Product of homotopies

In this file, we introduce definitions for the product of homotopies. We show that the products of relative homotopies are still relative homotopies. Finally, we specialize to the case of path homotopies, and provide the definition for the product of path classes. We show various lemmas associated with these products, such as the fact that path products commute with path composition, and that projection is the inverse of products.

Definitions

General homotopies

• ContinuousMap.Homotopy.pi homotopies: Let f and g be a family of functions indexed on I, such that for each i ∈ I, fᵢ and gᵢ are maps from A to Xᵢ. Let homotopies be a family of homotopies from fᵢ to gᵢ for each i. Then Homotopy.pi homotopies is the canonical homotopy from ∏ f to ∏ g, where ∏ f is the product map from A to Πi, Xᵢ, and similarly for ∏ g.

• ContinuousMap.HomotopyRel.pi homotopies: Same as ContinuousMap.Homotopy.pi, but all homotopies are done relative to some set S ⊆ A.

• ContinuousMap.Homotopy.prod F G is the product of homotopies F and G, where F is a homotopy between f₀ and f₁, G is a homotopy between g₀ and g₁. The result F × G is a homotopy between (f₀ × g₀) and (f₁ × g₁). Again, all homotopies are done relative to S.

• ContinuousMap.HomotopyRel.prod F G: Same as ContinuousMap.Homotopy.prod, but all homotopies are done relative to some set S ⊆ A.

Path products

• Path.Homotopic.pi The product of a family of path classes, where a path class is an equivalence class of paths up to path homotopy.

• Path.Homotopic.prod The product of two path classes.

def ContinuousMap.HomotopyRel.pi {I : Type u_1} {A : Type u_2} {X : I → Type u_3} [(i : I) → TopologicalSpace (X i)] [TopologicalSpace A] {f : (i : I) → C(A, X i)} {g : (i : I) → C(A, X i)} {S : Set A} (homotopies : (i : I) → (f i).HomotopyRel (g i) S) : (ContinuousMap.pi f).HomotopyRel (ContinuousMap.pi g) S

The relative product homotopy of homotopies between functions f and g

Equations

• ContinuousMap.HomotopyRel.pi homotopies = { toHomotopy := ContinuousMap.Homotopy.pi fun (i : I) => (homotopies i).toHomotopy, prop' := ⋯ }

@[simp]

theorem ContinuousMap.HomotopyRel.pi_apply {I : Type u_1} {A : Type u_2} {X : I → Type u_3} [(i : I) → TopologicalSpace (X i)] [TopologicalSpace A] {f : (i : I) → C(A, X i)} {g : (i : I) → C(A, X i)} {S : Set A} (homotopies : (i : I) → (f i).HomotopyRel (g i) S) (a : ↑unitInterval × A) (i : I) : (ContinuousMap.HomotopyRel.pi homotopies) a i = (homotopies i) a

@[simp]

theorem ContinuousMap.HomotopyRel.pi_toFun {I : Type u_1} {A : Type u_2} {X : I → Type u_3} [(i : I) → TopologicalSpace (X i)] [TopologicalSpace A] {f : (i : I) → C(A, X i)} {g : (i : I) → C(A, X i)} {S : Set A} (homotopies : (i : I) → (f i).HomotopyRel (g i) S) (a : ↑unitInterval × A) (i : I) : (ContinuousMap.HomotopyRel.pi homotopies) a i = (homotopies i) a

def ContinuousMap.Homotopy.prod {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ : C(A, α)} {f₁ : C(A, α)} {g₀ : C(A, β)} {g₁ : C(A, β)} (F : f₀.Homotopy f₁) (G : g₀.Homotopy g₁) : (f₀.prodMk g₀).Homotopy (f₁.prodMk g₁)

The product of homotopies F and G, where F takes f₀ to f₁ and G takes g₀ to g₁

Equations

• F.prod G = { toFun := fun (t : ↑unitInterval × A) => (F t, G t), continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯ }

@[simp]

theorem ContinuousMap.Homotopy.prod_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ : C(A, α)} {f₁ : C(A, α)} {g₀ : C(A, β)} {g₁ : C(A, β)} (F : f₀.Homotopy f₁) (G : g₀.Homotopy g₁) (t : ↑unitInterval × A) : (F.prod G) t = (F t, G t)

def ContinuousMap.HomotopyRel.prod {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ : C(A, α)} {f₁ : C(A, α)} {g₀ : C(A, β)} {g₁ : C(A, β)} {S : Set A} (F : f₀.HomotopyRel f₁ S) (G : g₀.HomotopyRel g₁ S) : (f₀.prodMk g₀).HomotopyRel (f₁.prodMk g₁) S

The relative product of homotopies F and G, where F takes f₀ to f₁ and G takes g₀ to g₁

Equations

• F.prod G = { toHomotopy := F.prod G.toHomotopy, prop' := ⋯ }

@[simp]

theorem ContinuousMap.HomotopyRel.prod_apply {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ : C(A, α)} {f₁ : C(A, α)} {g₀ : C(A, β)} {g₁ : C(A, β)} {S : Set A} (F : f₀.HomotopyRel f₁ S) (G : g₀.HomotopyRel g₁ S) (t : ↑unitInterval × A) : (F.prod G) t = (F t, G t)

@[simp]

theorem ContinuousMap.HomotopyRel.prod_toFun {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {A : Type u_3} [TopologicalSpace A] {f₀ : C(A, α)} {f₁ : C(A, α)} {g₀ : C(A, β)} {g₁ : C(A, β)} {S : Set A} (F : f₀.HomotopyRel f₁ S) (G : g₀.HomotopyRel g₁ S) (t : ↑unitInterval × A) : (F.prod G) t = (F t, G t)

def Path.Homotopic.piHomotopy {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as : (i : ι) → X i} {bs : (i : ι) → X i} (γ₀ : (i : ι) → Path (as i) (bs i)) (γ₁ : (i : ι) → Path (as i) (bs i)) (H : (i : ι) → (γ₀ i).Homotopy (γ₁ i)) : (Path.pi γ₀).Homotopy (Path.pi γ₁)

The product of a family of path homotopies. This is just a specialization of HomotopyRel.

Equations

• Path.Homotopic.piHomotopy γ₀ γ₁ H = ContinuousMap.HomotopyRel.pi H

def Path.Homotopic.pi {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as : (i : ι) → X i} {bs : (i : ι) → X i} (γ : (i : ι) → Path.Homotopic.Quotient (as i) (bs i)) : Path.Homotopic.Quotient as bs

The product of a family of path homotopy classes.

Equations

• Path.Homotopic.pi γ = Quotient.map Path.pi ⋯ (Quotient.choice γ)

theorem Path.Homotopic.pi_lift {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as : (i : ι) → X i} {bs : (i : ι) → X i} (γ : (i : ι) → Path (as i) (bs i)) : (Path.Homotopic.pi fun (i : ι) => ⟦γ i⟧) = ⟦Path.pi γ⟧

theorem Path.Homotopic.comp_pi_eq_pi_comp {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as : (i : ι) → X i} {bs : (i : ι) → X i} {cs : (i : ι) → X i} (γ₀ : (i : ι) → Path.Homotopic.Quotient (as i) (bs i)) (γ₁ : (i : ι) → Path.Homotopic.Quotient (bs i) (cs i)) : (Path.Homotopic.pi γ₀).comp (Path.Homotopic.pi γ₁) = Path.Homotopic.pi fun (i : ι) => (γ₀ i).comp (γ₁ i)

Composition and products commute. This is Path.trans_pi_eq_pi_trans descended to path homotopy classes.

@[reducible, inline]

abbrev Path.Homotopic.proj {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as : (i : ι) → X i} {bs : (i : ι) → X i} (i : ι) (p : Path.Homotopic.Quotient as bs) : Path.Homotopic.Quotient (as i) (bs i)

Abbreviation for projection onto the ith coordinate.

Equations

• Path.Homotopic.proj i p = p.mapFn { toFun := fun (p : (i : ι) → X i) => p i, continuous_toFun := ⋯ }

@[simp]

theorem Path.Homotopic.proj_pi {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as : (i : ι) → X i} {bs : (i : ι) → X i} (i : ι) (paths : (i : ι) → Path.Homotopic.Quotient (as i) (bs i)) : Path.Homotopic.proj i (Path.Homotopic.pi paths) = paths i

Lemmas showing projection is the inverse of pi.

@[simp]

theorem Path.Homotopic.pi_proj {ι : Type u_1} {X : ι → Type u_2} [(i : ι) → TopologicalSpace (X i)] {as : (i : ι) → X i} {bs : (i : ι) → X i} (p : Path.Homotopic.Quotient as bs) : (Path.Homotopic.pi fun (i : ι) => Path.Homotopic.proj i p) = p

def Path.Homotopic.prodHomotopy {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} {p₁ : Path a₁ a₂} {p₁' : Path a₁ a₂} {p₂ : Path b₁ b₂} {p₂' : Path b₁ b₂} (h₁ : p₁.Homotopy p₁') (h₂ : p₂.Homotopy p₂') : (p₁.prod p₂).Homotopy (p₁'.prod p₂')

The product of homotopies h₁ and h₂. This is HomotopyRel.prod specialized for path homotopies.

Equations

• Path.Homotopic.prodHomotopy h₁ h₂ = ContinuousMap.HomotopyRel.prod h₁ h₂

def Path.Homotopic.prod {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} (q₁ : Path.Homotopic.Quotient a₁ a₂) (q₂ : Path.Homotopic.Quotient b₁ b₂) : Path.Homotopic.Quotient (a₁, b₁) (a₂, b₂)

The product of path classes q₁ and q₂. This is Path.prod descended to the quotient.

Equations

• Path.Homotopic.prod q₁ q₂ = Quotient.map₂ Path.prod ⋯ q₁ q₂

theorem Path.Homotopic.prod_lift {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} (p₁ : Path a₁ a₂) (p₂ : Path b₁ b₂) : Path.Homotopic.prod ⟦p₁⟧ ⟦p₂⟧ = ⟦p₁.prod p₂⟧

theorem Path.Homotopic.comp_prod_eq_prod_comp {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ : α} {a₂ : α} {a₃ : α} {b₁ : β} {b₂ : β} {b₃ : β} (q₁ : Path.Homotopic.Quotient a₁ a₂) (q₂ : Path.Homotopic.Quotient b₁ b₂) (r₁ : Path.Homotopic.Quotient a₂ a₃) (r₂ : Path.Homotopic.Quotient b₂ b₃) : (Path.Homotopic.prod q₁ q₂).comp (Path.Homotopic.prod r₁ r₂) = Path.Homotopic.prod (q₁.comp r₁) (q₂.comp r₂)

Products commute with path composition. This is trans_prod_eq_prod_trans descended to the quotient.

@[reducible, inline]

abbrev Path.Homotopic.projLeft {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {c₁ : α × β} {c₂ : α × β} (p : Path.Homotopic.Quotient c₁ c₂) : Path.Homotopic.Quotient c₁.1 c₂.1

Abbreviation for projection onto the left coordinate of a path class.

Equations

• Path.Homotopic.projLeft p = p.mapFn { toFun := Prod.fst, continuous_toFun := ⋯ }

@[reducible, inline]

abbrev Path.Homotopic.projRight {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {c₁ : α × β} {c₂ : α × β} (p : Path.Homotopic.Quotient c₁ c₂) : Path.Homotopic.Quotient c₁.2 c₂.2

Abbreviation for projection onto the right coordinate of a path class.

Equations

• Path.Homotopic.projRight p = p.mapFn { toFun := Prod.snd, continuous_toFun := ⋯ }

@[simp]

theorem Path.Homotopic.projLeft_prod {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} (q₁ : Path.Homotopic.Quotient a₁ a₂) (q₂ : Path.Homotopic.Quotient b₁ b₂) : Path.Homotopic.projLeft (Path.Homotopic.prod q₁ q₂) = q₁

Lemmas showing projection is the inverse of product.

@[simp]

theorem Path.Homotopic.projRight_prod {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} (q₁ : Path.Homotopic.Quotient a₁ a₂) (q₂ : Path.Homotopic.Quotient b₁ b₂) : Path.Homotopic.projRight (Path.Homotopic.prod q₁ q₂) = q₂

@[simp]

theorem Path.Homotopic.prod_projLeft_projRight {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} (p : Path.Homotopic.Quotient (a₁, b₁) (a₂, b₂)) : Path.Homotopic.prod (Path.Homotopic.projLeft p) (Path.Homotopic.projRight p) = p