* cleaned up univalent_subcategory.hlean
* removed duplicates from move_to_lib.hlean
This commit is contained in:
Jonas Frey
Floris van Doorn
parent
593efa1bf2
commit
f89edc5403
2 changed files with 20 additions and 221 deletions
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@ -710,205 +710,6 @@ namespace misc
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end misc
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namespace category
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definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group :=
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begin
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fapply precategory.mk,
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{ exact λG H, G →g H },
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{ exact _ },
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{ exact λG H K ψ φ, ψ ∘g φ },
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{ exact λG, gid G },
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{ intros, apply homomorphism_eq, esimp },
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{ intros, apply homomorphism_eq, esimp },
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{ intros, apply homomorphism_eq, esimp }
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end
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definition precategory_AbGroup.{u} [instance] [constructor] : precategory.{u+1 u} AbGroup :=
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begin
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fapply precategory.mk,
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{ exact λG H, G →g H },
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{ exact _ },
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{ exact λG H K ψ φ, ψ ∘g φ },
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{ exact λG, gid G },
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{ intros, apply homomorphism_eq, esimp },
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{ intros, apply homomorphism_eq, esimp },
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{ intros, apply homomorphism_eq, esimp }
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end
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open iso
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definition Group_is_iso_of_is_equiv {G H : Group} (φ : G →g H) (H : is_equiv (group_fun φ)) :
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is_iso φ :=
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begin
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fconstructor,
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{ exact (isomorphism.mk φ H)⁻¹ᵍ },
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{ apply homomorphism_eq, rexact left_inv φ },
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{ apply homomorphism_eq, rexact right_inv φ }
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end
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definition Group_is_equiv_of_is_iso {G H : Group} (φ : G ⟶ H) (Hφ : is_iso φ) :
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is_equiv (group_fun φ) :=
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begin
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fapply adjointify,
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{ exact group_fun φ⁻¹ʰ },
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{ note p := right_inverse φ, exact ap010 group_fun p },
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{ note p := left_inverse φ, exact ap010 group_fun p }
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end
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definition Group_iso_equiv (G H : Group) : (G ≅ H) ≃ (G ≃g H) :=
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begin
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fapply equiv.MK,
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{ intro φ, induction φ with φ φi, constructor, exact Group_is_equiv_of_is_iso φ _ },
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{ intro v, induction v with φ φe, constructor, exact Group_is_iso_of_is_equiv φ _ },
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{ intro v, induction v with φ φe, apply isomorphism_eq, reflexivity },
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{ intro φ, induction φ with φ φi, apply iso_eq, reflexivity }
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end
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definition Group_props.{u} {A : Type.{u}} (v : (A → A → A) × (A → A) × A) : Prop.{u} :=
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begin
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induction v with m v, induction v with i o,
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fapply trunctype.mk,
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{ 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)) ×
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(Πa, m (i a) a = o) },
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{ apply is_trunc_of_imp_is_trunc, intro v, induction v with H v,
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have is_prop (Πa, m a o = a), from _,
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have is_prop (Πa, m o a = a), from _,
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have is_prop (Πa b c, m (m a b) c = m a (m b c)), from _,
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have is_prop (Πa, m (i a) a = o), from _,
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apply is_trunc_prod }
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end
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definition Group.sigma_char2.{u} : Group.{u} ≃
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Σ(A : Type.{u}) (v : (A → A → A) × (A → A) × A), Group_props v :=
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begin
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fapply equiv.MK,
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{ intro G, refine ⟨G, _⟩, induction G with G g, induction g with m s ma o om mo i mi,
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repeat (fconstructor; do 2 try assumption), },
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{ intro v, induction v with x v, induction v with y v, repeat induction y with x y,
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repeat induction v with x v, constructor, fconstructor, repeat assumption },
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{ intro v, induction v with x v, induction v with y v, repeat induction y with x y,
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repeat induction v with x v, reflexivity },
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{ intro v, repeat induction v with x v, reflexivity },
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end
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open is_trunc
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section
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local attribute group.to_has_mul group.to_has_inv [coercion]
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theorem inv_eq_of_mul_eq {A : Type} (G H : group A) (p : @mul A G ~2 @mul A H) :
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@inv A G ~ @inv A H :=
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begin
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have foo : Π(g : A), @inv A G g = (@inv A G g * g) * @inv A H g,
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from λg, !mul_inv_cancel_right⁻¹,
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cases G with Gs Gm Gh1 G1 Gh2 Gh3 Gi Gh4,
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cases H with Hs Hm Hh1 H1 Hh2 Hh3 Hi Hh4,
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change Gi ~ Hi, intro g, have p' : Gm ~2 Hm, from p,
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calc
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Gi g = Hm (Hm (Gi g) g) (Hi g) : foo
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... = Hm (Gm (Gi g) g) (Hi g) : by rewrite p'
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... = Hm G1 (Hi g) : by rewrite Gh4
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... = Gm G1 (Hi g) : by rewrite p'
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... = Hi g : Gh2
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end
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theorem one_eq_of_mul_eq {A : Type} (G H : group A)
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(p : @mul A (group.to_has_mul G) ~2 @mul A (group.to_has_mul H)) :
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@one A (group.to_has_one G) = @one A (group.to_has_one H) :=
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begin
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cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4,
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cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4,
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exact (Hh2 G1)⁻¹ ⬝ (p H1 G1)⁻¹ ⬝ Gh3 H1,
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end
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end
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open prod.ops
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definition group_of_Group_props.{u} {A : Type.{u}} {m : A → A → A} {i : A → A} {o : A}
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(H : Group_props (m, (i, o))) : group A :=
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⦃group, mul := m, inv := i, one := o, is_set_carrier := H.1,
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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⦄
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theorem Group_eq_equiv_lemma2 {A : Type} {m m' : A → A → A} {i i' : A → A} {o o' : A}
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(H : Group_props (m, (i, o))) (H' : Group_props (m', (i', o'))) :
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(m, (i, o)) = (m', (i', o')) ≃ (m ~2 m') :=
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begin
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have is_set A, from pr1 H,
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apply equiv_of_is_prop,
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{ intro p, exact apd100 (eq_pr1 p)},
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{ intro p, apply prod_eq (eq_of_homotopy2 p),
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apply prod_eq: esimp [Group_props] at *; esimp,
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{ apply eq_of_homotopy,
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exact inv_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p },
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{ exact one_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p }}
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end
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open sigma.ops
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theorem Group_eq_equiv_lemma {G H : Group}
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(p : (Group.sigma_char2 G).1 = (Group.sigma_char2 H).1) :
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((Group.sigma_char2 G).2 =[p] (Group.sigma_char2 H).2) ≃
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(is_mul_hom (equiv_of_eq (proof p qed : Group.carrier G = Group.carrier H))) :=
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begin
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refine !sigma_pathover_equiv_of_is_prop ⬝e _,
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induction G with G g, induction H with H h,
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esimp [Group.sigma_char2] at p, induction p,
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refine !pathover_idp ⬝e _,
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induction g with s m ma o om mo i mi, induction h with σ μ μa ε εμ με ι μι,
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exact Group_eq_equiv_lemma2 (Group.sigma_char2 (Group.mk G (group.mk s m ma o om mo i mi))).2.2
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(Group.sigma_char2 (Group.mk G (group.mk σ μ μa ε εμ με ι μι))).2.2
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end
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definition isomorphism.sigma_char (G H : Group) : (G ≃g H) ≃ Σ(e : G ≃ H), is_mul_hom e :=
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begin
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fapply equiv.MK,
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{ intro φ, exact ⟨equiv_of_isomorphism φ, to_respect_mul φ⟩ },
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{ intro v, induction v with e p, exact isomorphism_of_equiv e p },
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{ intro v, induction v with e p, induction e, reflexivity },
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{ intro φ, induction φ with φ H, induction φ, reflexivity },
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end
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definition Group_eq_equiv (G H : Group) : G = H ≃ (G ≃g H) :=
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begin
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refine (eq_equiv_fn_eq_of_equiv Group.sigma_char2 G H) ⬝e _,
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refine !sigma_eq_equiv ⬝e _,
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refine sigma_equiv_sigma_right Group_eq_equiv_lemma ⬝e _,
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transitivity (Σ(e : (Group.sigma_char2 G).1 ≃ (Group.sigma_char2 H).1),
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@is_mul_hom _ _ _ _ (to_fun e)), apply sigma_ua,
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exact !isomorphism.sigma_char⁻¹ᵉ
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end
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definition to_fun_Group_eq_equiv {G H : Group} (p : G = H)
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: Group_eq_equiv G H p ~ isomorphism_of_eq p :=
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begin
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induction p, reflexivity
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end
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definition Group_eq2 {G H : Group} {p q : G = H}
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(r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q :=
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begin
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apply eq_of_fn_eq_fn (Group_eq_equiv G H),
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apply isomorphism_eq,
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intro g, refine to_fun_Group_eq_equiv p g ⬝ r g ⬝ (to_fun_Group_eq_equiv q g)⁻¹,
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end
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definition Group_eq_equiv_Group_iso (G₁ G₂ : Group) : G₁ = G₂ ≃ G₁ ≅ G₂ :=
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Group_eq_equiv G₁ G₂ ⬝e (Group_iso_equiv G₁ G₂)⁻¹ᵉ
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definition category_Group.{u} : category Group.{u} :=
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category.mk precategory_Group
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begin
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intro G H,
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apply is_equiv_of_equiv_of_homotopy (Group_eq_equiv_Group_iso G H),
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intro p, induction p, fapply iso_eq, apply homomorphism_eq, reflexivity
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end
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definition category_AbGroup : category AbGroup :=
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category.mk precategory_AbGroup sorry
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definition Grp.{u} [constructor] : Category := category.Mk Group.{u} category_Group
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definition AbGrp [constructor] : Category := category.Mk AbGroup category_AbGroup
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end category
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namespace sphere
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-- definition constant_sphere_map_sphere {n m : ℕ} (H : n < m) (f : S n →* S m) :
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@ -4,20 +4,20 @@ open eq pointed sigma is_equiv equiv fiber algebra group is_trunc function prod
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namespace category
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section univ_functor
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section univ_subcat
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parameters {C : Precategory} {D : Category} (F : functor C D) (p : is_embedding F) (q : fully_faithful F)
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variables {a b : carrier C}
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include p q
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definition ab_eq_equiv_iso : a = b ≃ a ≅ b :=
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definition eq_equiv_iso_of_fully_faithful : a = b ≃ a ≅ b :=
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equiv.mk !ap !p -- a = b ≃ F a = F b
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⬝e equiv.mk iso_of_eq !iso_of_path_equiv -- F a = F b ≃ F a ≅ F b
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⬝e equiv.symm !iso_equiv_F_iso_F -- F a ≅ F b ≃ a ≅ b
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definition ab_equiv_homot_iso_of_eq : @ab_eq_equiv_iso a b ~ iso_of_eq :=
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definition eq_equiv_iso_of_fully_faithful_homot : @eq_equiv_iso_of_fully_faithful a b ~ iso_of_eq :=
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begin
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intro r,
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esimp [ab_eq_equiv_iso],
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esimp [eq_equiv_iso_of_fully_faithful],
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refine _ ⬝ left_inv (iso_equiv_F_iso_F F _ _) _,
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apply ap (inv (to_fun !iso_equiv_F_iso_F)),
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apply symm,
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@ -25,12 +25,12 @@ section univ_functor
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apply respect_refl
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end
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definition univ_domain : is_univalent C :=
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definition is_univalent_domain_of_fully_faithful_embedding : is_univalent C :=
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begin
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intros,
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apply homotopy_closed ab_eq_equiv_iso ab_equiv_homot_iso_of_eq
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apply homotopy_closed eq_equiv_iso_of_fully_faithful eq_equiv_iso_of_fully_faithful_homot
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end
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end univ_functor
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end univ_subcat
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definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group :=
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begin
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@ -165,14 +165,14 @@ repeat (fconstructor; assumption), assumption, intro b,
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open sigma.ops
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definition sigma_sigma_prod {A} {B C : A→Type} : (Σa, B a × C a) ≃ Σ p : (Σa, B a), C p.1 :=
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definition sigma_prod_equiv_sigma_sigma {A} {B C : A→Type} : (Σa, B a × C a) ≃ Σ p : (Σa, B a), C p.1 :=
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sigma_equiv_sigma_right (λa, !sigma.equiv_prod⁻¹ᵉ) ⬝e !sigma_assoc_equiv
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definition ab_group_group_comm (A : Type) : ab_group A ≃ Σ (g : group A), ∀ a b : A, a * b = b * a :=
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definition ab_group_equiv_group_comm (A : Type) : ab_group A ≃ Σ (g : group A), ∀ a b : A, a * b = b * a :=
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begin
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refine !ab_group.sigma_char ⬝e _,
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refine sigma_equiv_sigma_right AbGroup_Group_props ⬝e _,
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refine sigma_sigma_prod ⬝e _,
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refine sigma_prod_equiv_sigma_sigma ⬝e _,
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apply equiv.symm, apply sigma_equiv_sigma !group.sigma_char, intros,
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induction a, reflexivity
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end
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@ -264,9 +264,7 @@ induction p,
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definition to_fun_Group_eq_equiv {G H : Group} (p : G = H)
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: Group_eq_equiv G H p ~ isomorphism_of_eq p :=
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begin
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induction p, reflexivity
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end
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begin induction p, reflexivity end
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definition Group_eq2 {G H : Group} {p q : G = H}
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(r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q :=
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@ -299,7 +297,7 @@ apply is_trunc_equiv_closed, apply equiv.symm,
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apply group.sigma_char
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end
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definition ab_group_to_group (A : Type) (g : ab_group A) : group A := _
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definition ab_group_to_group (A : Type) (g : ab_group A) : group A := _
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definition group_comm_to_group (A : Type) : (Σ g : group A, ∀ (a b : A), a*b = b*a) → group A := pr1
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@ -314,17 +312,17 @@ induction a', fconstructor, intros, apply sigma_eq,
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apply is_prop.elim, intros, apply is_prop.elim, intros, apply is_prop.elim
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end
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definition th (A : Type) :
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@ab_group_to_group A ~ group_comm_to_group A ∘ ab_group_group_comm A :=
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definition ab_group_to_group_homot (A : Type) :
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@ab_group_to_group A ~ group_comm_to_group A ∘ ab_group_equiv_group_comm A :=
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begin intro, induction x, reflexivity end
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definition is_embedding_ab_group_to_group (A : Type) : is_embedding (@ab_group_to_group A) :=
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begin
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apply is_embedding_homotopy_closed_rev (th A), apply is_embedding_compose,
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apply is_embedding_homotopy_closed_rev (ab_group_to_group_homot A), apply is_embedding_compose,
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exact is_embedding_group_comm_to_group A, apply is_embedding_of_is_equiv
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end
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definition sigma_emb {A} {B C : A → Type} {f : Π a, B a → C a}
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definition is_embedding_total_of_is_embedding_fiber {A} {B C : A → Type} {f : Π a, B a → C a}
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: (∀ a, is_embedding (f a)) → is_embedding (total f) :=
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begin
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intro e, fapply is_embedding_of_is_prop_fiber, intro p, induction p with a c,
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@ -335,22 +333,22 @@ apply is_prop_fiber_of_is_embedding,
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apply is_trunc_equiv_closed -1 (H⁻¹ᵉ),
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end
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definition h2 : AbGroup_to_Group ~ Group_sigma⁻¹ ∘ total ab_group_to_group ∘ AbGroup_sigma :=
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definition AbGroup_to_Group_homot : AbGroup_to_Group ~ Group_sigma⁻¹ ∘ total ab_group_to_group ∘ AbGroup_sigma :=
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begin intro g, induction g, reflexivity end
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definition is_embedding_AbGroup_to_Group : is_embedding AbGroup_to_Group :=
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begin
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apply is_embedding_homotopy_closed_rev h2,
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apply is_embedding_homotopy_closed_rev AbGroup_to_Group_homot,
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apply is_embedding_compose,
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apply is_embedding_of_is_equiv,
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apply is_embedding_compose,
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apply sigma_emb is_embedding_ab_group_to_group,
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apply is_embedding_total_of_is_embedding_fiber is_embedding_ab_group_to_group,
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apply is_embedding_of_is_equiv
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end
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definition is_univalent_AbGroup : is_univalent precategory_AbGroup :=
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begin
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apply univ_domain AbGroup_to_Group is_embedding_AbGroup_to_Group, intros, apply is_equiv_id
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apply is_univalent_domain_of_fully_faithful_embedding AbGroup_to_Group is_embedding_AbGroup_to_Group, intros, apply is_equiv_id
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end
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definition category_AbGroup : category AbGroup := category.mk precategory_AbGroup is_univalent_AbGroup
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