Homotopy between paths

In this file, we define a Homotopy between two Paths. In addition, we define a relation Homotopic on Paths, and prove that it is an equivalence relation.

Definitions

• Path.Homotopy p₀ p₁ is the type of homotopies between paths p₀ and p₁
• Path.Homotopy.refl p is the constant homotopy between p and itself
• Path.Homotopy.symm F is the Path.Homotopy p₁ p₀ defined by reversing the homotopy
• Path.Homotopy.trans F G, where F : Path.Homotopy p₀ p₁, G : Path.Homotopy p₁ p₂ is the Path.Homotopy p₀ p₂ defined by putting the first homotopy on [0, 1/2] and the second on [1/2, 1]
• Path.Homotopy.hcomp F G, where F : Path.Homotopy p₀ q₀ and G : Path.Homotopy p₁ q₁ is a Path.Homotopy (p₀.trans p₁) (q₀.trans q₁)
• Path.Homotopic p₀ p₁ is the relation saying that there is a homotopy between p₀ and p₁
• Path.Homotopic.setoid x₀ x₁ is the setoid on Paths from Path.Homotopic
• Path.Homotopic.Quotient x₀ x₁ is the quotient type from Path x₀ x₀ by Path.Homotopic.setoid

@[reducible, inline]

abbrev Path.Homotopy {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p₀ p₁ : Path x₀ x₁) : Type u

The type of homotopies between two paths.

Equations

• p₀.Homotopy p₁ = p₀.HomotopyRel p₁.toContinuousMap {0, 1}

theorem Path.Homotopy.coeFn_injective {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} : Function.Injective DFunLike.coe

@[simp]

theorem Path.Homotopy.source {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑unitInterval) : F (t, 0) = x₀

@[simp]

theorem Path.Homotopy.target {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑unitInterval) : F (t, 1) = x₁

def Path.Homotopy.eval {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑unitInterval) : Path x₀ x₁

Evaluating a path homotopy at an intermediate point, giving us a Path.

Equations

• F.eval t = { toFun := ⇑(F.curry t), continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }

@[simp]

theorem Path.Homotopy.eval_zero {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : F.eval 0 = p₀

@[simp]

theorem Path.Homotopy.eval_one {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : F.eval 1 = p₁

def Path.Homotopy.refl {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : p.Homotopy p

Given a path p, we can define a Homotopy p p by F (t, x) = p x.

Equations

• Path.Homotopy.refl p = ContinuousMap.HomotopyRel.refl p.toContinuousMap {0, 1}

@[simp]

theorem Path.Homotopy.refl_apply {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) (x : ↑unitInterval × ↑unitInterval) : (refl p) x = p x.2

def Path.Homotopy.symm {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : p₁.Homotopy p₀

Given a Homotopy p₀ p₁, we can define a Homotopy p₁ p₀ by reversing the homotopy.

Equations

• F.symm = ContinuousMap.HomotopyRel.symm F

@[simp]

theorem Path.Homotopy.symm_apply {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (x : ↑unitInterval × ↑unitInterval) : F.symm x = F (unitInterval.symm x.1, x.2)

@[simp]

theorem Path.Homotopy.symm_symm {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : F.symm.symm = F

theorem Path.Homotopy.symm_bijective {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} : Function.Bijective symm

def Path.Homotopy.trans {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂) : p₀.Homotopy p₂

Given Homotopy p₀ p₁ and Homotopy p₁ p₂, we can define a Homotopy p₀ p₂ by putting the first homotopy on [0, 1/2] and the second on [1/2, 1].

Equations

• F.trans G = ContinuousMap.HomotopyRel.trans F G

theorem Path.Homotopy.trans_apply {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂) (x : ↑unitInterval × ↑unitInterval) : (F.trans G) x = if h : ↑x.1 ≤ 1 / 2 then F (⟨2 * ↑x.1, ⋯⟩, x.2) else G (⟨2 * ↑x.1 - 1, ⋯⟩, x.2)

theorem Path.Homotopy.symm_trans {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂) : (F.trans G).symm = G.symm.trans F.symm

def Path.Homotopy.cast {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) : q₀.Homotopy q₁

Casting a Homotopy p₀ p₁ to a Homotopy q₀ q₁ where p₀ = q₀ and p₁ = q₁.

Equations

• F.cast h₀ h₁ = ContinuousMap.HomotopyRel.cast F ⋯ ⋯

@[simp]

theorem Path.Homotopy.cast_apply {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) (a : ↑unitInterval × ↑unitInterval) : (F.cast h₀ h₁) a = F a

def Path.Homotopy.hcomp {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) : (p₀.trans p₁).Homotopy (q₀.trans q₁)

Suppose p₀ and q₀ are paths from x₀ to x₁, p₁ and q₁ are paths from x₁ to x₂. Furthermore, suppose F : Homotopy p₀ q₀ and G : Homotopy p₁ q₁. Then we can define a homotopy from p₀.trans p₁ to q₀.trans q₁.

Equations

• One or more equations did not get rendered due to their size.

theorem Path.Homotopy.hcomp_apply {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) (x : ↑unitInterval × ↑unitInterval) : (F.hcomp G) x = if h : ↑x.2 ≤ 1 / 2 then (F.eval x.1) ⟨2 * ↑x.2, ⋯⟩ else (G.eval x.1) ⟨2 * ↑x.2 - 1, ⋯⟩

theorem Path.Homotopy.hcomp_half {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) (t : ↑unitInterval) : (F.hcomp G) (t, ⟨1 / 2, ⋯⟩) = x₁

def Path.Homotopy.reparam {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) (f : ↑unitInterval → ↑unitInterval) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) : p.Homotopy (p.reparam f hf hf₀ hf₁)

Suppose p is a path, then we have a homotopy from p to p.reparam f by the convexity of I.

Equations

• One or more equations did not get rendered due to their size.

def Path.Homotopy.symm₂ {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) : p.symm.Homotopy q.symm

Suppose F : Homotopy p q. Then we have a Homotopy p.symm q.symm by reversing the second argument.

Equations

• F.symm₂ = { toFun := fun (x : ↑unitInterval × ↑unitInterval) => F (x.1, unitInterval.symm x.2), continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯, prop' := ⋯ }

@[simp]

theorem Path.Homotopy.symm₂_apply {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (x : ↑unitInterval × ↑unitInterval) : F.symm₂ x = F (x.1, unitInterval.symm x.2)

def Path.Homotopy.map {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y)) : (p.map ⋯).Homotopy (q.map ⋯)

Given F : Homotopy p q, and f : C(X, Y), we can define a homotopy from p.map f.continuous to q.map f.continuous.

Equations

• F.map f = { toFun := ⇑f ∘ ⇑F, continuous_toFun := ⋯, map_zero_left := ⋯, map_one_left := ⋯, prop' := ⋯ }

@[simp]

theorem Path.Homotopy.map_apply {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y)) (a✝ : ↑unitInterval × ↑unitInterval) : (F.map f) a✝ = (⇑f ∘ ⇑F) a✝

def Path.Homotopic {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p₀ p₁ : Path x₀ x₁) : Prop

Two paths p₀ and p₁ are Path.Homotopic if there exists a Homotopy between them.

Equations

• p₀.Homotopic p₁ = Nonempty (p₀.Homotopy p₁)

theorem Path.Homotopic.refl {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : p.Homotopic p

theorem Path.Homotopic.symm {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} ⦃p₀ p₁ : Path x₀ x₁⦄ (h : p₀.Homotopic p₁) : p₁.Homotopic p₀

theorem Path.Homotopic.trans {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} ⦃p₀ p₁ p₂ : Path x₀ x₁⦄ (h₀ : p₀.Homotopic p₁) (h₁ : p₁.Homotopic p₂) : p₀.Homotopic p₂

theorem Path.Homotopic.equivalence {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} : Equivalence Homotopic

theorem Path.Homotopic.map {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (h : p.Homotopic q) (f : C(X, Y)) : (p.map ⋯).Homotopic (q.map ⋯)

theorem Path.Homotopic.hcomp {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ p₁ : Path x₀ x₁} {q₀ q₁ : Path x₁ x₂} (hp : p₀.Homotopic p₁) (hq : q₀.Homotopic q₁) : (p₀.trans q₀).Homotopic (p₁.trans q₁)

def Path.Homotopic.setoid {X : Type u} [TopologicalSpace X] (x₀ x₁ : X) : Setoid (Path x₀ x₁)

The setoid on Paths defined by the equivalence relation Path.Homotopic. That is, two paths are equivalent if there is a Homotopy between them.

Equations

• Path.Homotopic.setoid x₀ x₁ = { r := Path.Homotopic, iseqv := ⋯ }

def Path.Homotopic.Quotient {X : Type u} [TopologicalSpace X] (x₀ x₁ : X) : Type u

The quotient on Path x₀ x₁ by the equivalence relation Path.Homotopic.

Equations

• Path.Homotopic.Quotient x₀ x₁ = Quotient (Path.Homotopic.setoid x₀ x₁)

instance Path.Homotopic.instInhabitedQuotientUnitUnit : Inhabited (Homotopic.Quotient () ())

Equations

• Path.Homotopic.instInhabitedQuotientUnitUnit = { default := Quotient.mk' (Path.refl ()) }

def Path.Homotopic.Quotient.comp {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} (P₀ : Homotopic.Quotient x₀ x₁) (P₁ : Homotopic.Quotient x₁ x₂) : Homotopic.Quotient x₀ x₂

The composition of path homotopy classes. This is Path.trans descended to the quotient.

Equations

• P₀.comp P₁ = Quotient.map₂ Path.trans ⋯ P₀ P₁

theorem Path.Homotopic.comp_lift {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} (P₀ : Path x₀ x₁) (P₁ : Path x₁ x₂) : ⟦P₀.trans P₁⟧ = Quotient.comp ⟦P₀⟧ ⟦P₁⟧

def Path.Homotopic.Quotient.mapFn {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (P₀ : Homotopic.Quotient x₀ x₁) (f : C(X, Y)) : Homotopic.Quotient (f x₀) (f x₁)

The image of a path homotopy class P₀ under a map f. This is Path.map descended to the quotient.

Equations

• P₀.mapFn f = Quotient.map (fun (q : Path x₀ x₁) => q.map ⋯) ⋯ P₀

theorem Path.Homotopic.map_lift {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (P₀ : Path x₀ x₁) (f : C(X, Y)) : ⟦P₀.map ⋯⟧ = Quotient.mapFn ⟦P₀⟧ f

theorem Path.Homotopic.hpath_hext {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ x₃ : X} {p₁ : Path x₀ x₁} {p₂ : Path x₂ x₃} (hp : ∀ (t : ↑unitInterval), p₁ t = p₂ t) : ⟦p₁⟧ ≍ ⟦p₂⟧

def Path.toHomotopyConst {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (p : Path x₀ x₁) : (ContinuousMap.const Y x₀).Homotopy (ContinuousMap.const Y x₁)

A path Path x₀ x₁ generates a homotopy between constant functions fun _ ↦ x₀ and fun _ ↦ x₁.

Equations

• p.toHomotopyConst = { toContinuousMap := p.comp ContinuousMap.fst, map_zero_left := ⋯, map_one_left := ⋯ }

@[simp]

theorem Path.toHomotopyConst_apply {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (p : Path x₀ x₁) (a✝ : ↑unitInterval × Y) : p.toHomotopyConst a✝ = p a✝.1

@[simp]

theorem ContinuousMap.homotopic_const_iff {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} [Nonempty Y] : (const Y x₀).Homotopic (const Y x₁) ↔ Joined x₀ x₁

Two constant continuous maps with nonempty domain are homotopic if and only if their values are joined by a path in the codomain.

def ContinuousMap.Homotopy.evalAt {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f g : C(X, Y)} (H : f.Homotopy g) (x : X) : Path (f x) (g x)

Given a homotopy H : f ∼ g, get the path traced by the point x as it moves from f x to g x.

Equations

• H.evalAt x = { toFun := fun (t : ↑unitInterval) => H (t, x), continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }

@[simp]

theorem ContinuousMap.Homotopy.evalAt_apply {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f g : C(X, Y)} (H : f.Homotopy g) (x : X) (t : ↑unitInterval) : (H.evalAt x) t = H (t, x)

@[simp]

theorem ContinuousMap.Homotopy.pathExtend_evalAt {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f g : C(X, Y)} (H : f.Homotopy g) (x : X) : ⇑(H.evalAt x).extend = fun (t : ℝ) => (H.extend t) x