feat(hott): use group isomorphisms instead of equality between groups
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
8db4676c46
commit
1135d80266
4 changed files with 303 additions and 44 deletions
260
hott/algebra/group_theory.hlean
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260
hott/algebra/group_theory.hlean
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Basic group theory
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This file will be rewritten in the future, when we develop are more systematic notation for
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describing homomorphisms
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-/
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import algebra.category.category algebra.hott
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open eq algebra pointed function is_trunc pi category equiv is_equiv
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set_option class.force_new true
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namespace group
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definition pointed_Group [instance] (G : Group) : pointed G := pointed.mk one
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definition pType_of_Group [reducible] (G : Group) : Type* := pointed.mk' G
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definition Set_of_Group (G : Group) : Set := trunctype.mk G _
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definition Group_of_CommGroup [coercion] [constructor] (G : CommGroup) : Group :=
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Group.mk G _
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definition comm_group_Group_of_CommGroup [instance] [constructor] (G : CommGroup)
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: comm_group (Group_of_CommGroup G) :=
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begin esimp, exact _ end
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definition group_pType_of_Group [instance] (G : Group) : group (pType_of_Group G) :=
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Group.struct G
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/- group homomorphisms -/
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definition is_homomorphism [class] [reducible]
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{G₁ G₂ : Type} [group G₁] [group G₂] (φ : G₁ → G₂) : Type :=
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Π(g h : G₁), φ (g * h) = φ g * φ h
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section
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variables {G G₁ G₂ G₃ : Type} {g h : G₁} (ψ : G₂ → G₃) {φ₁ φ₂ : G₁ → G₂} (φ : G₁ → G₂)
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[group G] [group G₁] [group G₂] [group G₃]
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[is_homomorphism ψ] [is_homomorphism φ₁] [is_homomorphism φ₂] [is_homomorphism φ]
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definition respect_mul /- φ -/ : Π(g h : G₁), φ (g * h) = φ g * φ h :=
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by assumption
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theorem respect_one /- φ -/ : φ 1 = 1 :=
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mul.right_cancel
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(calc
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φ 1 * φ 1 = φ (1 * 1) : respect_mul φ
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... = φ 1 : ap φ !one_mul
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... = 1 * φ 1 : one_mul)
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theorem respect_inv /- φ -/ (g : G₁) : φ g⁻¹ = (φ g)⁻¹ :=
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eq_inv_of_mul_eq_one (!respect_mul⁻¹ ⬝ ap φ !mul.left_inv ⬝ !respect_one)
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definition is_embedding_homomorphism /- φ -/ (H : Π{g}, φ g = 1 → g = 1) : is_embedding φ :=
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begin
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apply function.is_embedding_of_is_injective,
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intro g g' p,
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apply eq_of_mul_inv_eq_one,
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apply H,
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refine !respect_mul ⬝ _,
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rewrite [respect_inv φ, p],
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apply mul.right_inv
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end
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definition is_homomorphism_compose {ψ : G₂ → G₃} {φ : G₁ → G₂}
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(H1 : is_homomorphism ψ) (H2 : is_homomorphism φ) : is_homomorphism (ψ ∘ φ) :=
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λg h, ap ψ !respect_mul ⬝ !respect_mul
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definition is_homomorphism_id (G : Type) [group G] : is_homomorphism (@id G) :=
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λg h, idp
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end
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structure homomorphism (G₁ G₂ : Group) : Type :=
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(φ : G₁ → G₂)
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(p : is_homomorphism φ)
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infix ` →g `:55 := homomorphism
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definition group_fun [unfold 3] [coercion] := @homomorphism.φ
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definition homomorphism.struct [instance] [priority 2000] {G₁ G₂ : Group} (φ : G₁ →g G₂)
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: is_homomorphism φ :=
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homomorphism.p φ
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variables {G G₁ G₂ G₃ : Group} {g h : G₁} {ψ : G₂ →g G₃} {φ₁ φ₂ : G₁ →g G₂} (φ : G₁ →g G₂)
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definition to_respect_mul /- φ -/ (g h : G₁) : φ (g * h) = φ g * φ h :=
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respect_mul φ g h
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theorem to_respect_one /- φ -/ : φ 1 = 1 :=
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respect_one φ
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theorem to_respect_inv /- φ -/ (g : G₁) : φ g⁻¹ = (φ g)⁻¹ :=
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respect_inv φ g
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definition to_is_embedding_homomorphism /- φ -/ (H : Π{g}, φ g = 1 → g = 1) : is_embedding φ :=
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is_embedding_homomorphism φ @H
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definition is_set_homomorphism [instance] (G₁ G₂ : Group) : is_set (G₁ →g G₂) :=
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begin
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have H : G₁ →g G₂ ≃ Σ(f : G₁ → G₂), Π(g₁ g₂ : G₁), f (g₁ * g₂) = f g₁ * f g₂,
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begin
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fapply equiv.MK,
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{ intro φ, induction φ, constructor, assumption},
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{ intro v, induction v, constructor, assumption},
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{ intro v, induction v, reflexivity},
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{ intro φ, induction φ, reflexivity}
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end,
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apply is_trunc_equiv_closed_rev, exact H
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end
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local attribute group_pType_of_Group pointed.mk' [reducible]
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definition pmap_of_homomorphism [constructor] /- φ -/ : pType_of_Group G₁ →* pType_of_Group G₂ :=
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pmap.mk φ (respect_one φ)
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definition homomorphism_eq (p : group_fun φ₁ ~ group_fun φ₂) : φ₁ = φ₂ :=
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begin
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induction φ₁ with φ₁ q₁, induction φ₂ with φ₂ q₂, esimp at p, induction p,
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exact ap (homomorphism.mk φ₁) !is_prop.elim
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end
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/- categorical structure of groups + homomorphisms -/
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definition homomorphism_compose [constructor] [trans] (ψ : G₂ →g G₃) (φ : G₁ →g G₂) : G₁ →g G₃ :=
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homomorphism.mk (ψ ∘ φ) (is_homomorphism_compose _ _)
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definition homomorphism_id [constructor] [refl] (G : Group) : G →g G :=
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homomorphism.mk (@id G) (is_homomorphism_id G)
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infixr ` ∘g `:75 := homomorphism_compose
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notation 1 := homomorphism_id _
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structure isomorphism (A B : Group) :=
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(to_hom : A →g B)
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(is_equiv_to_hom : is_equiv to_hom)
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infix ` ≃g `:25 := isomorphism
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attribute isomorphism.to_hom [coercion]
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attribute isomorphism.is_equiv_to_hom [instance]
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definition equiv_of_isomorphism [constructor] (φ : G₁ ≃g G₂) : G₁ ≃ G₂ :=
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equiv.mk φ _
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definition isomorphism_of_equiv [constructor] (φ : G₁ ≃ G₂)
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(p : Πg₁ g₂, φ (g₁ * g₂) = φ g₁ * φ g₂) : G₁ ≃g G₂ :=
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isomorphism.mk (homomorphism.mk φ p) !to_is_equiv
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definition eq_of_isomorphism {G₁ G₂ : Group} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
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Group_eq (equiv_of_isomorphism φ) (respect_mul φ)
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definition isomorphism_of_eq {G₁ G₂ : Group} (φ : G₁ = G₂) : G₁ ≃g G₂ :=
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isomorphism_of_equiv (equiv_of_eq (ap Group.carrier φ))
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begin intros, induction φ, reflexivity end
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definition to_ginv [constructor] (φ : G₁ ≃g G₂) : G₂ →g G₁ :=
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homomorphism.mk φ⁻¹
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abstract begin
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intro g₁ g₂, apply eq_of_fn_eq_fn' φ,
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rewrite [respect_mul φ, +right_inv φ]
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end end
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definition isomorphism.refl [refl] [constructor] (G : Group) : G ≃g G :=
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isomorphism.mk 1 !is_equiv_id
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definition isomorphism.symm [symm] [constructor] (φ : G₁ ≃g G₂) : G₂ ≃g G₁ :=
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isomorphism.mk (to_ginv φ) !is_equiv_inv
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definition isomorphism.trans [trans] [constructor] (φ : G₁ ≃g G₂) (ψ : G₂ ≃g G₃) : G₁ ≃g G₃ :=
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isomorphism.mk (ψ ∘g φ) !is_equiv_compose
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postfix `⁻¹ᵍ`:(max + 1) := isomorphism.symm
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infixl ` ⬝g `:75 := isomorphism.trans
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-- TODO
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-- definition Group_univalence (G₁ G₂ : Group) : (G₁ ≃g G₂) ≃ (G₁ = G₂) :=
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-- begin
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-- fapply equiv.MK,
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-- { exact eq_of_isomorphism},
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-- { intro p, apply transport _ p, reflexivity},
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-- { intro p, induction p, esimp, },
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-- { }
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-- end
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/- category of groups -/
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definition precategory_group [constructor] : precategory Group :=
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precategory.mk homomorphism
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@homomorphism_compose
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@homomorphism_id
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(λG₁ G₂ G₃ G₄ φ₃ φ₂ φ₁, homomorphism_eq (λg, idp))
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(λG₁ G₂ φ, homomorphism_eq (λg, idp))
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(λG₁ G₂ φ, homomorphism_eq (λg, idp))
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-- TODO
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-- definition category_group : category Group :=
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-- category.mk precategory_group
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-- begin
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-- intro G₁ G₂,
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-- fapply adjointify,
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-- { intro φ, fapply Group_eq, },
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-- { },
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-- { }
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-- end
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/- given an equivalence A ≃ B we can transport a group structure on A to a group structure on B -/
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section
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parameters {A B : Type} (f : A ≃ B) [group A]
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definition group_equiv_mul (b b' : B) : B := f (f⁻¹ᶠ b * f⁻¹ᶠ b')
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definition group_equiv_one : B := f one
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definition group_equiv_inv (b : B) : B := f (f⁻¹ᶠ b)⁻¹
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local infix * := group_equiv_mul
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local postfix ^ := group_equiv_inv
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local notation 1 := group_equiv_one
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theorem group_equiv_mul_assoc (b₁ b₂ b₃ : B) : (b₁ * b₂) * b₃ = b₁ * (b₂ * b₃) :=
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by rewrite [↑group_equiv_mul, +left_inv f, mul.assoc]
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theorem group_equiv_one_mul (b : B) : 1 * b = b :=
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by rewrite [↑group_equiv_mul, ↑group_equiv_one, left_inv f, one_mul, right_inv f]
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theorem group_equiv_mul_one (b : B) : b * 1 = b :=
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by rewrite [↑group_equiv_mul, ↑group_equiv_one, left_inv f, mul_one, right_inv f]
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theorem group_equiv_mul_left_inv (b : B) : b^ * b = 1 :=
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by rewrite [↑group_equiv_mul, ↑group_equiv_one, ↑group_equiv_inv,
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+left_inv f, mul.left_inv]
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definition group_equiv_closed : group B :=
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⦃group,
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mul := group_equiv_mul,
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mul_assoc := group_equiv_mul_assoc,
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one := group_equiv_one,
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one_mul := group_equiv_one_mul,
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mul_one := group_equiv_mul_one,
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inv := group_equiv_inv,
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mul_left_inv := group_equiv_mul_left_inv,
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is_set_carrier := is_trunc_equiv_closed 0 f⦄
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end
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definition trivial_group_of_is_contr (G : Group) [H : is_contr G] : G ≃g G0 :=
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begin
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fapply isomorphism_of_equiv,
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{ apply equiv_unit_of_is_contr},
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{ intros, reflexivity}
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end
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definition trivial_group_of_is_contr' (G : Group) [H : is_contr G] : G = G0 :=
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eq_of_isomorphism (trivial_group_of_is_contr G)
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end group
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@ -8,7 +8,7 @@ homotopy groups of a pointed space
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import .trunc_group types.trunc .group_theory
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open nat eq pointed trunc is_trunc algebra
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open nat eq pointed trunc is_trunc algebra group function equiv unit
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namespace eq
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@ -64,8 +64,7 @@ namespace eq
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exact loopn_pequiv_loopn k (pequiv_of_eq begin rewrite [trunc_index.zero_add] end)
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end
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open equiv unit
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theorem trivial_homotopy_of_is_set (A : Type*) [H : is_set A] (n : ℕ) : πg[n+1] A = G0 :=
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theorem trivial_homotopy_of_is_set (A : Type*) [H : is_set A] (n : ℕ) : πg[n+1] A ≃g G0 :=
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begin
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apply trivial_group_of_is_contr,
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apply is_trunc_trunc_of_is_trunc,
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@ -80,32 +79,15 @@ namespace eq
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definition ghomotopy_group_succ_out (A : Type*) (n : ℕ) : πg[n +1] A = π₁ Ω[n] A := idp
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definition ghomotopy_group_succ_in (A : Type*) (n : ℕ) : πg[succ n +1] A = πg[n +1] Ω A :=
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definition ghomotopy_group_succ_in (A : Type*) (n : ℕ) : πg[succ n +1] A ≃g πg[n +1] Ω A :=
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begin
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fapply Group_eq,
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fapply isomorphism_of_equiv,
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{ apply equiv_of_eq, exact ap (ptrunc 0) (loop_space_succ_eq_in A (succ n))},
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{ exact abstract [irreducible] begin refine trunc.rec _, intro p, refine trunc.rec _, intro q,
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rewrite [▸*,-+tr_eq_cast_ap, +trunc_transport], refine !trunc_transport ⬝ _, apply ap tr,
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apply loop_space_succ_eq_in_concat end end},
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end
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definition homotopy_group_add (A : Type*) (n m : ℕ) : πg[n+m +1] A = πg[n +1] Ω[m] A :=
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begin
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revert A, induction m with m IH: intro A,
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{ reflexivity},
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{ esimp [iterated_ploop_space, nat.add], refine !ghomotopy_group_succ_in ⬝ _, refine !IH ⬝ _,
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exact ap (ghomotopy_group n) !loop_space_succ_eq_in⁻¹}
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end
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theorem trivial_homotopy_add_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ)
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(H : is_set (Ω[n] A)) : πg[m+n+1] A = G0 :=
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!homotopy_group_add ⬝ !trivial_homotopy_of_is_set
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theorem trivial_homotopy_le_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ) (H1 : n ≤ m)
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(H2 : is_set (Ω[n] A)) : πg[m+1] A = G0 :=
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obtain (k : ℕ) (p : n + k = m), from le.elim H1,
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ap (λx, πg[x+1] A) (p⁻¹ ⬝ add.comm n k) ⬝ trivial_homotopy_add_of_is_set_loop_space k H2
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definition phomotopy_group_functor [constructor] (n : ℕ) {A B : Type*} (f : A →* B)
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: π*[n] A →* π*[n] B :=
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ptrunc_functor 0 (apn n f)
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@ -140,7 +122,39 @@ namespace eq
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apply ap tr, apply apn_con
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end
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open group function
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definition homotopy_group_homomorphism [constructor] (n : ℕ) {A B : Type*} (f : A →* B)
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: πg[n+1] A →g πg[n+1] B :=
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begin
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fconstructor,
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{ exact phomotopy_group_functor (n+1) f},
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{ apply phomotopy_group_functor_mul}
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end
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definition homotopy_group_isomorphism_of_pequiv [constructor] (n : ℕ) {A B : Type*} (f : A ≃* B)
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: πg[n+1] A ≃g πg[n+1] B :=
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begin
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apply isomorphism.mk (homotopy_group_homomorphism n f),
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esimp, apply is_equiv_trunc_functor, apply is_equiv_apn,
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end
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definition homotopy_group_add (A : Type*) (n m : ℕ) : πg[n+m +1] A ≃g πg[n +1] Ω[m] A :=
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begin
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revert A, induction m with m IH: intro A,
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{ reflexivity},
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{ esimp [iterated_ploop_space, nat.add], refine !ghomotopy_group_succ_in ⬝g _, refine !IH ⬝g _,
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apply homotopy_group_isomorphism_of_pequiv,
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exact pequiv_of_eq !loop_space_succ_eq_in⁻¹}
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end
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theorem trivial_homotopy_add_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ)
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(H : is_set (Ω[n] A)) : πg[m+n+1] A ≃g G0 :=
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!homotopy_group_add ⬝g !trivial_homotopy_of_is_set
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theorem trivial_homotopy_le_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ) (H1 : n ≤ m)
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(H2 : is_set (Ω[n] A)) : πg[m+1] A ≃g G0 :=
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obtain (k : ℕ) (p : n + k = m), from le.elim H1,
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isomorphism_of_eq (ap (λx, πg[x+1] A) (p⁻¹ ⬝ add.comm n k)) ⬝g
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trivial_homotopy_add_of_is_set_loop_space k H2
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/- some homomorphisms -/
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@ -162,14 +176,6 @@ namespace eq
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exact ap tr !con_inv
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end
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definition homotopy_group_homomorphism [constructor] (n : ℕ) {A B : Type*} (f : A →* B)
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: πg[n+1] A →g πg[n+1] B :=
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begin
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fconstructor,
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{ exact phomotopy_group_functor (n+1) f},
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{ apply phomotopy_group_functor_mul}
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end
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notation `π→g[`:95 n:0 ` +1] `:0 f:95 := homotopy_group_homomorphism n f
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end eq
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@ -77,11 +77,4 @@ namespace algebra
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(resp_mul : Π(g h : G), f (g * h) = f g * f h) : G = H :=
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Group_eq_of_eq (ua f) (λg h, !cast_ua ⬝ resp_mul g h ⬝ ap011 mul !cast_ua⁻¹ !cast_ua⁻¹)
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definition trivial_group_of_is_contr (G : Group) [H : is_contr G] : G = G0 :=
|
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begin
|
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fapply Group_eq,
|
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{ apply equiv_unit_of_is_contr},
|
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{ intros, reflexivity}
|
||||
end
|
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end algebra
|
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|
|
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@ -7,8 +7,8 @@ Declaration of the circle
|
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-/
|
||||
|
||||
import .sphere
|
||||
import types.bool types.int.hott types.equiv
|
||||
import algebra.homotopy_group algebra.hott .connectedness
|
||||
import types.int.hott
|
||||
import algebra.homotopy_group .connectedness
|
||||
|
||||
open eq susp bool sphere_index is_equiv equiv is_trunc is_conn pi algebra
|
||||
|
||||
|
|
@ -293,7 +293,7 @@ namespace circle
|
|||
preserve_binary_of_inv_preserve base_eq_base_equiv concat (@add ℤ _) decode_add p q
|
||||
|
||||
--the carrier of π₁(S¹) is the set-truncation of base = base.
|
||||
open algebra trunc
|
||||
open algebra trunc group
|
||||
|
||||
definition fg_carrier_equiv_int : π[1](S¹.) ≃ ℤ :=
|
||||
trunc_equiv_trunc 0 base_eq_base_equiv ⬝e @(trunc_equiv 0 ℤ) proof _ qed
|
||||
|
|
@ -301,16 +301,16 @@ namespace circle
|
|||
definition con_comm_base (p q : base = base) : p ⬝ q = q ⬝ p :=
|
||||
eq_of_fn_eq_fn base_eq_base_equiv (by esimp;rewrite [+encode_con,add.comm])
|
||||
|
||||
definition fundamental_group_of_circle : π₁(S¹.) = gℤ :=
|
||||
definition fundamental_group_of_circle : π₁(S¹.) ≃g gℤ :=
|
||||
begin
|
||||
apply (Group_eq fg_carrier_equiv_int),
|
||||
apply (isomorphism_of_equiv fg_carrier_equiv_int),
|
||||
intros g h,
|
||||
induction g with g', induction h with h',
|
||||
apply encode_con,
|
||||
end
|
||||
|
||||
open nat
|
||||
definition homotopy_group_of_circle (n : ℕ) : πg[n+1 +1] S¹. = G0 :=
|
||||
definition homotopy_group_of_circle (n : ℕ) : πg[n+1 +1] S¹. ≃g G0 :=
|
||||
begin
|
||||
refine @trivial_homotopy_add_of_is_set_loop_space S¹. 1 n _,
|
||||
apply is_trunc_equiv_closed_rev, apply base_eq_base_equiv
|
||||
|
|
|
|||
Loading…
Reference in a new issue