +142,37 @@ namespace category
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equivalence.mk F is_equivalence.mk
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end
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section
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parameters {C D : Precategory} (F : C ⇒ D)
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[H₁ : fully_faithful F] [H₂ : split_essentially_surjective F]
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include H₁ H₂
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definition inverse_of_fully_faithful_of_split_essentially_surjective [constructor] : D ⇒ C :=
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begin
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fapply functor.mk,
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{ exact λd, (H₂ d).1},
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{ intro d d' g, apply (to_fun_hom F)⁻¹ᶠ, refine to_inv (H₂ d').2 ∘ g ∘ to_hom (H₂ d).2},
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{ intro d, apply inv_eq_of_eq, rewrite [id_left, respect_id, to_left_inverse]},
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{ intros d₁ d₂ d₃ g f, apply inv_eq_of_eq,
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rewrite [respect_comp, +right_inv (to_fun_hom F), +assoc', comp_inverse_cancel_left]}
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end
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definition is_equivalence_of_fully_faithful_of_split_essentially_surjective [constructor]
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: is_equivalence F :=
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begin
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fapply is_equivalence.mk,
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{ exact inverse_of_fully_faithful_of_split_essentially_surjective},
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{ fapply natural_iso.mk',
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{ intro c, esimp, apply reflect_iso F, exact (H₂ (F c)).2},
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intro c c' f, esimp, apply eq_of_fn_eq_fn' (to_fun_hom F),
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rewrite [+respect_comp, +right_inv (to_fun_hom F), comp_inverse_cancel_left]},
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{ fapply natural_iso.mk',
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{ intro c, esimp, exact (H₂ c).2},
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intro c c' f, esimp, rewrite [right_inv (to_fun_hom F), comp_inverse_cancel_left]}
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end
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end
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variables {C D E : Precategory} {F : C ⇒ D}
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--TODO: add variants
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@ -141,8 +183,8 @@ namespace category
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apply eq_inverse_of_comp_eq_id, apply counit_unit_eq
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end
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definition fully_faithful_of_is_equivalence (F : C ⇒ D) [H : is_equivalence F]
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: fully_faithful F :=
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definition fully_faithful_of_is_equivalence [instance] [constructor] (F : C ⇒ D)
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[H : is_equivalence F] : fully_faithful F :=
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begin
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intro c c',
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fapply adjointify,
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@ -178,10 +220,13 @@ namespace category
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[H : is_isomorphism F] : is_equiv (to_fun_ob F) :=
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pr2 H
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definition is_fully_faithful_of_is_isomorphism [instance] [unfold 4] (F : C ⇒ D)
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definition fully_faithful_of_is_isomorphism [unfold 4] (F : C ⇒ D)
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[H : is_isomorphism F] : fully_faithful F :=
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pr1 H
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section
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local attribute fully_faithful_of_is_isomorphism [instance]
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definition strict_inverse [constructor] (F : C ⇒ D) [H : is_isomorphism F] : D ⇒ C :=
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begin
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fapply functor.mk,
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@ -215,9 +260,10 @@ namespace category
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{ rewrite [adj], rewrite [▸*,respect_inv_of_eq F]},
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{ rewrite [adj,▸*,respect_hom_of_eq F]}},
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end
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end
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definition is_equivalence_of_is_isomorphism [instance] [constructor] (F : C ⇒ D) [H : is_isomorphism F]
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: is_equivalence F :=
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definition is_equivalence_of_is_isomorphism [instance] [constructor] (F : C ⇒ D)
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[is_isomorphism F] : is_equivalence F :=
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begin
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fapply is_equivalence.mk,
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{ apply F⁻¹ˢ},
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@ -369,6 +415,33 @@ namespace category
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: c ⟶ c' ≃ H c ⟶ H c' :=
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equiv.mk (to_fun_hom (isomorphism.to_functor H)) _
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/- weak equivalences -/
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theorem is_prop_is_weak_equivalence [instance] (F : C ⇒ D) : is_prop (is_weak_equivalence F) :=
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by unfold is_weak_equivalence; exact _
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definition is_weak_equivalence_of_is_equivalence [instance] (F : C ⇒ D) [is_equivalence F]
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: is_weak_equivalence F :=
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(_, _)
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definition fully_faithful_of_is_weak_equivalence.mk [instance] (F : C ⇒ D)
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[H : is_weak_equivalence F] : fully_faithful F :=
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pr1 H
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definition essentially_surjective_of_is_weak_equivalence.mk [instance] (F : C ⇒ D)
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[H : is_weak_equivalence F] : essentially_surjective F :=
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pr2 H
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definition is_weak_equivalence_compose (G : D ⇒ E) (F : C ⇒ D)
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[H : is_weak_equivalence G] [K : is_weak_equivalence F] : is_weak_equivalence (G ∘f F) :=
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(fully_faithful_compose G F, essentially_surjective_compose G F)
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definition weak_equivalence.mk [constructor] (F : C ⇒ D) (H : is_weak_equivalence F) : C ≃w D :=
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weak_equivalence.mk' C 1 F
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definition weak_equivalence.symm [unfold 3] : C ≃w D → D ≃w C
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| (@weak_equivalence.mk' _ _ X F₁ F₂ H₁ H₂) := weak_equivalence.mk' X F₂ F₁
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/- TODO
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definition is_equiv_isomorphism_of_eq [constructor] (C D : Precategory)
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: is_equiv (@isomorphism_of_eq C D) :=
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@ -401,6 +474,8 @@ namespace category
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definition is_equivalence_equiv_is_weak_equivalence [constructor] {C D : Category}
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(F : C ⇒ D) : is_equivalence F ≃ is_weak_equivalence F :=
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sorry
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-- weak_equivalence.trans
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-/
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||||
|
||||
|
||||
|
|
|
|||
|
|
@ -179,6 +179,13 @@ namespace iso
|
|||
variable {C}
|
||||
definition iso_eq {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' :=
|
||||
by (cases f; cases f'; apply (iso_mk_eq p))
|
||||
|
||||
definition iso_pathover {X : Type} {x₁ x₂ : X} {p : x₁ = x₂} {a : X → ob} {b : X → ob}
|
||||
{f₁ : a x₁ ≅ b x₁} {f₂ : a x₂ ≅ b x₂} (q : to_hom f₁ =[p] to_hom f₂) : f₁ =[p] f₂ :=
|
||||
begin
|
||||
cases f₁, cases f₂, esimp at q, induction q, apply pathover_idp_of_eq,
|
||||
exact ap (iso.mk _) !is_prop.elim
|
||||
end
|
||||
variable [C]
|
||||
|
||||
-- The structure for isomorphism can be characterized up to equivalence by a sigma type.
|
||||
|
|
@ -225,6 +232,41 @@ namespace iso
|
|||
p ▸ !iso.refl = iso_of_eq p :=
|
||||
by cases p; reflexivity
|
||||
|
||||
definition hom_pathover {X : Type} {x₁ x₂ : X} {p : x₁ = x₂} {a b : X → ob}
|
||||
{f₁ : a x₁ ⟶ b x₁} {f₂ : a x₂ ⟶ b x₂} (q : hom_of_eq (ap b p) ∘ f₁ = f₂ ∘ hom_of_eq (ap a p)) :
|
||||
f₁ =[p] f₂ :=
|
||||
begin
|
||||
induction p, apply pathover_idp_of_eq, exact !id_left⁻¹ ⬝ q ⬝ !id_right
|
||||
end
|
||||
|
||||
definition hom_pathover_constant_left {X : Type} {x₁ x₂ : X} {p : x₁ = x₂} {a : ob} {b : X → ob}
|
||||
{f₁ : a ⟶ b x₁} {f₂ : a ⟶ b x₂} (q : hom_of_eq (ap b p) ∘ f₁ = f₂) : f₁ =[p] f₂ :=
|
||||
hom_pathover (q ⬝ !id_right⁻¹ ⬝ ap (λx, _ ∘ hom_of_eq x) !ap_constant⁻¹)
|
||||
|
||||
definition hom_pathover_constant_right {X : Type} {x₁ x₂ : X} {p : x₁ = x₂} {a : X → ob} {b : ob}
|
||||
{f₁ : a x₁ ⟶ b} {f₂ : a x₂ ⟶ b} (q : f₁ = f₂ ∘ hom_of_eq (ap a p)) : f₁ =[p] f₂ :=
|
||||
hom_pathover (ap (λx, hom_of_eq x ∘ _) !ap_constant ⬝ !id_left ⬝ q)
|
||||
|
||||
definition hom_pathover_id_left {p : a = b} {c : ob → ob} {f₁ : a ⟶ c a} {f₂ : b ⟶ c b}
|
||||
(q : hom_of_eq (ap c p) ∘ f₁ = f₂ ∘ hom_of_eq p) : f₁ =[p] f₂ :=
|
||||
hom_pathover (q ⬝ ap (λx, _ ∘ hom_of_eq x) !ap_id⁻¹)
|
||||
|
||||
definition hom_pathover_id_right {p : a = b} {c : ob → ob} {f₁ : c a ⟶ a} {f₂ : c b ⟶ b}
|
||||
(q : hom_of_eq p ∘ f₁ = f₂ ∘ hom_of_eq (ap c p)) : f₁ =[p] f₂ :=
|
||||
hom_pathover (ap (λx, hom_of_eq x ∘ _) !ap_id ⬝ q)
|
||||
|
||||
definition hom_pathover_id_left_id_right {p : a = b} {f₁ : a ⟶ a} {f₂ : b ⟶ b}
|
||||
(q : hom_of_eq p ∘ f₁ = f₂ ∘ hom_of_eq p) : f₁ =[p] f₂ :=
|
||||
hom_pathover_id_left (ap (λx, hom_of_eq x ∘ _) !ap_id ⬝ q)
|
||||
|
||||
definition hom_pathover_id_left_constant_right {p : a = b} {f₁ : a ⟶ c} {f₂ : b ⟶ c}
|
||||
(q : f₁ = f₂ ∘ hom_of_eq p) : f₁ =[p] f₂ :=
|
||||
hom_pathover_constant_right (q ⬝ ap (λx, _ ∘ hom_of_eq x) !ap_id⁻¹)
|
||||
|
||||
definition hom_pathover_constant_left_id_right {p : a = b} {f₁ : c ⟶ a} {f₂ : c ⟶ b}
|
||||
(q : hom_of_eq p ∘ f₁ = f₂) : f₁ =[p] f₂ :=
|
||||
hom_pathover_constant_left (ap (λx, hom_of_eq x ∘ _) !ap_id ⬝ q)
|
||||
|
||||
section
|
||||
open funext
|
||||
variables {X : Type} {x y : X} {F G : X → ob}
|
||||
|
|
|
|||
|
|
@ -74,6 +74,14 @@ namespace category
|
|||
definition comp_id_eq_id_comp (f : hom a b) : f ∘ id = id ∘ f := !id_right ⬝ !id_left⁻¹
|
||||
definition id_comp_eq_comp_id (f : hom a b) : id ∘ f = f ∘ id := !id_left ⬝ !id_right⁻¹
|
||||
|
||||
definition hom_whisker_left (g : b ⟶ c) (p : f = f') : g ∘ f = g ∘ f' :=
|
||||
ap (λx, g ∘ x) p
|
||||
|
||||
definition hom_whisker_right (g : c ⟶ a) (p : f = f') : f ∘ g = f' ∘ g :=
|
||||
ap (λx, x ∘ g) p
|
||||
|
||||
/- many variants of hom_pathover are defined in .iso and .functor.basic -/
|
||||
|
||||
definition left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
|
||||
calc i = i ∘ id : by rewrite id_right
|
||||
... = id : by rewrite H
|
||||
|
|
|
|||
|
|
@ -151,7 +151,7 @@ Every file is in the folder [homotopy](homotopy/homotopy.md)
|
|||
- 8.4 (Fiber sequences and the long exact sequence): Mostly in [homotopy.chain_complex](homotopy/chain_complex.hlean), [homotopy.LES_of_homotopy_groups](homotopy/LES_of_homotopy_groups.hlean). Definitions 8.4.1 and 8.4.2 in [types.pointed](types/pointed.hlean), Corollary 8.4.8 in [homotopy.homotopy_group](homotopy/homotopy_group.hlean).
|
||||
- 8.5 (The Hopf fibration): [hit.pushout](hit/pushout.hlean) (Lemma 8.5.3), [hopf](homotopy/hopf.hlean) (The Hopf construction, Lemmas 8.5.5 and 8.5.7), [susp](homotopy/susp.hlean) (Definition 8.5.6), [circle](homotopy/circle.hlean) (multiplication on the circle, Lemma 8.5.8), [join](homotopy/join.hlean) (join is associative, Lemma 8.5.9), [complex_hopf](homotopy/complex_hopf.hlean) (the H-space structure on the circle and the complex Hopf fibration, i.e. Theorem 8.5.1), [sphere2](homotopy/sphere2.hlean) (Corollary 8.5.2)
|
||||
- 8.6 (The Freudenthal suspension theorem): [connectedness](homotopy/connectedness.hlean) (Lemma 8.6.1), [wedge](homotopy/wedge.hlean) (Wedge connectivity, Lemma 8.6.2). Corollary 8.6.14 is proven directly in [freudenthal](homotopy/freudenthal.hlean), however, we don't prove Theorem 8.6.4. Stability of iterated suspensions is also in [freudenthal](homotopy/freudenthal.hlean). The homotopy groups of spheres in this section are computed in [sphere2](homotopy/sphere2.hlean).
|
||||
- 8.7 (The van Kampen theorem): [vankampen](homotopy/vankampen.hlean) (the pushout of Groupoids is formalized in [algebra.category.constructions.pushout](algebra/category/constructions/pushout.hlean) with some definitions in [algebra.graph](algebra/graph.hlean))
|
||||
- 8.7 (The van Kampen theorem): [vankampen](homotopy/vankampen.hlean) (the pushout of Groupoids is formalized in [algebra.category.constructions.pushout](algebra/category/constructions/pushout.hlean), including the universal property of this pushout. Some preliminary definitions for this pushout are in [algebra.graph](algebra/graph.hlean))
|
||||
- 8.8 (Whitehead’s theorem and Whitehead’s principle): 8.8.1 and 8.8.2 at the bottom of [types.trunc](types/trunc.hlean), 8.8.3 in [homotopy_group](homotopy/homotopy_group.hlean). [Rest to be moved]
|
||||
- 8.9 (A general statement of the encode-decode method): [types.eq](types/eq.hlean).
|
||||
- 8.10 (Additional Results): Theorem 8.10.3 is formalized in [homotopy.EM](homotopy/EM.hlean).
|
||||
|
|
|
|||
|
|
@ -11,7 +11,7 @@ import hit.trunc types.equiv cubical.square
|
|||
|
||||
open equiv sigma sigma.ops eq trunc is_trunc pi is_equiv fiber prod
|
||||
|
||||
variables {A B : Type} (f : A → B) {b : B}
|
||||
variables {A B C : Type} (f : A → B) {b : B}
|
||||
|
||||
/- the image of a map is the (-1)-truncated fiber -/
|
||||
definition image' [constructor] (f : A → B) (b : B) : Type := ∥ fiber f b ∥
|
||||
|
|
@ -273,6 +273,33 @@ namespace function
|
|||
{ intro H, induction H with g p, reflexivity},
|
||||
end
|
||||
|
||||
definition is_embedding_compose (g : B → C) (f : A → B)
|
||||
(H₁ : is_embedding g) (H₂ : is_embedding f) : is_embedding (g ∘ f) :=
|
||||
begin
|
||||
intros, apply @(is_equiv.homotopy_closed (ap g ∘ ap f)),
|
||||
{ apply is_equiv_compose},
|
||||
symmetry, exact ap_compose g f
|
||||
end
|
||||
|
||||
definition is_surjective_compose (g : B → C) (f : A → B)
|
||||
(H₁ : is_surjective g) (H₂ : is_surjective f) : is_surjective (g ∘ f) :=
|
||||
begin
|
||||
intro c, induction H₁ c with b p, induction H₂ b with a q,
|
||||
exact image.mk a (ap g q ⬝ p)
|
||||
end
|
||||
|
||||
definition is_split_surjective_compose (g : B → C) (f : A → B)
|
||||
(H₁ : is_split_surjective g) (H₂ : is_split_surjective f) : is_split_surjective (g ∘ f) :=
|
||||
begin
|
||||
intro c, induction H₁ c with b p, induction H₂ b with a q,
|
||||
exact fiber.mk a (ap g q ⬝ p)
|
||||
end
|
||||
|
||||
definition is_injective_compose (g : B → C) (f : A → B)
|
||||
(H₁ : Π⦃b b'⦄, g b = g b' → b = b') (H₂ : Π⦃a a'⦄, f a = f a' → a = a')
|
||||
⦃a a' : A⦄ (p : g (f a) = g (f a')) : a = a' :=
|
||||
H₂ (H₁ p)
|
||||
|
||||
/-
|
||||
The definitions
|
||||
is_surjective_of_is_equiv
|
||||
|
|
|
|||
|
|
@ -131,7 +131,7 @@ namespace EM
|
|||
begin
|
||||
intro x, induction x using EM.elim,
|
||||
{ exact Point X},
|
||||
{ note p := e⁻¹ᵉ g, exact p},
|
||||
{ exact e⁻¹ᵉ g},
|
||||
{ exact inv_preserve_binary e concat mul r g h}
|
||||
end
|
||||
|
||||
|
|
|
|||
|
|
@ -314,6 +314,7 @@ namespace susp
|
|||
{ reflexivity}
|
||||
end
|
||||
|
||||
-- TODO: rename to psusp_adjoint_loop (also in above lemmas)
|
||||
definition susp_adjoint_loop [constructor] (X Y : Type*) : psusp X →* Y ≃ X →* Ω Y :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
|
|
|
|||
|
|
@ -8,7 +8,7 @@ import hit.pushout hit.prop_trunc algebra.category.constructions.pushout
|
|||
algebra.category.constructions.fundamental_groupoid algebra.category.functor.equivalence
|
||||
|
||||
open eq pushout category functor sum iso paths set_quotient is_trunc trunc pi quotient
|
||||
is_equiv fiber equiv
|
||||
is_equiv fiber equiv function
|
||||
|
||||
namespace pushout
|
||||
section
|
||||
|
|
@ -16,12 +16,14 @@ namespace pushout
|
|||
parameters {S TL : Type.{u}} -- do we want to put these in different universe levels?
|
||||
{BL : Type.{v}} {TR : Type.{w}} (k : S → TL) (f : TL → BL) (g : TL → TR)
|
||||
[ksurj : is_surjective k]
|
||||
include ksurj
|
||||
|
||||
definition pushout_of_sum [unfold 6] (x : BL + TR) : pushout f g :=
|
||||
quotient.class_of _ x
|
||||
|
||||
local notation `F` := fundamental_groupoid_functor f
|
||||
local notation `G` := fundamental_groupoid_functor g
|
||||
include ksurj
|
||||
|
||||
local notation `F` := Π₁⇒ f
|
||||
local notation `G` := Π₁⇒ g
|
||||
local notation `C` := Groupoid_bpushout k F G
|
||||
local notation `R` := bpushout_prehom_index k F G
|
||||
local notation `Q` := bpushout_hom_rel_index k F G
|
||||
|
|
@ -321,6 +323,8 @@ namespace pushout
|
|||
trunc 0 (pushout_of_sum x = pushout_of_sum y) ≃ @hom C _ x y :=
|
||||
pushout_teq_equiv x (pushout_of_sum y)
|
||||
|
||||
--Groupoid_pushout k F G
|
||||
|
||||
definition decode_point_comp [unfold 8] {x₁ x₂ x₃ : BL + TR}
|
||||
(c₂ : @hom C _ x₂ x₃) (c₁ : @hom C _ x₁ x₂) :
|
||||
decode_point (c₂ ∘ c₁) = tconcat (decode_point c₁) (decode_point c₂) :=
|
||||
|
|
@ -330,7 +334,7 @@ namespace pushout
|
|||
apply decode_list_append
|
||||
end
|
||||
|
||||
definition vankampen_functor [constructor] : C ⇒ fundamental_groupoid (pushout f g) :=
|
||||
definition vankampen_functor [constructor] : C ⇒ Π₁ (pushout f g) :=
|
||||
begin
|
||||
fconstructor,
|
||||
{ exact pushout_of_sum},
|
||||
|
|
@ -362,5 +366,14 @@ namespace pushout
|
|||
{ exact essentially_surjective_vankampen_functor}
|
||||
end
|
||||
|
||||
definition fundamental_groupoid_bpushout [constructor] :
|
||||
Groupoid_bpushout k (Π₁⇒ f) (Π₁⇒ g) ≃w Π₁ (pushout f g) :=
|
||||
weak_equivalence.mk vankampen_functor is_weak_equivalence_vankampen_functor
|
||||
|
||||
end
|
||||
|
||||
definition fundamental_groupoid_pushout [constructor] {TL BL TR : Type}
|
||||
(f : TL → BL) (g : TL → TR) : Groupoid_bpushout (@id TL) (Π₁⇒ f) (Π₁⇒ g) ≃w Π₁ (pushout f g) :=
|
||||
fundamental_groupoid_bpushout (@id TL) f g
|
||||
|
||||
end pushout
|
||||
|
|
|
|||
|
|
@ -276,7 +276,7 @@ namespace pointed
|
|||
end
|
||||
|
||||
prefix `Ω→`:(max+5) := ap1
|
||||
notation `Ω→[`:95 n:0 `] `:0 f:95 := apn n f
|
||||
notation `Ω→[`:95 n:0 `]`:0 := apn n
|
||||
|
||||
/- categorical properties of pointed homotopies -/
|
||||
|
||||
|
|
|
|||
|
|
@ -191,6 +191,7 @@ namespace sigma
|
|||
: bc =[p] ⟨p ▸ bc.1, p ▸D bc.2⟩ :=
|
||||
by induction p; induction bc; apply idpo
|
||||
|
||||
-- TODO: interchange sigma_pathover and sigma_pathover'
|
||||
definition sigma_pathover (p : a = a') (u : Σ(b : B a), C a b) (v : Σ(b : B a'), C a' b)
|
||||
(r : u.1 =[p] v.1) (s : u.2 =[apd011 C p r] v.2) : u =[p] v :=
|
||||
begin
|
||||
|
|
@ -198,6 +199,21 @@ namespace sigma
|
|||
esimp [apd011] at s, induction s using idp_rec_on, apply idpo
|
||||
end
|
||||
|
||||
definition sigma_pathover' (p : a = a') (u : Σ(b : B a), C a b) (v : Σ(b : B a'), C a' b)
|
||||
(r : u.1 =[p] v.1) (s : pathover (λx, C x.1 x.2) u.2 (sigma_eq p r) v.2) : u =[p] v :=
|
||||
begin
|
||||
induction u, induction v, esimp at *, induction r,
|
||||
induction s using idp_rec_on, apply idpo
|
||||
end
|
||||
|
||||
definition sigma_pathover_nondep {B : Type} {C : A → B → Type} (p : a = a')
|
||||
(u : Σ(b : B), C a b) (v : Σ(b : B), C a' b)
|
||||
(r : u.1 = v.1) (s : pathover (λx, C (prod.pr1 x) (prod.pr2 x)) u.2 (prod.prod_eq p r) v.2) : u =[p] v :=
|
||||
begin
|
||||
induction p, induction u, induction v, esimp at *, induction r,
|
||||
induction s using idp_rec_on, apply idpo
|
||||
end
|
||||
|
||||
/-
|
||||
TODO:
|
||||
* define the projections from the type u =[p] v
|
||||
|
|
|
|||
|
|
@ -717,6 +717,7 @@ namespace trunc
|
|||
tconcat (loop_ptrunc_pequiv n A p) (loop_ptrunc_pequiv n A q) :=
|
||||
encode_con p q
|
||||
|
||||
-- rename
|
||||
definition iterated_loop_ptrunc_pequiv (n : ℕ₋₂) (k : ℕ) (A : Type*) :
|
||||
Ω[k] (ptrunc (n+k) A) ≃* ptrunc n (Ω[k] A) :=
|
||||
begin
|
||||
|
|
|
|||
Loading…
Reference in a new issue