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* cleaned up univalent_subcategory.hlean

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* removed duplicates from move_to_lib.hlean
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Jonas Frey 2017-09-14 15:36:35 -04:00 committed by Floris van Doorn
parent 593efa1bf2
commit f89edc5403
2 changed files with 20 additions and 221 deletions
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199
move_to_lib.hlean
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@ -710,205 +710,6 @@ namespace misc
end misc end misc
namespace category
definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group :=
begin
fapply precategory.mk,
{ exact λG H, G →g H },
{ exact _ },
{ exact λG H K ψ φ, ψ ∘g φ },
{ exact λG, gid G },
{ intros, apply homomorphism_eq, esimp },
{ intros, apply homomorphism_eq, esimp },
{ intros, apply homomorphism_eq, esimp }
end
definition precategory_AbGroup.{u} [instance] [constructor] : precategory.{u+1 u} AbGroup :=
begin
fapply precategory.mk,
{ exact λG H, G →g H },
{ exact _ },
{ exact λG H K ψ φ, ψ ∘g φ },
{ exact λG, gid G },
{ intros, apply homomorphism_eq, esimp },
{ intros, apply homomorphism_eq, esimp },
{ intros, apply homomorphism_eq, esimp }
end
open iso
definition Group_is_iso_of_is_equiv {G H : Group} (φ : G →g H) (H : is_equiv (group_fun φ)) :
is_iso φ :=
begin
fconstructor,
{ exact (isomorphism.mk φ H)⁻¹ᵍ },
{ apply homomorphism_eq, rexact left_inv φ },
{ apply homomorphism_eq, rexact right_inv φ }
end
definition Group_is_equiv_of_is_iso {G H : Group} (φ : G ⟶ H) (Hφ : is_iso φ) :
is_equiv (group_fun φ) :=
begin
fapply adjointify,
{ exact group_fun φ⁻¹ʰ },
{ note p := right_inverse φ, exact ap010 group_fun p },
{ note p := left_inverse φ, exact ap010 group_fun p }
end
definition Group_iso_equiv (G H : Group) : (G ≅ H) ≃ (G ≃g H) :=
begin
fapply equiv.MK,
{ intro φ, induction φ with φ φi, constructor, exact Group_is_equiv_of_is_iso φ _ },
{ intro v, induction v with φ φe, constructor, exact Group_is_iso_of_is_equiv φ _ },
{ intro v, induction v with φ φe, apply isomorphism_eq, reflexivity },
{ intro φ, induction φ with φ φi, apply iso_eq, reflexivity }
end
definition Group_props.{u} {A : Type.{u}} (v : (A → A → A) × (A → A) × A) : Prop.{u} :=
begin
induction v with m v, induction v with i o,
fapply trunctype.mk,
{ exact is_set A × (Πa, m a o = a) × (Πa, m o a = a) × (Πa b c, m (m a b) c = m a (m b c)) ×
(Πa, m (i a) a = o) },
{ apply is_trunc_of_imp_is_trunc, intro v, induction v with H v,
have is_prop (Πa, m a o = a), from _,
have is_prop (Πa, m o a = a), from _,
have is_prop (Πa b c, m (m a b) c = m a (m b c)), from _,
have is_prop (Πa, m (i a) a = o), from _,
apply is_trunc_prod }
end
definition Group.sigma_char2.{u} : Group.{u} ≃
Σ(A : Type.{u}) (v : (A → A → A) × (A → A) × A), Group_props v :=
begin
fapply equiv.MK,
{ intro G, refine ⟨G, _⟩, induction G with G g, induction g with m s ma o om mo i mi,
repeat (fconstructor; do 2 try assumption), },
{ intro v, induction v with x v, induction v with y v, repeat induction y with x y,
repeat induction v with x v, constructor, fconstructor, repeat assumption },
{ intro v, induction v with x v, induction v with y v, repeat induction y with x y,
repeat induction v with x v, reflexivity },
{ intro v, repeat induction v with x v, reflexivity },
end
open is_trunc
section
local attribute group.to_has_mul group.to_has_inv [coercion]
theorem inv_eq_of_mul_eq {A : Type} (G H : group A) (p : @mul A G ~2 @mul A H) :
@inv A G ~ @inv A H :=
begin
have foo : Π(g : A), @inv A G g = (@inv A G g * g) * @inv A H g,
from λg, !mul_inv_cancel_right⁻¹,
cases G with Gs Gm Gh1 G1 Gh2 Gh3 Gi Gh4,
cases H with Hs Hm Hh1 H1 Hh2 Hh3 Hi Hh4,
change Gi ~ Hi, intro g, have p' : Gm ~2 Hm, from p,
calc
Gi g = Hm (Hm (Gi g) g) (Hi g) : foo
... = Hm (Gm (Gi g) g) (Hi g) : by rewrite p'
... = Hm G1 (Hi g) : by rewrite Gh4
... = Gm G1 (Hi g) : by rewrite p'
... = Hi g : Gh2
end
theorem one_eq_of_mul_eq {A : Type} (G H : group A)
(p : @mul A (group.to_has_mul G) ~2 @mul A (group.to_has_mul H)) :
@one A (group.to_has_one G) = @one A (group.to_has_one H) :=
begin
cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4,
cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4,
exact (Hh2 G1)⁻¹ ⬝ (p H1 G1)⁻¹ ⬝ Gh3 H1,
end
end
open prod.ops
definition group_of_Group_props.{u} {A : Type.{u}} {m : A → A → A} {i : A → A} {o : A}
(H : Group_props (m, (i, o))) : group A :=
⦃group, mul := m, inv := i, one := o, is_set_carrier := H.1,
mul_one := H.2.1, one_mul := H.2.2.1, mul_assoc := H.2.2.2.1, mul_left_inv := H.2.2.2.2⦄
theorem Group_eq_equiv_lemma2 {A : Type} {m m' : A → A → A} {i i' : A → A} {o o' : A}
(H : Group_props (m, (i, o))) (H' : Group_props (m', (i', o'))) :
(m, (i, o)) = (m', (i', o')) ≃ (m ~2 m') :=
begin
have is_set A, from pr1 H,
apply equiv_of_is_prop,
{ intro p, exact apd100 (eq_pr1 p)},
{ intro p, apply prod_eq (eq_of_homotopy2 p),
apply prod_eq: esimp [Group_props] at *; esimp,
{ apply eq_of_homotopy,
exact inv_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p },
{ exact one_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p }}
end
open sigma.ops
theorem Group_eq_equiv_lemma {G H : Group}
(p : (Group.sigma_char2 G).1 = (Group.sigma_char2 H).1) :
((Group.sigma_char2 G).2 =[p] (Group.sigma_char2 H).2) ≃
(is_mul_hom (equiv_of_eq (proof p qed : Group.carrier G = Group.carrier H))) :=
begin
refine !sigma_pathover_equiv_of_is_prop ⬝e _,
induction G with G g, induction H with H h,
esimp [Group.sigma_char2] at p, induction p,
refine !pathover_idp ⬝e _,
induction g with s m ma o om mo i mi, induction h with σ μ μa ε εμ με ι μι,
exact Group_eq_equiv_lemma2 (Group.sigma_char2 (Group.mk G (group.mk s m ma o om mo i mi))).2.2
(Group.sigma_char2 (Group.mk G (group.mk σ μ μa ε εμ με ι μι))).2.2
end
definition isomorphism.sigma_char (G H : Group) : (G ≃g H) ≃ Σ(e : G ≃ H), is_mul_hom e :=
begin
fapply equiv.MK,
{ intro φ, exact ⟨equiv_of_isomorphism φ, to_respect_mul φ⟩ },
{ intro v, induction v with e p, exact isomorphism_of_equiv e p },
{ intro v, induction v with e p, induction e, reflexivity },
{ intro φ, induction φ with φ H, induction φ, reflexivity },
end
definition Group_eq_equiv (G H : Group) : G = H ≃ (G ≃g H) :=
begin
refine (eq_equiv_fn_eq_of_equiv Group.sigma_char2 G H) ⬝e _,
refine !sigma_eq_equiv ⬝e _,
refine sigma_equiv_sigma_right Group_eq_equiv_lemma ⬝e _,
transitivity (Σ(e : (Group.sigma_char2 G).1 ≃ (Group.sigma_char2 H).1),
@is_mul_hom _ _ _ _ (to_fun e)), apply sigma_ua,
exact !isomorphism.sigma_char⁻¹ᵉ
end
definition to_fun_Group_eq_equiv {G H : Group} (p : G = H)
: Group_eq_equiv G H p ~ isomorphism_of_eq p :=
begin
induction p, reflexivity
end
definition Group_eq2 {G H : Group} {p q : G = H}
(r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q :=
begin
apply eq_of_fn_eq_fn (Group_eq_equiv G H),
apply isomorphism_eq,
intro g, refine to_fun_Group_eq_equiv p g ⬝ r g ⬝ (to_fun_Group_eq_equiv q g)⁻¹,
end
definition Group_eq_equiv_Group_iso (G₁ G₂ : Group) : G₁ = G₂ ≃ G₁ ≅ G₂ :=
Group_eq_equiv G₁ G₂ ⬝e (Group_iso_equiv G₁ G₂)⁻¹ᵉ
definition category_Group.{u} : category Group.{u} :=
category.mk precategory_Group
begin
intro G H,
apply is_equiv_of_equiv_of_homotopy (Group_eq_equiv_Group_iso G H),
intro p, induction p, fapply iso_eq, apply homomorphism_eq, reflexivity
end
definition category_AbGroup : category AbGroup :=
category.mk precategory_AbGroup sorry
definition Grp.{u} [constructor] : Category := category.Mk Group.{u} category_Group
definition AbGrp [constructor] : Category := category.Mk AbGroup category_AbGroup
end category
namespace sphere namespace sphere
-- definition constant_sphere_map_sphere {n m : ℕ} (H : n < m) (f : S n →* S m) : -- definition constant_sphere_map_sphere {n m : ℕ} (H : n < m) (f : S n →* S m) :

40
univalent_subcategory.hlean
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@ -4,20 +4,20 @@ open eq pointed sigma is_equiv equiv fiber algebra group is_trunc function prod
namespace category namespace category
section univ_functor section univ_subcat
parameters {C : Precategory} {D : Category} (F : functor C D) (p : is_embedding F) (q : fully_faithful F) parameters {C : Precategory} {D : Category} (F : functor C D) (p : is_embedding F) (q : fully_faithful F)
variables {a b : carrier C} variables {a b : carrier C}
include p q include p q
definition ab_eq_equiv_iso : a = b ≃ a ≅ b := definition eq_equiv_iso_of_fully_faithful : a = b ≃ a ≅ b :=
equiv.mk !ap !p -- a = b ≃ F a = F b equiv.mk !ap !p -- a = b ≃ F a = F b
⬝e equiv.mk iso_of_eq !iso_of_path_equiv -- F a = F b ≃ F a ≅ F b ⬝e equiv.mk iso_of_eq !iso_of_path_equiv -- F a = F b ≃ F a ≅ F b
⬝e equiv.symm !iso_equiv_F_iso_F -- F a ≅ F b ≃ a ≅ b ⬝e equiv.symm !iso_equiv_F_iso_F -- F a ≅ F b ≃ a ≅ b
definition ab_equiv_homot_iso_of_eq : @ab_eq_equiv_iso a b ~ iso_of_eq := definition eq_equiv_iso_of_fully_faithful_homot : @eq_equiv_iso_of_fully_faithful a b ~ iso_of_eq :=
begin begin
intro r, intro r,
esimp [ab_eq_equiv_iso], esimp [eq_equiv_iso_of_fully_faithful],
refine _ ⬝ left_inv (iso_equiv_F_iso_F F _ _) _, refine _ ⬝ left_inv (iso_equiv_F_iso_F F _ _) _,
apply ap (inv (to_fun !iso_equiv_F_iso_F)), apply ap (inv (to_fun !iso_equiv_F_iso_F)),
apply symm, apply symm,
@ -25,12 +25,12 @@ section univ_functor
apply respect_refl apply respect_refl
end end
definition univ_domain : is_univalent C := definition is_univalent_domain_of_fully_faithful_embedding : is_univalent C :=
begin begin
intros, intros,
apply homotopy_closed ab_eq_equiv_iso ab_equiv_homot_iso_of_eq apply homotopy_closed eq_equiv_iso_of_fully_faithful eq_equiv_iso_of_fully_faithful_homot
end end
end univ_functor end univ_subcat
definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group := definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group :=
begin begin
@ -165,14 +165,14 @@ repeat (fconstructor; assumption), assumption, intro b,
open sigma.ops open sigma.ops
definition sigma_sigma_prod {A} {B C : A→Type} : (Σa, B a × C a) ≃ Σ p : (Σa, B a), C p.1 := definition sigma_prod_equiv_sigma_sigma {A} {B C : A→Type} : (Σa, B a × C a) ≃ Σ p : (Σa, B a), C p.1 :=
sigma_equiv_sigma_right (λa, !sigma.equiv_prod⁻¹ᵉ) ⬝e !sigma_assoc_equiv sigma_equiv_sigma_right (λa, !sigma.equiv_prod⁻¹ᵉ) ⬝e !sigma_assoc_equiv
definition ab_group_group_comm (A : Type) : ab_group A ≃ Σ (g : group A), ∀ a b : A, a * b = b * a := definition ab_group_equiv_group_comm (A : Type) : ab_group A ≃ Σ (g : group A), ∀ a b : A, a * b = b * a :=
begin begin
refine !ab_group.sigma_char ⬝e _, refine !ab_group.sigma_char ⬝e _,
refine sigma_equiv_sigma_right AbGroup_Group_props ⬝e _, refine sigma_equiv_sigma_right AbGroup_Group_props ⬝e _,
refine sigma_sigma_prod ⬝e _, refine sigma_prod_equiv_sigma_sigma ⬝e _,
apply equiv.symm, apply sigma_equiv_sigma !group.sigma_char, intros, apply equiv.symm, apply sigma_equiv_sigma !group.sigma_char, intros,
induction a, reflexivity induction a, reflexivity
end end
@ -264,9 +264,7 @@ induction p,
definition to_fun_Group_eq_equiv {G H : Group} (p : G = H) definition to_fun_Group_eq_equiv {G H : Group} (p : G = H)
: Group_eq_equiv G H p ~ isomorphism_of_eq p := : Group_eq_equiv G H p ~ isomorphism_of_eq p :=
begin begin induction p, reflexivity end
induction p, reflexivity
end
definition Group_eq2 {G H : Group} {p q : G = H} definition Group_eq2 {G H : Group} {p q : G = H}
(r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q := (r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q :=
@ -314,17 +312,17 @@ induction a', fconstructor, intros, apply sigma_eq,
apply is_prop.elim, intros, apply is_prop.elim, intros, apply is_prop.elim apply is_prop.elim, intros, apply is_prop.elim, intros, apply is_prop.elim
end end
definition th (A : Type) : definition ab_group_to_group_homot (A : Type) :
@ab_group_to_group A ~ group_comm_to_group A ∘ ab_group_group_comm A := @ab_group_to_group A ~ group_comm_to_group A ∘ ab_group_equiv_group_comm A :=
begin intro, induction x, reflexivity end begin intro, induction x, reflexivity end
definition is_embedding_ab_group_to_group (A : Type) : is_embedding (@ab_group_to_group A) := definition is_embedding_ab_group_to_group (A : Type) : is_embedding (@ab_group_to_group A) :=
begin begin
apply is_embedding_homotopy_closed_rev (th A), apply is_embedding_compose, apply is_embedding_homotopy_closed_rev (ab_group_to_group_homot A), apply is_embedding_compose,
exact is_embedding_group_comm_to_group A, apply is_embedding_of_is_equiv exact is_embedding_group_comm_to_group A, apply is_embedding_of_is_equiv
end end
definition sigma_emb {A} {B C : A → Type} {f : Π a, B a → C a} definition is_embedding_total_of_is_embedding_fiber {A} {B C : A → Type} {f : Π a, B a → C a}
: (∀ a, is_embedding (f a)) → is_embedding (total f) := : (∀ a, is_embedding (f a)) → is_embedding (total f) :=
begin begin
intro e, fapply is_embedding_of_is_prop_fiber, intro p, induction p with a c, intro e, fapply is_embedding_of_is_prop_fiber, intro p, induction p with a c,
@ -335,22 +333,22 @@ apply is_prop_fiber_of_is_embedding,
apply is_trunc_equiv_closed -1 (H⁻¹ᵉ), apply is_trunc_equiv_closed -1 (H⁻¹ᵉ),
end end
definition h2 : AbGroup_to_Group ~ Group_sigma⁻¹ ∘ total ab_group_to_group ∘ AbGroup_sigma := definition AbGroup_to_Group_homot : AbGroup_to_Group ~ Group_sigma⁻¹ ∘ total ab_group_to_group ∘ AbGroup_sigma :=
begin intro g, induction g, reflexivity end begin intro g, induction g, reflexivity end
definition is_embedding_AbGroup_to_Group : is_embedding AbGroup_to_Group := definition is_embedding_AbGroup_to_Group : is_embedding AbGroup_to_Group :=
begin begin
apply is_embedding_homotopy_closed_rev h2, apply is_embedding_homotopy_closed_rev AbGroup_to_Group_homot,
apply is_embedding_compose, apply is_embedding_compose,
apply is_embedding_of_is_equiv, apply is_embedding_of_is_equiv,
apply is_embedding_compose, apply is_embedding_compose,
apply sigma_emb is_embedding_ab_group_to_group, apply is_embedding_total_of_is_embedding_fiber is_embedding_ab_group_to_group,
apply is_embedding_of_is_equiv apply is_embedding_of_is_equiv
end end
definition is_univalent_AbGroup : is_univalent precategory_AbGroup := definition is_univalent_AbGroup : is_univalent precategory_AbGroup :=
begin begin
apply univ_domain AbGroup_to_Group is_embedding_AbGroup_to_Group, intros, apply is_equiv_id apply is_univalent_domain_of_fully_faithful_embedding AbGroup_to_Group is_embedding_AbGroup_to_Group, intros, apply is_equiv_id
end end
definition category_AbGroup : category AbGroup := category.mk precategory_AbGroup is_univalent_AbGroup definition category_AbGroup : category AbGroup := category.mk precategory_AbGroup is_univalent_AbGroup